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73 | Through this post I am going to explain How Linear Regression works? Let us start with what is regression and how it works? Regression is widely used for prediction and forecasting in field of machine learning. Focus of regression is on the relationship between dependent and one or more independent variables. The “dependent variable” represents the output or effect, or is tested to see if it is the effect. The “independent variables” represent the inputs or causes, or are tested to see if they are the cause. Regression analysis helps to understand how the value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are kept unchanged. In the regression, dependent variable is estimated as function of independent variables which is called regression function. Regression model involves following variables.
- Independent variables X.
- Dependent variable Y
- Unknown parameter θ
In the regression model Y is function of (X,θ). There are many techniques for regression analysis, but here we will consider linear regression.
In the Linear regression, dependent variable(Y) is the linear combination of the independent variables(X). Here regression function is known as hypothesis which is defined as below.
hθ(X) = f(X,θ)
Suppose we have only one independent variable(x), then our hypothesis is defined as below.
The goal is to find some values of θ(known as coefficients), so we can minimize the difference between real and predicted values of dependent variable(y). If we take the values of all θ are zeros, then our predicted value will be zero. Cost function is used as measurement factor of linear regression model and it calculates average squared error for m observations. Cost function is denoted by J(θ) and defined as below.
As we can see from the above formula, if cost is large then, predicted value is far from the real value and if cost is small then, predicted value is nearer to real value. Therefor, we have to minimize cost to meet more accurate prediction.
Linear regression in R
R is language and environment for statistical computing. R has powerful and comprehensive features for fitting regression models. We will discuss about how linear regression works in R. In R, basic function for fitting linear model is lm(). The format is
fit <- lm(formula, data)
where formula describes model(in our case linear model) and data describes which data are used to fit model. The resulting object(fit in this case) is a list that contains information about the fitted model. The formula typically written as
Y ~ x1 + x2 + … + xk
where ~ separates the dependent variable(y) on the left from independent variables(x1, x2, ….. , xk) from right, and the independent variables are separated by + signs. let’s see simple regression example(example is from book R in action). We have the dataset women which contains height and weight for a set of 15 women ages 30 to 39. we want to predict weight from height. R code to fit this model is as below.
>fit <-lm(weight ~ height, data=women) >summary(fit)
Output of the summary function gives information about the object fit. Output is as below
Call: lm(formula = weight ~ height, data = women) Residuals: Min 1Q Median 3Q Max -1.7333 -1.1333 -0.3833 0.7417 3.1167 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -87.51667 5.93694 -14.74 1.71e-09 *** height 3.45000 0.09114 37.85 1.09e-14 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.525 on 13 degrees of freedom Multiple R-squared: 0.991, Adjusted R-squared: 0.9903 F-statistic: 1433 on 1 and 13 DF, p-value: 1.091e-14
Let’s understand the output. Values of coefficients(θs) are -87.51667 and 3.45000, hence prediction equation for model is as below
Weight = -87.52 + 3.45*height
In the output, residual standard error is cost which is 1.525. Now, we will look at real values of weight of 15 women first and then will look at predicted values. Actual values of weight of 15 women are as below
Output 115 117 120 123 126 129 132 135 139 142 146 150 154 159 164
Predicted values of 15 women are as below
Output 1 2 3 4 5 6 7 8 9 112.5833 116.0333 119.4833 122.9333 126.3833 129.8333 133.2833 136.7333 140.1833 10 11 12 13 14 15 143.6333 147.0833 150.5333 153.9833 157.4333 160.8833
We can see that predicted values are nearer to the actual values.Finally, we understand what is regression, how it works and regression in R.
Here, I want to beware you from the misunderstanding about correlation and causation. In the regression, dependent variable is correlated with the independent variable. This means, as the value of the independent variable changes, value of the dependent variable also changes. But, this does not mean that independent variable cause to change the value of dependent variable. Causation implies correlation , but reverse is not true. For example, smoking causes the lung cancer and smoking is correlated with alcoholism. Many discussions are there on this topic. if we go deep into than one blog is not enough to explain this.But, we will keep in mind that we will consider correlation between dependent variable and independent variable in regression.
In the next blog, I will discuss about the real world business problem and how to use regression into it.
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Would you like to understand the value of predictive analysis when applied on web analytics data to help improve your understanding relationship between different variables? So register now for our Upcoming Webinar: How to perform predictive analysis on your web analytics tool data. Get More Info & Book Your Seat Now! | http://www.tatvic.com/blog/linear-regression-using-r/ | 13 |
67 | Least common multiple
In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. If either a or b is 0, LCM(a, b) is defined to be zero.
The LCM of more than two integers is also well-defined: it is the smallest integer that is divisible by each of them.
A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and 2 as well.
What is the LCM of 4 and 6?
Multiples of 4 are:
- 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, ...
and the multiples of 6 are:
- 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...
Common multiples of 4 and 6 are simply the numbers that are in both lists:
- 12, 24, 36, 48, 60, 72, ....
So, from this list of the first few common multiples of the numbers 4 and 6, their least common multiple is 12.
When adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. For instance,
where the denominator 42 was used because it is the least common multiple of 21 and 6.
Computing the least common multiple
Reduction by the greatest common divisor
Many school age children are taught the term greatest common factor (GCF) instead of the greatest common divisor(GCD); therefore, for those familiar with the concept of GCF, substitute GCF when GCD is used below.
The following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor (GCD):
This formula is also valid when exactly one of a and b is 0, since gcd(a, 0) = |a|.
Because gcd(a, b) is a divisor of both a and b, it's more efficient to compute the LCM by dividing before multiplying:
This reduces the size of one input for both the division and the multiplication, and reduces the required storage needed for intermediate results (overflow in the a×b computation). Because gcd(a, b) is a divisor of both a and b, the division is guaranteed to yield an integer, so the intermediate result can be stored in an integer. Done this way, the previous example becomes:
Finding least common multiples by prime factorization
The unique factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number.
Here we have the composite number 90 made up of one atom of the prime number 2, two atoms of the prime number 3 and one atom of the prime number 5.
This knowledge can be used to find the LCM of a set of numbers.
Example: Find the value of lcm(8,9,21).
First, factor out each number and express it as a product of prime number powers.
The lcm will be the product of multiplying the highest power of each prime number together. The highest power of the three prime numbers 2, 3, and 7 is 23, 32, and 71, respectively. Thus,
This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization, but is useful for illustrating concepts.
This method can be illustrated using a Venn diagram as follows. Find the prime factorization of each of the two numbers. Put the prime factors into a Venn diagram with one circle for each of the two numbers, and all factors they share in common in the intersection. To find the LCM, just multiply all of the prime numbers in the diagram.
Here is an example:
- 48 = 2 × 2 × 2 × 2 × 3,
- 180 = 2 × 2 × 3 × 3 × 5,
and what they share in common is two "2"s and a "3":
- Least common multiple = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720
- Greatest common divisor = 2 × 2 × 3 = 12
This also works for the greatest common divisor (GCD), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the GCD of 48 and 180 is 2 × 2 × 3 = 12.
A simple algorithm
This method works as easily for finding the LCM of several integers.
Let there be a finite sequence of positive integers X = (x1, x2, ..., xn), n > 1. The algorithm proceeds in steps as follows: on each step m it examines and updates the sequence X(m) = (x1(m), x2(m), ..., xn(m)), X(1) = X. The purpose of the examination is to pick up the least (perhaps, one of many) element of the sequence X(m). Assuming xk0(m) is the selected element, the sequence X(m+1) is defined as
- xk(m+1) = xk(m), k ≠ k0
- xk0(m+1) = xk0(m) + xk0.
In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X(m) to X(m+1) unchanged.
The algorithm stops when all elements in sequence X(m) are equal. Their common value L is exactly LCM(X). (For a proof and an interactive simulation see reference below, Algorithm for Computing the LCM.)
A method using a table
This method works for any number of factors. One begins by listing all of the numbers vertically in a table (in this example 4, 7, 12, 21, and 42):
The process begins by dividing all of the factors by 2. If any of them divides evenly, write 2 at the top of the table and the result of division by 2 of each factor in the space to the right of each factor and below the 2. If a number does not divide evenly, just rewrite the number again. If 2 does not divide evenly into any of the numbers, try 3.
Now, check if 2 divides again:
Once 2 no longer divides, divide by 3. If 3 no longer divides, try 5 and 7. Keep going until all of the numbers have been reduced to 1.
Now, multiply the numbers on the top and you have the LCM. In this case, it is 2 × 2 × 3 × 7 = 84. You will get to the LCM the quickest if you use prime numbers and start from the lowest prime, 2.
Fundamental theorem of arithmetic
where the exponents n2, n3, ... are non-negative integers; for example, 84 = 22 31 50 71 110 130 ...
Given two integers and their least common multiple and greatest common divisor are given by the formulas
In fact, any rational number can be written uniquely as the product of primes if negative exponents are allowed. When this is done, the above formulas remain valid. Using the same examples as above:
The positive integers may be partially ordered by divisibility: if a divides b (i.e. if b is an integer multiple of a) write a ≤ b (or equivalently, b ≥ a). (Forget the usual magnitude-based definition of ≤ in this section - it isn't used.)
Under this ordering, the positive integers become a lattice with meet given by the gcd and join given by the lcm. The proof is straightforward, if a bit tedious; it amounts to checking that lcm and gcd satisfy the axioms for meet and join. Putting the lcm and gcd into this more general context establishes a duality between them:
- If a formula involving integer variables, gcd, lcm, ≤ and ≥ is true, then the formula obtained by switching gcd with lcm and switching ≥ with ≤ is also true. (Remember ≤ is defined as divides).
The following pairs of dual formulas are special cases of general lattice-theoretic identities.
This identity is self-dual:
Let D be the product of ω(D) distinct prime numbers (i.e. D is squarefree).
where the absolute bars || denote the cardinality of a set.
The LCM in commutative rings
The least common multiple can be defined generally over commutative rings as follows: Let a and b be elements of a commutative ring R. A common multiple of a and b is an element m of R such that both a and b divide m (i.e. there exist elements x and y of R such that ax = m and by = m). A least common multiple of a and b is a common multiple that is minimal in the sense that for any other common multiple n of a and b, m divides n.
In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are associates. In a unique factorization domain, any two elements have a least common multiple. In a principal ideal domain, the least common multiple of a and b can be characterised as a generator of the intersection of the ideals generated by a and b (the intersection of a collection of ideals is always an ideal). In principal ideal domains, one can even talk about the least common multiple of arbitrary collections of elements: it is a generator of the intersection of the ideals generated by the elements of the collection.
See also
- Crandall, Richard; Pomerance, Carl (2001), Prime Numbers: A Computational Perspective, New York: Springer, ISBN 0-387-94777-9
- Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0-19-853171-5
- Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea
- Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950
- Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766 | http://en.wikipedia.org/wiki/Least_common_multiple | 13 |
55 | Geometry is heavily tested on the GRE Math section, and a thorough review of geometrical concepts is essential to a high score. Consider the following problem:
“If the length of an edge of a cube X is twice the length of an edge of cube Y, what is the ratio of the volume of cube Y to the volume of cube X?”
The easiest way to solve this is to pick a number for the initial edge length and plug it into the problem. For instance, let’s say cube X is a 4x4x4 cube. Cube X would have a volume of 64. Cube Y would have to be a 2x2x2 cube, since 2 is half of 4, and it would have a volume of 8. The ratio of the volume of cube Y to the volume of cube X would thus be 8 to 64, or 1/8.
However, you really should have known that to begin with. Imagine that cube X had edges that were three times as long as those of Cube Y. Then Cube X would now be a 6x6x6 cube if Cube Y remains a 2x2x2 cube, and the volume ratio would be 8 to 216, or 1/27. Notice something? 8 is 2 ^3, and 27 is 3^3. If the ratio of the sides is 1:4, the ratio of the volumes will be 1:64. If the ratio of the sides is 1:5, the ratio of the volumes will be 1:125. Since these are cubes, you just cube the ratios. 1^3 is 1, and 4^3 is 64; 5^3 is 125. If you know this simple property of the relationship between length and volume, it will take a problem that would take 30 seconds to solve and turn it into a problem that takes 5 seconds to solve. On a timed exam, that could be the difference between getting another, harder question right or wrong. Memorizing these kinds of mathematical facts is something that the GRE test writers expect top scorers to do, and they write the questions so that they can be solved quickly if you know them. It also pays to memorize the squares and cubes of the numbers 1 through 12.
So with cubes, you cube the ratio of the sides. What about squares? If you guessed that you square the ratio of the side lengths in order to get the ratio of the areas, you’d be right, as you can see from a quick demonstration. If the original square has side lengths of 1 and the new square has side lengths of 2, the side ratio is 1:2 and the area ratio is 1:4. If the new square has side lengths of 3, then the side ratio is 1:3 and the area ratio is 1:9. If the new square has side lengths of 4, then the side ratio is 1:4 and the area ratio is 1:16, and so on. Sure enough, you just square the original ratio.
So now you know about cubes and squares, but what about tesseracts? “Tessawhats?” you say? A tesseract is to a cube as a cube is to a square, just as a cube is to a square what a square is to a line. Still confused? Let me explain it this way: say you draw a line a foot long running from east to west. This line only exists in one dimension: east-west. Then, you decide to square it by adding three more lines: two perpendicular to it running north to south and one parallel to it running east to west. This square exists in two dimensions: east-west and north-south. Now you decide to turn the square into a cube by adding lines in the up-down dimension, so that each edge of the original square is now the edge of another square emanating from it. This cube exists in three spatial dimensions: east-west, north-south, and up-down. Now you take this cube you’ve made and decide to square it…in a fourth spacial dimension.
What is this fourth dimension? Who knows. We live in a world in which we experience only three spacial dimensions, so it is impossible for us to imagine what a four dimensional object would look like. That hasn’t stopped mathematicians from naming four-dimensional objects, and this hypercube I’ve just described to you is called a tesseract. As you know, even though a cube is a three dimensional object, it is possible to draw a cube on a piece of paper in only two dimensions by using perspective and all those other artistic illusions. Likewise, some have attempted to render tesseracts in three dimensions in order to give some approximation of what they might look like. Having never seen an actual tesseract, though, you might still find these representations confusing.
In terms of doing calculations, though, tesseracts are simple as can be. For a square with side lengths of 1 and another square with side lengths of 2, the ratio of side lengths is 1:2^1 (since sides are 1 dimensional), or 1:2, and the ratio of areas will be 1:2^2 (since squares are 2 dimensional) or 1:4. For a cube with side lengths of 1 and another cube with side lengths of 2, the ratio of volumes is 1:2^3 (since cubes are 3 dimensional), or 1:8. So, for a tesseract with side lengths of 1 and another tesseract with side lengths of 2, the ratio of hypervolumes(?) is 1:2^4 (since tesseracts are 4 dimensional), or 1:16. It just follows the pattern. Try not to think about it too much.
If you’re having trouble with tesseracts, don’t worry. They’re not on the test. I just wrote about them to mess with your head.
Remember, if you ever want extra help getting ready for the GRE, you can always study with experts like me through Test Masters. Until then, happy studying! | http://www.newgre.org/preparation/sample-math-problem-hip-square-or-cube/ | 13 |
59 | In the preceding posts, I mentioned infinite products as approximations for π. These may be seen geometrically as exhaustion methods, where the area of a polygon approaches the circular area alternately from above, from below, from above, from below, etc.
There are also integral representations of pi. In such integral representations, π appears in the quantitative value of the integral of a mathematical function. Visually, this is often represented as the area delimited by the bounds of the function. However, the relation with the circle is lost, when viewed under Cartesian coordinates. For example, the graph of the simplest instance of the Cauchy-Lorentz distribution, f(x)=1/(1+x²), "has nothing at all to do with circles or geometry in any obvious way" as quoted from last Pi-day Sunday function from Matt Springer's Built on Facts blog.
In order to view the role of the circle in integral representations of π, we need to switch to alternative ways to visualize math functions. As an example, let's take the constant function y=f(x)=2. The function f maps an element x from a domain to the element y of the target. In this case, for every x, the target y has the constant value 2. With Cartesian coordinates, we are used to represent this function as a horizontal straight line, like in Figure 1a (click on the figure to view it enlarged). If however we write it as R=f(r)=2, where the function f maps any circle of radius r of the domain to a target circle of radius R=2, the same function can be viewed as a circle of constant radius, like in Figure 1b. So the same function f can be equally well viewed as a straight line or as a circle (x, y, r or R are only dummy variables).
Now if we take another example, the linear function, y=f(x)=2x, we are often used to view it in Cartesian coordinates as a straight line with slope 2, like in Figure 1c. In the circular representation R=f(r)=2r, this works however differently. Because we are relating circles of the input domain to other circles of the target, for each circle of radius r, we need to draw the target circle of radius 2r. A single line won't do. For one value of r, we need to draw two circles. If we use blue circles for elements of the input domain and red circles for elements of the target, we could visualize it for successive values of r as an animation like in Figure 1d. In that way, we view the progression of the target circle as the input circle becomes larger.
Unlike the Cartesian representation which shows the progression of a function in a static graph, this circular representation needs a dynamic or recurrent process to get grip of the progression of the function. Therefore it isn't very adapted for illustrations in print media. On the other hand, it has the advantage of keeping track of the geometrical form of the circle. And that's exactly what we need in order to perceive the circular nature when π shows up in mathematical functions. The relation of the integral of the Cauchy-Lorentz distribution f(r)=1/(1+r²) with the circle can then be seen with the help of the geometric counterparts of arithmetic operations like addition, squaring and dividing. A convenient procedure is illustrated in the successive steps of Figure 2.
Step 1. Draw the input circle of radius r and the reference circle of radius unity.
Step 2. Determine r².
Step 5. Find the target ring related to the input ring ranging over [r, r + dr]. This yields a ring of width dr/(1+r²). The location of this ring depends on the relative progression rates of r and r² (I've not yet found a straightforward explanation for this determination).
Step 6. Integrate dr/(1+r²) for r running over all space. For r becoming larger and larger, the summed area tends towards the area of a circle of radius 1. For the positive half plane, this corresponds to the π/2 value found analytically.
The tricky step seems to be the way how to relate the progression between r and 1/(1+r²) in steps 5 and 6. One can verify for example the value of the integral at intermediate steps. For the integral from r=0 to 1, the value in the positive half plane must be π/4, which can be verified on the figure below.
In order to gain more insight on π, it could be of interest to develop skills for this circular representation. | http://commonsensequantum.blogspot.com/2010/04/keeping-track-of-circle-for-integral.html | 13 |
707 | |This is the print version of Geometry
You won't see this message or any elements not part of the book's content when you print or preview this page.
Part I- Euclidean Geometry
Chapter 1: Points, Lines, Line Segments and Rays
Points and lines are two of the most fundamental concepts in Geometry, but they are also the most difficult to define. We can describe intuitively their characteristics, but there is no set definition for them: they, along with the plane, are the undefined terms of geometry. All other geometric definitions and concepts are built on the undefined ideas of the point, line and plane. Nevertheless, we shall try to define them.
A point is an exact location in space. Points are dimensionless. That is, a point has no width, length, or height. We locate points relative to some arbitrary standard point, often called the "origin". Many physical objects suggest the idea of a point. Examples include the tip of a pencil, the corner of a cube, or a dot on a sheet of paper.
As for a line segment, we specify a line with two points. Starting with the corresponding line segment, we find other line segments that share at least two points with the original line segment. In this way we extend the original line segment indefinitely. The set of all possible line segments findable in this way constitutes a line. A line extends indefinitely in a single dimension. Its length, having no limit, is infinite. Like the line segments that constitute it, it has no width or height. You may specify a line by specifying any two points within the line. For any two points, only one line passes through both points. On the other hand, an unlimited number of lines pass through any single point.
We construct a ray similarly to the way we constructed a line, but we extend the line segment beyond only one of the original two points. A ray extends indefinitely in one direction, but ends at a single point in the other direction. That point is called the end-point of the ray. Note that a line segment has two end-points, a ray one, and a line none.
A point exists in zero dimensions. A line exists in one dimension, and we specify a line with two points. A plane exists in two dimensions. We specify a plane with three points. Any two of the points specify a line. All possible lines that pass through the third point and any point in the line make up a plane. In more obvious language, a plane is a flat surface that extends indefinitely in its two dimensions, length and width. A plane has no height.
Space exists in three dimensions. Space is made up of all possible planes, lines, and points. It extends indefinitely in all directions.
Mathematics can extend space beyond the three dimensions of length, width, and height. We then refer to "normal" space as 3-dimensional space. A 4-dimensional space consists of an infinite number of 3-dimensional spaces. Etc.
[How we label and reference points, lines, and planes.]
Chapter 2: Angles
An angle is the union of two rays with a common endpoint, called the vertex. The angles formed by vertical and horizontal lines are called right angles; lines, segments, or rays that intersect in right angles are said to be perpendicular.
Angles, for our purposes, can be measured in either degrees (from 0 to 360) or radians (from 0 to ). Angles length can be determined by measuring along the arc they map out on a circle. In radians we consider the length of the arc of the circle mapped out by the angle. Since the circumference of a circle is , a right angle is radians. In degrees, the circle is 360 degrees, and so a right angle would be 90 degrees.
Angles are named in several ways.
- By naming the vertex of the angle (only if there is only one angle formed at that vertex; the name must be non-ambiguous)
- By naming a point on each side of the angle with the vertex in between.
- By placing a small number on the interior of the angle near the vertex.
Classification of Angles by Degree Measure
- an angle is said to be acute if it measures between 0 and 90 degrees, exclusive.
- an angle is said to be right if it measures 90 degrees.
- notice the small box placed in the corner of a right angle, unless the box is present it is not assumed the angle is 90 degrees.
- all right angles are congruent
- an angle is said to be obtuse if it measures between 90 and 180 degrees, exclusive.
Special Pairs of Angles
- adjacent angles
- adjacent angles are angles with a common vertex and a common side.
- adjacent angles have no interior points in common.
- complementary angles
- complementary angles are two angles whose sum is 90 degrees.
- complementary angles may or may not be adjacent.
- if two complementary angles are adjacent, then their exterior sides are perpendicular.
- supplementary angles
- two angles are said to be supplementary if their sum is 180 degrees.
- supplementary angles need not be adjacent.
- if supplementary angles are adjacent, then the sides they do not share form a line.
- linear pair
- if a pair of angles is both adjacent and supplementary, they are said to form a linear pair.
- vertical angles
- angles with a common vertex whose sides form opposite rays are called vertical angles.
- vertical angles are congruent.
Side-Side-Side (SSS) (Postulate 12) If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Side-Angle-Side (SAS) (Postulate 13)
If two sides and the included angle of a second triangle, then the two triangles are congruent.
If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then two triangles are congruent.
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.
NO - Angle-Side-Side (ASS)
The "ASS" postulate does not work, unlike the other ones. A way that students can remember this is that "ass" is not a nice word, so we don't use it in geometry (since it does not work).
There are two approaches to furthering knowledge: reasoning from known ideas and synthesizing observations. In inductive reasoning you observe the world, and attempt to explain based on your observations. You start with no prior assumptions. Deductive reasoning consists of logical assertions from known facts.
What you need to know
Before one can start to understand logic, and thereby begin to prove geometric theorems, one must first know a few vocabulary words and symbols.
Conditional: a conditional is something which states that one statement implies another. A conditional contains two parts: the condition and the conclusion, where the former implies the latter. A conditional is always in the form "If statement 1, then statement 2." In most mathematical notation, a conditional is often written in the form p ⇒ q, which is read as "If p, then q" where p and q are statements.
Converse: the converse of a logical statement is when the conclusion becomes the condition and vice versa; i.e., p ⇒ q becomes q ⇒ p. For example, the converse of the statement "If someone is a woman, then they are a human" would be "If someone is a human, then they are a woman." The converse of a conditional does not necessarily have the same truth value as the original, though it sometimes does, as will become apparent later.
AND: And is a logical operator which is true only when both statements are true. For example, the statement "Diamond is the hardest substance known to man AND a diamond is a metal" is false. While the former statement is true, the latter is not. However, the statement "Diamond is the hardest substance known to man AND diamonds are made of carbon" would be true, because both parts are true.
OR: If two statements are joined together by "or," then the truth of the "or" statement is dependant upon whether one or both of the statements from which it is composed is true. For example, the statement "Tuesday is the day after Monday OR Thursday is the day after Saturday" would have a truth value of "true," because even though the latter statement is false, the former is true.
NOT: If a statement is preceded by "NOT," then it is evaluating the opposite truth value of that statement. The symbol for "NOT" is For example, if the statement p is "Elvis is dead," then ¬p would be "Elvis is not dead." The concept of "NOT" can cause some confusion when it relates to statements which contain the word "all." For example, if r is "¬". "All men have hair," then ¬r would be "All men do not have hair" or "No men have hair." Do not confuse this with "Not all men have hair" or "Some men have hair." The "NOT" should apply to the verb in the statement: in this case, "have." ¬p can also be written as NOT p or ~p. NOT p may also be referred to as the "negation of p."
Inverse: The inverse of a conditional says that the negation of the condition implies the negation of the conclusion. For example, the inverse of p ⇒ q is ¬p ⇒ ¬q. Like a converse, an inverse does not necessarily have the same truth value as the original conditional.
Biconditional: A biconditional is conditional where the condition and the conclusion imply one another. A biconditional starts with the words "if and only if." For example, "If and only if p, then q" means both that p implies q and that q implies p.
Premise: A premise is a statement whose truth value is known initially. For example, if one were to say "If today is Thursday, then the cafeteria will serve burritos," and one knew that what day it was, then the premise would be "Today is Thursday" or "Today is not Thursday."
⇒: The symbol which denotes a conditional. p ⇒ q is read as "if p, then q."
Iff: Iff is a shortened form of "if and only if." It is read as "if and only if."
⇔: The symbol which denotes a biconditonal. p ⇔ q is read as "If and only if p, then q."
∴: The symbol for "therefore." p ∴ q means that one knows that p is true (p is true is the premise), and has logically concluded that q must also be true.
∧: The symbol for "and."
∨: The symbol for "or."
There are a few forms of deductive logic. One of the most common deductive logical arguments is modus ponens, which states that:
- p ⇒ q
- p ∴ q
- (If p, then q)
- (p, therefore q)
An example of modus ponens:
- If I stub my toe, then I will be in pain.
- I stub my toe.
- Therefore, I am in pain.
Another form of deductive logic is modus tollens, which states the following.
- p ⇒ q
- ¬q ∴ ¬p
- (If p, then q)
- (not q, therefore not p)
Modus tollens is just as valid a form of logic as modus ponens. The following is an example which uses modus tollens.
- If today is Thursday, then the cafeteria will be serving burritos.
- The cafeteria is not serving burritos, therefore today is not Thursday.
Another form of deductive logic is known as the If-Then Transitive Property. Simply put, it means that there can be chains of logic where one thing implies another thing. The If-Then Transitive Property states:
- p ⇒ q
- (q ⇒ r) ∴ (p ⇒ r)
- (If p, then q)
- ((If q, then r), therefore (if p, then r))
For example, consider the following chain of if-then statements.
- If today is Thursday, then the cafeteria will be serving burritos.
- If the cafeteria will be serving burritos, then I will be happy.
- Therefore, if today is Thursday, then I will be happy.
Inductive reasoning is a logical argument which does not definitely prove a statement, but rather assumes it. Inductive reasoning is used often in life. Polling is an example of the use of inductive reasoning. If one were to poll one thousand people, and 300 of those people selected choice A, then one would infer that 30% of any population might also select choice A. This would be using inductive logic, because it does not definitively prove that 30% of any population would select choice A.
Because of this factor of uncertainty, inductive reasoning should be avoided when possible when attempting to prove geometric properties.
Truth tables are a way that one can display all the possibilities that a logical system may have when given certain premises. The following is a truth table with two premises (p and q), which shows the truth value of some basic logical statements. (NOTE: T = true; F = false)
|p||q||¬p||¬q||p ⇒ q||p ⇔ q||p ∧ q||p ∨ q|
Unlike science which has theories, mathematics has a definite notion of proof. Mathematics applies deductive reasoning to create a series of logical statements which show that one thing implies another.
Consider a triangle, which we define as a shape with three vertices joined by three lines. We know that we can arbitrarily pick some point on a page, and make that into a vertex. We repeat that process and pick a second point. Using a ruler, we can connect these two points. We now make a third point, and using the ruler connect it to each of the other points. We have constructed a triangle.
In mathematics we formalize this process into axioms, and carefully lay out the sequence of statements to show what follows. All definitions are clearly defined. In modern mathematics, we are always working within some system where various axioms hold.
The most common form of explicit proof in highschool geometry is a two column proof consists of five parts: the given, the proposition, the statement column, the reason column, and the diagram (if one is given).
Example of a Two-Column Proof
Now, suppose a problem tells you to solve for , showing all steps made to get to the answer. A proof shows how this is done:
Prove: x = 1
|Property of subtraction|
We use "Given" as the first reason, because it is "given" to us in the problem.
Written proofs (also known as informal proofs, paragraph proofs, or 'plans for proof') are written in paragraph form. Other than this formatting difference, they are similar to two-column proofs.
Sometimes it is helpful to start with a written proof, before formalizing the proof in two-column form. If you're having trouble putting your proof into two column form, try "talking it out" in a written proof first.
Example of a Written Proof
We are given that x + 1 = 2, so if we subtract one from each side of the equation (x + 1 - 1 = 2 - 1), then we can see that x = 1 by the definition of subtraction.
A flowchart proof or more simply a flow proof is a graphical representation of a two-column proof. Each set of statement and reasons are recorded in a box and then arrows are drawn from one step to another. This method shows how different ideas come together to formulate the proof.
Postulates in geometry are very similar to axioms, self-evident truths, and beliefs in logic, political philosophy and personal decision-making. The five postulates of Euclidean Geometry define the basic rules governing the creation and extension of geometric figures with ruler and compass. Together with the five axioms (or "common notions") and twenty-three definitions at the beginning of Euclid's Elements, they form the basis for the extensive proofs given in this masterful compilation of ancient Greek geometric knowledge. They are as follows:
- A straight line may be drawn from any given point to any other.
- A straight line may be extended to any finite length.
- A circle may be described with any given point as its center and any distance as its radius.
- All right angles are congruent.
- If a straight line intersects two other straight lines, and so makes the two interior angles on one side of it together less than two right angles, then the other straight lines will meet at a point if extended far enough on the side on which the angles are less than two right angles.
Postulate 5, the so-called Parallel Postulate was the source of much annoyance, probably even to Euclid, for being so relatively prolix. Mathematicians have a peculiar sense of aesthetics that values simplicity arising from simplicity, with the long complicated proofs, equations and calculations needed for rigorous certainty done behind the scenes, and to have such a long sentence amidst such other straightforward, intuitive statements seems awkward. As a result, many mathematicians over the centuries have tried to prove the results of the Elements without using the Parallel Postulate, but to no avail. However, in the past two centuries, assorted non-Euclidean geometries have been derived based on using the first four Euclidean postulates together with various negations of the fifth.
Chapter 7. Vertical Angles
Vertical angles are a pair of angles with a common vertex whose sides form opposite rays. An extensively useful fact about vertical angles is that they are congruent. Aside from saying that any pair of vertical angles "obviously" have the same measure by inspection, we can prove this fact with some simple algebra and an observation about supplementary angles. Let two lines intersect at a point, and angles A1 and A2 be a pair of vertical angles thus formed. At the point of intersection, two other angles are also formed, and we'll call either one of them B1 without loss of generality. Since B1 and A1 are supplementary, we can say that the measure of B1 plus the measure of A1 is 180. Similarly, the measure of B1 plus the measure of A2 is 180. Thus the measure of A1 plus the measure of B1 equals the measure of A2 plus the measure of B1, by substitution. Then by subracting the measure of B1 from each side of this equality, we have that the measure of A1 equals the measure of A2.
Parallel Lines in a Plane
Two coplanar lines are said to be parallel if they never intersect. For any given point on the first line, its distance to the second line is equal to the distance between any other point on the first line and the second line. The common notation for parallel lines is "||" (a double pipe); it is not unusual to see "//" as well. If line m is parallel to line n, we write "m || n". Lines in a plane either coincide, intersect in a point, or are parallel. Controversies surrounding the Parallel Postulate lead to the development of non-Euclidean geometries.
Parallel Lines and Special Pairs of Angles
When two (or more) parallel lines are cut by a transversal, the following angle relationships hold:
- corresponding angles are congruent
- alternate exterior angles are congruent
- same-side interior angles are supplementary
Theorems Involving Parallel Lines
- If a line in a plane is perpendicular to one of two parallel lines, it is perpendicular to the other line as well.
- If a line in a plane is parallel to one of two parallel lines, it is parallel to both parallel lines.
- If three or more parallel lines are intersected by two or more transversals, then they divide the transversals proportionally.
Congruent shapes are the same size with corresponding lengths and angles equal. In other words, they are exactly the same size and shape. They will fit on top of each other perfectly. Therefore if you know the size and shape of one you know the size and shape of the others. For example:
Each of the above shapes is congruent to each other. The only difference is in their orientation, or the way they are rotated. If you traced them onto paper and cut them out, you could see that they fit over each other exactly.
Having done this, right away we can see that, though the angles correspond in size and position, the sides do not. Therefore it is proved the triangles are not congruent.
Similar shapes are like congruent shapes in that they must be the same shape, but they don't have to be the same size. Their corresponding angles are congruent and their corresponding sides are in proportion.
Methods of Determining Congruence
Two triangles are congruent if:
- each pair of corresponding sides is congruent
- two pairs of corresponding angles are congruent and a pair of corresponding sides are congruent
- two pairs of corresponding sides and the angles included between them are congruent
Tips for Proofs
Commonly used prerequisite knowledge in determining the congruence of two triangles includes:
- by the reflexive property, a segment is congruent to itself
- vertical angles are congruent
- when parallel lines are cut by a transversal corresponding angles are congruent
- when parallel lines are cut by a transversal alternate interior angles are congruent
- midpoints and bisectors divide segments and angles into two congruent parts
For two triangles to be similar, all 3 corresponding angles must be congruent, and all three sides must be proportionally equal. Two triangles are similar if...
- Two angles of each triangle are congruent.
- The acute angle of a right triangle is congruent to the acute angle of another right triangle.
- The two triangles are congruent. Note here that congruency implies similarity.
A quadrilateral is a polygon that has four sides.
Special Types of Quadrilaterals
- A parallelogram is a quadrilateral having two pairs of parallel sides.
- A square, a rhombus, and a rectangle are all examples of parallelograms.
- A rhombus is a quadrilateral of which all four sides are the same length.
- A rectangle is a parallelogram of which all four angles are 90 degrees.
- A square is a quadrilateral of which all four sides are of the same length, and all four angles are 90 degrees.
- A square is a rectangle, a rhombus, and a parallelogram.
- A trapezoid is a quadrilateral which has two parallel sides (U.S.)
- U.S. usage: A trapezium is a quadrilateral which has no parallel sides.
- U.K usage: A trapezium is a quadrilateral with two parallel sides (same as US trapezoid definition).
- A kite is an quadrilateral with two pairs of congruent adjacent sides.
One of the most important properties used in proofs is that the sum of the angles of the quadrilateral is always 360 degrees. This can easily be proven too:
If you draw a random quadrilateral, and one of its diagonals, you'll split it up into two triangles. Given that the sum of the angles of a triangle is 180 degrees, you can sum them up, and it'll give 360 degrees.
A parallelogram is a geometric figure with two pairs of parallel sides. Parallelograms are a special type of quadrilateral. The opposite sides are equal in length and the opposite angles are also equal. The area is equal to the product of any side and the distance between that side and the line containing the opposite side.
Properties of Parallelograms
The following properties are common to all parallelograms (parallelogram, rhombus, rectangle, square)
- both pairs of opposite sides are parallel
- both pairs of opposite sides are congruent
- both pairs of opposite angles are congruent
- the diagonals bisect each other
- A rhombus is a parallelogram with four congruent sides.
- The diagonals of a rhombus are perpendicular.
- Each diagonal of a rhombus bisects two angles the rhombus.
- A rhombus may or may not be a square.
- A square is a parallelogram with four right angles and four congruent sides.
- A square is both a rectangle and a rhombus and inherits all of their properties.
A Trapezoid (American English) or Trapezium (British English) is a quadrilateral that has two parallel sides and two non parallel sides.
Some properties of trapezoids:
- The interior angles sum to 360° as in any quadrilateral.
- The parallel sides are unequal.
- Each of the parallel sides is called a base (b) of the trapezoid. The two angles that join one base are called 'base angles'.
- If the two non-parallel sides are equal, the trapezoid is called an isosceles trapezoid.
- In an isosceles trapezoid, each pair of base angles are equal.
- If one pair of base angles of a trapezoid are equal, the trapezoid is isosceles.
- A line segment connecting the midpoints of the non-parallel sides is called the median (m) of the trapeziod.
- The median of a trapezoid is equal to one half the sum of the bases (called b1 and b2).
- A line segment perpendicular to the bases is called an altitude (h) of the trapezoid.
The area (A) of a trapezoid is equal to the product of an altitude and the median.
Recall though that the median is half of the sum of the bases.
Substituting for m, we get:
A circle is a set of all points in a plane that are equidistant from a single point; that single point is called the centre of the circle and the distance between any point on circle and the centre is called radius of the circle.
a chord is an internal segment of a circle that has both of its endpoints on the circumference of the circle.
- the diameter of a circle is the largest chord possible
a secant of a circle is any line that intersects a circle in two places.
- a secant contains any chord of the circle
a tangent to a circle is a line that intersects a circle in exactly one point, called the point of tangency.
- at the point of tangency the tangent line and the radius of the circle are perpendicular
Chapter 16. Circles/Arcs
An arc is a segment of the perimeter of a given circle. The measure of an arc is measured as an angle, this could be in radians or degrees (more on radians later). The exact measure of the arc is determined by the measure of the angle formed when a line is drawn from the center of the circle to each end point. As an example the circle below has an arc cut out of it with a measure of 30 degrees.
As I mentioned before an arc can be measured in degrees or radians. A radian is merely a different method for measuring an angle. If we take a unit circle (which has a radius of 1 unit), then if we take an arc with the length equal to 1 unit, and draw line from each endpoint to the center of the circle the angle formed is equal to 1 radian. this concept is displayed below, in this circle an arc has been cut off by an angle of 1 radian, and therefore the length of the arc is equal to because the radius is 1.
From this definition we can say that on the unit circle a single radian is equal to radians because the perimeter of a unit circle is equal to . Another useful property of this definition that will be extremely useful to anyone who studies arcs is that the length of an arc is equal to its measure in radians multiplied by the radius of the circle.
Converting to and from radians is a fairly simple process. 2 facts are required to do so, first a circle is equal to 360 degrees, and it is also equal to . using these 2 facts we can form the following formula:
, thus 1 degree is equal to radians.
From here we can simply multiply by the number of degrees to convert to radians. for example if we have 20 degrees and want to convert to radians then we proceed as follows:
The same sort of argument can be used to show the formula for getting 1 radian.
, thus 1 radian is equal to
A tangent is a line in the same plane as a given circle that meets that circle in exactly one point. That point is called the point of tangency. A tangent cannot pass through a circle; if it does, it is classified as a chord. A secant is a line containing a chord.
A common tangent is a line tangent to two circles in the same plane. If the tangent does not intersect the line containing and connecting the centers of the circles, it is an external tangent. If it does, it is an internal tangent.
Two circles are tangent to one another if in a plane they intersect the same tangent in the same point.
Sector of a circle
A sector of a circle can be thought of as a pie piece. In the picture below, a sector of the circle is shaded yellow.
To find the area of a sector, find the area of the whole circle and then multiply by the angle of the sector over 360 degrees.
A more intuitive approach can be used when the sector is half the circle. In this case the area of the sector would just be the area of the circle divided by 2.
- See Angle
Addition Property of Equality
For any real numbers a, b, and c, if a = b, then a + c = b + c.
A figure is an angle if and only if it is composed of two rays which share a common endpoint. Each of these rays (or segments, as the case may be) is known as a side of the angle (For example, in the illustration at right), and the common point is known as the angle's vertex (point B in the illustration). Angles are measured by the difference of their slopes. The units for angle measure are radians and degrees. Angles may be classified by their degree measure.
- Acute Angle: an angle is an acute angle if and only if it has a measure of less than 90°
- Right Angle: an angle is an right angle if and only if it has a measure of exactly 90°
- Obtuse Angle: an angle is an obtuse angle if and only if it has a measure of greater than 90°
Angle Addition Postulate
If P is in the interior of an angle , then
Center of a circle
Point P is the center of circle C if and only if all points in circle C are equidistant from point P and point P is contained in the same plane as circle C.
A collection of points is said to be a circle with a center at point P and a radius of some distance r if and only if it is the collection of all points which are a distance of r away from point P and are contained by a plane which contain point P.
A polygon is said to be concave if and only if it contains at least one interior angle with a measure greater than 180° exclusively and less than 360° exclusively.
Two angles formed by a transversal intersecting with two lines are corresponding angles if and only if one is on the inside of the two lines, the other is on the outside of the two lines, and both are on the same side of the transversal.
Corresponding Angles Postulate
If two lines cut by a transversal are parallel, then their corresponding angles are congruent.
Corresponding Parts of Congruent Triangles are Congruent Postulate
The Corresponding Parts of Congruent Triangles are Congruent Postulate (CPCTC) states:
- If ∆ABC ≅ ∆XYZ, then all parts of ∆ABC are congruent to their corresponding parts in ∆XYZ. For example:
- ∠ABC ≅ ∠XYZ
- ∠BCA ≅ ∠YZX
- ∠CAB ≅ ∠ZXY
CPCTC also applies to all other parts of the triangles, such as a triangle's altitude, median, circumcenter, et al.
A line segment is the diameter of a circle if and only if it is a chord of the circle which contains the circle's center.
- See Circle
and if they cross they are congruent
A collection of points is a line if and only if the collection of points is perfectly straight (aligned), is infinitely long, and is infinitely thin. Between any two points on a line, there exists an infinite number of points which are also contained by the line. Lines are usually written by two points in the line, such as line AB, or
A collection of points is a line segment if and only if it is perfectly straight, is infinitely thin, and has a finite length. A line segment is measured by the shortest distance between the two extreme points on the line segment, known as endpoints. Between any two points on a line segment, there exists an infinite number of points which are also contained by the line segment.
Two lines or line segments are said to be parallel if and only if the lines are contained by the same plane and have no points in common if continued infinitely.
Two planes are said to be parallel if and only if the planes have no points in common when continued infinitely.
Two lines that intersect at a 90° angle.
Given a line, and a point P not in line , then there is one and only one line that goes through point P perpendicular to
An object is a plane if and only if it is a two-dimensional object which has no thickness or curvature and continues infinitely. A plane can be defined by three points. A plane may be considered to be analogous to a piece of paper.
A point is a zero-dimensional mathematical object representing a location in one or more dimensions. A point has no size; it has only location.
A polygon is a closed plane figure composed of at least 3 straight lines. Each side has to intersect another side at their respective endpoints, and that the lines intersecting are not collinear.
The radius of a circle is the distance between any given point on the circle and the circle's center.
- See Circle
A ray is a straight collection of points which continues infinitely in one direction. The point at which the ray stops is known as the ray's endpoint. Between any two points on a ray, there exists an infinite number of points which are also contained by the ray.
The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the point's coordinate. The distance between two points is the absolute value of the difference between the two coordinates of the two points.
Geometry/Synthetic versus analytic geometry
- Two and Three-Dimensional Geometry and Other Geometric Figures
Perimeter and Arclength
Perimeter of Circle
The circles perimeter can be calculated using the following formula
where and the radius of the circle.
Perimeter of Polygons
The perimeter of a polygon with number of sides abbreviated can be caculated using the following formula
Arclength of Circles
The arclength of a given circle with radius can be calculated using
where is the angle given in radians.
Arclength of Curves
If a curve in have a parameter form for , then the arclength can be calculated using the following fomula
Derivation of formula can be found using differential geometry on infinitely small triangles.
Area of Circles
The method for finding the area of a circle is
Where π is a constant roughly equal to 3.14159265358978 and r is the radius of the circle; a line drawn from any point on the circle to its center.
Area of Triangles
Three ways of calculating the area inside of a triangle are mentioned here.
If one of the sides of the triangle is chosen as a base, then a height for the triangle and that particular base can be defined. The height is a line segment perpendicular to the base or the line formed by extending the base and the endpoints of the height are the corner point not on the base and a point on the base or line extending the base. Let B = the length of the side chosen as the base. Let
h = the distance between the endpoints of the height segment which is perpendicular to the base. Then the area of the triangle is given by:
This method of calculating the area is good if the value of a base and its corresponding height in the triangle is easily determined. This is particularly true if the triangle is a right triangle, and the lengths of the two sides sharing the 90o angle can be determined.
- , also known as Heron's Formula
If the lengths of all three sides of a triangle are known, Hero's formula may be used to calculate the area of the triangle. First, the semiperimeter, s, must be calculated by dividing the sum of the lengths of all three sides by 2. For a triangle having side lengths a, b, and c :
Then the triangle's area is given by:
If the triangle is needle shaped, that is, one of the sides is very much shorter than the other two then it can be difficult to compute the area because the precision needed is greater than that available in the calculator or computer that is used. In otherwords Heron's formula is numerically unstable. Another formula that is much more stable is:
where , , and have been sorted so that .
In a triangle with sides length a, b, and c and angles A, B, and C opposite them,
This formula is true because in the formula . It is useful because you don't need to find the height from an angle in a separate step, and is also used to prove the law of sines (divide all terms in the above equation by a*b*c and you'll get it directly!)
Area of Rectangles
The area calculation of a rectangle is simple and easy to understand. One of the sides is chosen as the base, with a length b. An adjacent side is then the height, with a length h, because in a rectangle the adjacent sides are perpendicular to the side chosen as the base. The rectangle's area is given by:
Sometimes, the baselength may be referred to as the length of the rectangle, l, and the height as the width of the rectangle, w. Then the area formula becomes:
Regardless of the labels used for the sides, it is apparent that the two formulas are equivalent.
Of course, the area of a square with sides having length s would be:
Area of Parallelograms
The area of a parallelogram can be determined using the equation for the area of a rectangle. The formula is:
A is the area of a parallelogram. b is the base. h is the height.
The height is a perpendicular line segment that connects one of the vertices to its opposite side (the base).
Area of Rhombus
Remember in a rombus all sides are equal in length.
and represent the diagonals.
Area of Trapezoids
The area of a trapezoid is derived from taking the arithmetic mean of its two parallel sides to form a rectangle of equal area.
Where and are the lengths of the two parallel bases.
Area of Kites
The area of a kite is based on splitting the kite into four pieces by halving it along each diagonal and using these pieces to form a rectangle of equal area.
Where a and b are the diagonals of the kite.
Alternatively, the kite may be divided into two halves, each of which is a triangle, by the longer of its diagonals, a. The area of each triangle is thus
Where b is the other (shorter) diagonal of the kite. And the total area of the kite (which is composed of two identical such triangles) is
Which is the same as
Areas of other Quadrilaterals
The areas of other quadrilaterals are slightly more complex to calculate, but can still be found if the quadrilateral is well-defined. For example, a quadrilateral can be divided into two triangles, or some combination of triangles and rectangles. The areas of the constituent polygons can be found and added up with arithmetic.
Volume is like area expanded out into 3 dimensions. Area deals with only 2 dimensions. For volume we have to consider another dimension. Area can be thought of as how much space some drawing takes up on a flat piece of paper. Volume can be thought of as how much space an object takes up.
|Common equations for volume:|
|A cube:||s = length of a side|
|A rectangular prism:||l = length, w = width, h = height|
|A cylinder (circular prism):||r = radius of circular face, h = height|
|Any prism that has a constant cross sectional area along the height:||A = area of the base, h = height|
|A sphere:||r = radius of sphere
which is the integral of the Surface Area of a sphere
|An ellipsoid:||a, b, c = semi-axes of ellipsoid|
|A pyramid:||A = area of the base, h = height of pyramid|
|A cone (circular-based pyramid):||r = radius of circle at base, h = distance from base to tip
(The units of volume depend on the units of length - if the lengths are in meters, the volume will be in cubic meters, etc.)
The volume of any solid whose cross sectional areas are all the same is equal to that cross sectional area times the distance the centroid(the center of gravity in a physical object) would travel through the solid.
If two solids are contained between two parallel planes and every plane parallel to these two plane has equal cross sections through these two solids, then their volumes are equal.
A Polygon is a two-dimensional figure, meaning all of the lines in the figure are contained within one plane. They are classified by the number of angles, which is also the number of sides.
One key point to note is that a polygon must have at least three sides. Normally, three to ten sided figures are referred to by their names (below), while figures with eleven or more sides is an n-gon, where n is the number of sides. Hence a forty-sided polygon is called a 40-gon.
A polygon with three angles and sides.
A polygon with four angles and sides.
A polygon with five angles and sides.
A polygon with six angles and sides.
A polygon with seven angles and sides.
A polygon with eight angles and sides.
A polygon with nine angles and sides.
A polygon with ten angles and sides.
For a list of n-gon names, go to and scroll to the bottom of the page.
Polygons are also classified as convex or concave. A convex polygon has interior angles less than 180 degrees, thus all triangles are convex. If a polygon has at least one internal angle greater than 180 degrees, then it is concave. An easy way to tell if a polygon is concave is if one side can be extended and crosses the interior of the polygon. Concave polygons can be divided into several convex polygons by drawing diagonals. Regular polygons are polygons in which all sides and angles are congruent.
A triangle is a type of polygon having three sides and, therefore, three angles. The triangle is a closed figure formed from three straight line segments joined at their ends. The points at the ends can be called the corners, angles, or vertices of the triangle. Since any given triangle lies completely within a plane, triangles are often treated as two-dimensional geometric figures. As such, a triangle has no volume and, because it is a two-dimensionally closed figure, the flat part of the plane inside the triangle has an area, typically referred to as the area of the triangle. Triangles are always convex polygons.
A triangle must have at least some area, so all three corner points of a triangle cannot lie in the same line. The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. The preceding statement is sometimes called the Triangle Inequality.
Certain types of triangles
Categorized by angle
The sum of the interior angles in a triangle always equals 180o. This means that no more than one of the angles can be 90o or more. All three angles can all be less than 90oin the triangle; then it is called an acute triangle. One of the angles can be 90o and the other two less than 90o; then the triangle is called a right triangle. Finally, one of the angles can be more than 90o and the other two less; then the triangle is called an obtuse triangle.
Categorized by sides
If all three of the sides of a triangle are of different length, then the triangle is called a scalene triangle.
If two of the sides of a triangle are of equal length, then it is called an isoceles triangle. In an isoceles triangle, the angle between the two equal sides can be more than, equal to, or less than 90o. The other two angles are both less than 90o.
If all three sides of a triangle are of equal length, then it is called an equilateral triangle and all three of the interior angles must be 60o, making it equilangular. Because the interior angles are all equal, all equilateral triangles are also the three-sided variety of a regular polygon and they are all similar, but might not be congruent. However, polygons having four or more equal sides might not have equal interior angles, might not be regular polygons, and might not be similar or congruent. Of course, pairs of triangles which are not equilateral might be similar or congruent.
Opposite corners and sides in triangles
If one of the sides of a triangle is chosen, the interior angles of the corners at the side's endpoints can be called adjacent angles. The corner which is not one of these endpoints can be called the corner opposite to the side. The interior angle whose vertex is the opposite corner can be called the angle opposite to the side.
Likewise, if a corner or its angle is chosen, then the two sides sharing an endpoint at that corner can be called adjacent sides. The side not having this corner as one of its two endpoints can be called the side opposite to the corner.
The sides or their lengths of a triangle are typically labeled with lower case letters. The corners or their corresponding angles can be labeled with capital letters. The triangle as a whole can be labeled by a small triangle symbol and its corner points. In a triangle, the largest interior angle is opposite to longest side, and vice versa.
Any triangle can be divided into two right triangles by taking the longest side as a base, and extending a line segment from the opposite corner to a point on the base such that it is perpendicular to the base. Such a line segment would be considered the height or altitude ( h ) for that particular base ( b ). The two right triangles resulting from this division would both share the height as one of its sides. The interior angles at the meeting of the height and base would be 90o for each new right triangle. For acute triangles, any of the three sides can act as the base and have a corresponding height. For more information on right triangles, see Right Triangles and Pythagorean Theorem.
Area of Triangles
If base and height of a triangle are known, then the area of the triangle can be calculated by the formula:
( is the symbol for area)
Ways of calculating the area inside of a triangle are further discussed under Area.
The centroid is constructed by drawing all the medians of the triangle. All three medians intersect at the same point: this crossing point is the centroid. Centroids are always inside a triangle. They are also the centre of gravity of the triangle.
The three angle bisectors of the triangle intersect at a single point, called the incentre. Incentres are always inside the triangle. The three sides are equidistant from the incentre. The incentre is also the centre of the inscribed circle (incircle) of a triangle, or the interior circle which touches all three sides of the triangle.
The circumcentre is the intersection of all three perpendicular bisectors. Unlike the incentre, it is outside the triangle if the triangle is obtuse. Acute triangles always have circumcentres inside, while the circumcentre of a right triangle is the midpoint of the hypotenuse. The vertices of the triangle are equidistant from the circumcentre. The circumcentre is so called because it is the centre of the circumcircle, or the exterior circle which touches all three vertices of the triangle.
The orthocentre is the crossing point of the three altitudes. It is always inside acute triangles, outside obtuse triangles, and on the right vertex of the right-angled triangle.
Please note that the centres of an equilateral triangle are always the same point.
Right Triangles and Pythagorean Theorem
Right triangles are triangles in which one of the interior angles is 90o. A 90o angle is called a right angle. Right triangles are sometimes called right-angled triangles. The other two interior angles are complementary, i.e. their sum equals 90o. Right triangles have special properties which make it easier to conceptualize and calculate their parameters in many cases.
The side opposite of the right angle is called the hypotenuse. The sides adjacent to the right angle are the legs. When using the Pythagorean Theorem, the hypotenuse or its length is often labeled with a lower case c. The legs (or their lengths) are often labeled a and b.
Either of the legs can be considered a base and the other leg would be considered the height (or altitude), because the right angle automatically makes them perpendicular. If the lengths of both the legs are known, then by setting one of these sides as the base ( b ) and the other as the height ( h ), the area of the right triangle is very easy to calculate using this formula:
This is intuitively logical because another congruent right triangle can be placed against it so that the hypotenuses are the same line segment, forming a rectangle with sides having length b and width h. The area of the rectangle is b × h, so either one of the congruent right triangles forming it has an area equal to half of that rectangle.
Right triangles can be neither equilateral, acute, nor obtuse triangles. Isosceles right triangles have two 45° angles as well as the 90° angle. All isosceles right triangles are similar since corresponding angles in isosceles right triangles are equal. If another triangle can be divided into two right triangles (see Triangle), then the area of the triangle may be able to be determined from the sum of the two constituent right triangles. Also the Pythagorean theorem can be used for non right triangles. a2+b2=c2-2c
For history regarding the Pythagorean Theorem, see Pythagorean theorem. The Pythagorean Theorem states that:
- In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's take a right triangle as shown here and set c equal to the length of the hypotenuse and set a and b each equal to the lengths of the other two sides. Then the Pythagorean Theorem can be stated as this equation:
Using the Pythagorean Theorem, if the lengths of any two of the sides of a right triangle are known and it is known which side is the hypotenuse, then the length of the third side can be determined from the formula.
Sine, Cosine, and Tangent for Right Triangles
Sine, Cosine, and Tangent are all functions of an angle, which are useful in right triangle calculations. For an angle designated as θ, the sine function is abbreviated as sin θ, the cosine function is abbreviated as cos θ, and the tangent function is abbreviated as tan θ. For any
angle θ, sin θ, cos θ, and tan θ are each single determined values and if θ is a known value, sin θ, cos θ, and tan θ can be looked up in a table or found with a calculator. There is a table listing these function values at the end of this section. For an angle between listed values, the sine, cosine, or tangent of that angle can be estimated from the values in the table. Conversely, if a number is known to be the sine, cosine, or tangent of a angle, then such tables could be used in reverse to find (or estimate) the value of a corresponding angle.
These three functions are related to right triangles in the following ways:
In a right triangle,
- the sine of a non-right angle equals the length of the leg opposite that angle divided by the length of the hypotenuse.
- the cosine of a non-right angle equals the length of the leg adjacent to it divided by the length of the hypotenuse.
- the tangent of a non-right angle equals the length of the leg opposite that angle divided by the length of the leg adjacent to it.
For any value of θ where cos θ ≠ 0,
If one considers the diagram representing a right triangle with the two non-right angles θ1and θ2, and the side lengths a,b,c as shown here:
For the functions of angle θ1:
Analogously, for the functions of angle θ2:
Table of sine, cosine, and tangent for angles θ from 0 to 90°
|θ in degrees||θ in radians||sin θ||cos θ||tan θ|
General rules for important angles:
Polyominoes are shapes made from connecting unit squares together, though certain connections are not allowed.
A domino is the shape made from attaching unit squares so that they share one full edge. The term polyomino is based on the word domino. There is only one possible domino.
Tromino↑Jump back a section
A polymino made from four squares is called a tetromino. There are five possible combinations and two reflections:
A polymino made from five squares is called a pentomino. There are twelve possible pentominoes, excluding mirror images and rotations.
Ellipses are sometimes called ovals. Ellipses contain two foci. The sum of the distance from a point on the ellipse to one focus and that same point to the other focus is constant
Area Shapes Extended into 3rd Dimension
Geometry/Area Shapes Extended into 3rd Dimension
Area Shapes Extended into 3rd Dimension Linearly to a Line or Point
Geometry/Area Shapes Extended into 3rd Dimension Linearly to a Line or Point
Ellipsoids and Spheres
Geometry/Ellipsoids and Spheres
Suppose you are an astronomer in America. You observe an exciting event (say, a supernova) in the sky and would like to tell your colleagues in Europe about it. Suppose the supernova appeared at your zenith. You can't tell astronomers in Europe to look at their zenith because their zenith points in a different direction. You might tell them which constellation to look in. This might not work, though, because it might be too hard to find the supernova by searching an entire constellation. The best solution would be to give them an exact position by using a coordinate system.
On Earth, you can specify a location using latitude and longitude. This system works by measuring the angles separating the location from two great circles on Earth (namely, the equator and the prime meridian). Coordinate systems in the sky work in the same way.
The equatorial coordinate system is the most commonly used. The equatorial system defines two coordinates: right ascension and declination, based on the axis of the Earth's rotation. The declination is the angle of an object north or south of the celestial equator. Declination on the celestial sphere corresponds to latitude on the Earth. The right ascension of an object is defined by the position of a point on the celestial sphere called the vernal equinox. The further an object is east of the vernal equinox, the greater its right ascension.
A coordinate system is a system designed to establish positions with respect to given reference points. The coordinate system consists of one or more reference points, the styles of measurement (linear measurement or angular measurement) from those reference points, and the directions (or axes) in which those measurements will be taken. In astronomy, various coordinate systems are used to precisely define the locations of astronomical objects.
Latitude and longitude are used to locate a certain position on the Earth's surface. The lines of latitude (horizontal) and the lines of longitude (vertical) make up an invisible grid over the earth. Lines of latitude are called parallels. Lines of longitude aren't completely straight (they run from the exact point of the north pole to the exact point of the south pole) so they are called meridians. 0 degrees latitude is the Earth's middle, called the equator. O degrees longitude was tricky because there really is no middle of the earth vertically. It was finally agreed that the observatory in Greenwich, U.K. would be 0 degrees longitude due to its significant roll in scientific discoveries and creating latitude and longitude. 0 degrees longitude is called the prime meridian.
Latitude and longitude are measured in degrees. One degree is about 69 miles. There are sixty minutes (') in a degree and sixty seconds (") in a minute. These tiny units make GPS's (Global Positioning Systems) much more exact.
There are a few main lines of latitude:the Arctic Circle, the Antarctic Circle, the Tropic of Cancer, and the Tropic of Capricorn. The Antarctic Circle is 66.5 degrees south of the equator and it marks the temperate zone from the Antarctic zone. The Arctic Circle is an exact mirror in the north. The Tropic of Cancer separates the tropics from the temperate zone. It is 23.5 degrees north of the equator. It is mirrored in the south by the Tropic of Capricorn.
Horizontal coordinate system
One of the simplest ways of placing a star on the night sky is the coordinate system based on altitude or azimuth, thus called the Alt-Az or horizontal coordinate system. The reference circles for this system are the horizon and the celestial meridian, both of which may be most easily graphed for a given location using the celestial sphere.
In simplest terms, the altitude is the angle made from the position of the celestial object (e.g. star) to the point nearest it on the horizon. The azimuth is the angle from the northernmost point of the horizon (which is also its intersection with the celestial meridian) to the point on the horizon nearest the celestial object. Usually azimuth is measured eastwards from due north. So east has az=90°, south has az=180°, west has az=270° and north has az=360° (or 0°). An object's altitude and azimuth change as the earth rotates.
Equatorial coordinate system
The equatorial coordinate system is another system that uses two angles to place an object on the sky: right ascension and declination.
Ecliptic coordinate system
The ecliptic coordinate system is based on the ecliptic plane, i.e., the plane which contains our Sun and Earth's average orbit around it, which is tilted at 23°26' from the plane of Earth's equator. The great circle at which this plane intersects the celestial sphere is the ecliptic, and one of the coordinates used in the ecliptic coordinate system, the ecliptic latitude, describes how far an object is to ecliptic north or to ecliptic south of this circle. On this circle lies the point of the vernal equinox (also called the first point of Aries); ecliptic longitude is measured as the angle of an object relative to this point to ecliptic east. Ecliptic latitude is generally indicated by φ, whereas ecliptic longitude is usually indicated by λ.
Galactic coordinate system
As a member of the Milky Way Galaxy, we have a clear view of the Milky Way from Earth. Since we are inside the Milky Way, we don't see the galaxy's spiral arms, central bulge and so forth directly as we do for other galaxies. Instead, the Milky Way completely encircles us. We see the Milky Way as a band of faint starlight forming a ring around us on the celestial sphere. The disk of the galaxy forms this ring, and the bulge forms a bright patch in the ring. You can easily see the Milky Way's faint band from a dark, rural location.
Our galaxy defines another useful coordinate system — the galactic coordinate system. This system works just like the others we've discussed. It also uses two coordinates to specify the position of an object on the celestial sphere. The galactic coordinate system first defines a galactic latitude, the angle an object makes with the galactic equator. The galactic equator has been selected to run through the center of the Milky Way's band. The second coordinate is galactic longitude, which is the angular separation of the object from the galaxy's "prime meridian," the great circle that passes through the Galactic center and the galactic poles. The galactic coordinate system is useful for describing an object's position with respect to the galaxy's center. For example, if an object has high galactic latitude, you might expect it to be less obstructed by interstellar dust.
Transformations between coordinate systems
One can use the principles of spherical trigonometry as applied to triangles on the celestial sphere to derive formulas for transforming coordinates in one system to those in another. These formulas generally rely on the spherical law of cosines, known also as the cosine rule for sides. By substituting various angles on the celestial sphere for the angles in the law of cosines and by thereafter applying basic trigonometric identities, most of the formulas necessary for coordinate transformations can be found. The law of cosines is stated thus:
To transform from horizontal to equatorial coordinates, the relevant formulas are as follows:
where RA is the right ascension, Dec is the declination, LST is the local sidereal time, Alt is the altitude, Az is the azimuth, and Lat is the observer's latitude. Using the same symbols and formulas, one can also derive formulas to transform from equatorial to horizontal coordinates:
Transformation from equatorial to ecliptic coordinate systems can similarly be accomplished using the following formulas:
where RA is the right ascension, Dec is the declination, φ is the ecliptic latitude, λ is the ecliptic longitude, and ε is the tilt of Earth's axis relative to the ecliptic plane. Again, using the same formulas and symbols, new formulas for transforming ecliptic to equatorial coordinate systems can be found:
- Traditional Geometry:
A topological space is a set X, and a collection of subsets of X, C such that both the empty set and X are contained in C and the union of any subcollection of sets in C and the intersection of any finite subcollection of sets in C are also contained within C. The sets in C are called open sets. Their complements relative to X are called closed sets.
Given two topological spaces, X and Y, a map f from X to Y is continuous if for every open set U of Y, f−1(U) is an open set of X.
Hyperbolic and Elliptic Geometry
There are precisely three different classes of three-dimensional constant-curvature geometry: Euclidean, hyperbolic and elliptic geometry. The three geometries are all built on the same first four axioms, but each has a unique version of the fifth axiom, also known as the parallel postulate. The 1868 Essay on an Interpretation of Non-Euclidean Geometry by Eugenio Beltrami (1835 - 1900) proved the logical consistency of the two Non-Euclidean geometries, hyperbolic and elliptic.
The Parallel Postulate
The parallel postulate is as follows for the corresponding geometries.
Euclidean geometry: Playfair's version: "Given a line l and a point P not on l, there exists a unique line m through P that is parallel to l." Euclid's version: "Suppose that a line l meets two other lines m and n so that the sum of the interior angles on one side of l is less than 180°. Then m and n intersect in a point on that side of l." These two versions are equivalent; though Playfair's may be easier to conceive, Euclid's is often useful for proofs.
Hyperbolic geometry: Given an arbitrary infinite line l and any point P not on l, there exist two or more distinct lines which pass through P and are parallel to l.
Elliptic geometry: Given an arbitrary infinite line l and any point P not on l, there does not exist a line which passes through P and is parallel to l.
Hyperbolic geometry is also known as saddle geometry or Lobachevskian geometry. It differs in many ways to Euclidean geometry, often leading to quite counter-intuitive results. Some of these remarkable consequences of this geometry's unique fifth postulate include:
1. The sum of the three interior angles in a triangle is strictly less than 180°. Moreover, the angle sums of two distinct triangles are not necessarily the same.
2. Two triangles with the same interior angles have the same area.
Models of Hyperbolic Space
The following are four of the most common models used to describe hyperbolic space.
1. The Poincaré Disc Model. Also known as the conformal disc model. In it, the hyperbolic plane is represented by the interior of a circle, and lines are represented by arcs of circles that are orthogonal to the boundary circle and by diameters of the boundary circle. Preserves hyperbolic angles.
2. The Klein Model. Also known as the Beltrami-Klein model or projective disc model. In it, the hyperbolic plane is represented by the interior of a circle, and lines are represented by chords of the circle. This model gives a misleading visual representation of the magnitude of angles.
3. The Poincaré Half-Plane Model. The hyperbolic plane is represented by one-half of the Euclidean plane, as defined by a given Euclidean line l, where l is not considered part of the hyperbolic space. Lines are represented by half-circles orthogonal to l or rays perpendicular to l. Preserves hyperbolic angles.
4. The Lorentz Model. Spheres in Lorentzian four-space. The hyperbolic plane is represented by a two-dimensional hyperboloid of revolution embedded in three-dimensional Minkowski space.
Based on this geometry's definition of the fifth axiom, what does parallel mean? The following definitions are made for this geometry. If a line l and a line m do not intersect in the hyperbolic plane, but intersect at the plane's boundary of infinity, then l and m are said to be parallel. If a line p and a line q neither intersect in the hyperbolic plane nor at the boundary at infinity, then p and q are said to be ultraparallel.
The Ultraparallel Theorem
For any two lines m and n in the hyperbolic plane such that m and n are ultraparallel, there exists a unique line l that is perpendicular to both m and n.
Elliptic geometry differs in many ways to Euclidean geometry, often leading to quite counter-intuitive results. For example, directly from this geometry's fifth axiom we have that there exist no parallel lines. Some of the other remarkable consequences of the parallel postulate include: The sum of the three interior angles in a triangle is strictly greater than 180°.
Models of Elliptic Space
Spherical geometry gives us perhaps the simplest model of elliptic geometry. Points are represented by points on the sphere. Lines are represented by circles through the points.
- Euclid's First Four Postulates
- Euclid's Fifth Postulate
- Incidence Geometry
- Projective and Affine Planes (necessary?)
- Axioms of Betweenness
- Pasch and Crossbar
- Axioms of Congruence
- Continuity (necessary?)
- Hilbert Planes
- Neutral Geometry
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- Modern geometry
- An Alternative Way and Alternative Geometric Means of Calculating the Area of a Circle =
Geometry/An Alternative Way and Alternative Geometric Means of Calculating the Area of a Circle | http://en.m.wikibooks.org/wiki/Geometry/Print_version | 13 |
50 | The Origin of Asteroidsby Dr. Walt Brown
(This article has been reproduced with permission from the Center for Scientific Creation. The original article can be found here.)
NOTE - In order to fully understand the content of this article (and it’s companion article The Origin of Comets), you should read the book, In the Beginning by Dr. Walt Brown. This book fully explains Dr. Brown’s Hydroplate Theory which is the foundation upon which this article is written. In fact, this “article” is actually a chapter in the book, In the Beginning. Members of the 4th Day Alliance can download the complete PDF copy of this chapter by clicking here.
Figure 156: Asteroid Ida and Its Moon, Dactyl. In 1993, the Galileo spacecraft, heading toward Jupiter, took this picture 2,000 miles from asteroid Ida. To the surprise of most, Ida had a moon (about 1 mile in diameter) orbiting 60 miles away! Both Ida and Dactyl are composed of earthlike rock. We now know of 68 other asteroids that have moons.1 According to the laws of orbital mechanics (described in the preceding chapter), capturing a moon in space is unbelievably difficult—unless both the asteroid and a nearby potential moon had very similar speeds and directions and unless gases surrounded the asteroid during capture. If so, the asteroid, its moon, and each gas molecule were probably coming from the same place and were launched at about the same time. Within a million years, passing bodies would have stripped the moons away, so these asteroid-moon captures must have been recent.
From a distance, large asteroids look like big rocks. However, many show, by their low density, that they contain either much empty space or something light, such as water ice.2 Also, the best close-up pictures of an asteroid show millions of smaller rocks on its surface. Therefore, asteroids are flying rock piles held together by gravity. Ida, about 35 miles long, does not have enough gravity to squeeze itself into a spherical shape.
SUMMARY: The fountains of the great deep launched rocks as well as muddy water. As rocks moved farther from Earth, Earth’s gravity became less significant to them, and the gravity of nearby rocks became increasingly significant. Consequently, many rocks, assisted by their mutual gravity and surrounding clouds of water vapor, merged to become asteroids. Isolated rocks in space are meteoroids. Drag forces caused by water vapor and thrust forces produced by the radiometer effect concentrated asteroids in what is now the asteroid belt. All the so-called “mavericks of the solar system” (asteroids, meteoroids, and comets) resulted from the explosive events at the beginning of the flood.
Asteroids, also called minor planets, are rocky bodies orbiting the Sun. The orbits of most asteroids lie between those of Mars and Jupiter, a region called the asteroid belt. The largest asteroid, Ceres, is almost 600 miles in diameter and has about one-third the volume of all other asteroids combined. Orbits of almost 30,000 asteroids have been calculated. Many more asteroids have been detected, some less than 20 feet in diameter. A few that cross the Earth’s orbit would do great damage if they ever collided with Earth.
Two explanations are given for the origin of asteroids: (1) they were produced by an exploded planet, and (2) a planet failed to evolve completely. Experts recognize the problems with each explanation and are puzzled. The hydroplate theory offers a simple and complete—but quite different—solution that also answers other questions.
Meteorites, Meteors, and MeteoroidsIn space, solid bodies smaller than an asteroid but larger than a molecule are called “meteoroids.” They are renamed “meteors” as they travel through Earth’s atmosphere, and “meteorites” if they hit the ground.
Exploded-Planet Explanation. Smaller asteroids are more numerous than larger asteroids, a pattern typical of fragmented bodies. Seeing this pattern led to the early belief that asteroids are remains of an exploded planet. Later, scientists realized that all the fragments combined would not make up one small planet.3 Besides, too much energy is needed to explode and scatter even the smallest planet.
Failed-Planet Explanation. The most popular explanation today for asteroids is that they are bodies that did not merge to become a planet. Never explained is how, in nearly empty space, matter merged to become these rocky bodies in the first place,4 why rocky bodies started to form a planet but stopped,5 or why it happened only between the orbits of Mars and Jupiter. Also, because only vague explanations have been given for how planets formed, any claim to understand how one planet failed to form lacks credibility. In general, orbiting rocks do not merge to become either planets or asteroids. Special conditions are required, as explained on page 267 and Endnote 23 on page 288.] Today, collisions and near collisions fragment and scatter asteroids, just the opposite of this “failed-planet explanation.” In fact, during the 4,600,000,000 years evolutionists say asteroids have existed, asteroids would have had so many collisions that they should be much more fragmented than they are today.6
Hydroplate Explanation. Asteroids are composed of rocks expelled from Earth. The size distribution of asteroids does show that at least part of a planet fragmented. Although an energy source is not available to explode and disperse an entire Earth-size planet, the eruption of so much supercritical water from the subterranean chambers could have launched one 2,300th of the Earth—the mass of all asteroids combined. Astronomers have tried to describe the exploded planet, not realizing they were standing on the remaining 99.95% of it—too close to see it.7
As flood waters escaped from the subterranean chambers, pillars, forced to carry more and more of the weight of the overlying crust, were crushed. Also, the almost 10-mile-high walls of the rupture were unstable, because rock is not strong enough to support a cliff more than 5 miles high. As lower portions of the walls were crushed, large blocks8 were swept up and launched by the jetting fountains. Unsupported rock in the top 5 miles then fragmented. The smaller the rock, the faster it accelerated and the farther it went, just as a rapidly flowing stream carries smaller dirt particles faster and farther.
Water droplets in the fountains partially evaporated and quickly froze. Large rocks had large spheres of influence which grew as the rocks traveled away from Earth. Larger rocks became “seeds” around which other rocks and ice collected as spheres of influence expanded. Because of all the evaporated water vapor and the resulting aerobraking, even more mass concentrated around the “seeds.”Clumps of rocks became asteroids.
Question 1: Why did some clumps of rocks and ice in space become asteroids and others become comets?Imagine living in a part of the world where heavy frost settled each night, but the Sun shone daily. After many decades, would the countryside be buried in hundreds of feet of frost?
The answer depends on several things besides the obvious need for a large source of water. If dark rocks initially covered the ground, the Sun would heat them during the day, so frost from the previous night would tend to evaporate. However, if the sunlight was dim or the frost was thick (thereby reflecting more sunlight during the day), little frost would evaporate. More frost would accumulate the next night. Frost thickness would increase every 24 hours.
Now imagine living on a newly formed asteroid. Its spin would give you day-night cycles. After sunset, surface temperatures would plummet toward nearly absolute zero (-460°F), because asteroids do not have enough gravity to hold an atmosphere for long. With little atmosphere to insulate the asteroid, the day’s heat would quickly radiate, unimpeded, into outer space. Conversely, when the Sun rose, its rays would have little atmosphere to warm, so temperatures at the asteroid’s surface would rise rapidly.
As the fountains of the great deep launched rocks and water droplets, evaporation in space dispersed an “ocean” of water molecules and other gases in the inner solar system. Gas molecules that struck the cold side of your spinning asteroid would become frost.9 Sunlight would usually be dim on rocks in larger, more elongated orbits. Therefore, little frost would evaporate during the day, and the frost’s thickness would increase. Your “world” would become a comet. However, if your “world” orbited relatively near the Sun, its rays would evaporate each night’s frost, so your “world” would remain an asteroid.
Heavier rocks could not be launched with as much velocity as smaller particles (dirt, water droplets, and smaller rocks). The heavier rocks merged to become asteroids, while the smaller particles, primarily water, merged to become comets, which generally have larger orbits. No “sharp line” separates asteroids and comets.
PREDICTION 33Asteroids are rock piles, often with ice acting as a weak “glue” inside. Large rocks that began the capture process are nearer the centers of asteroids. Comets, which are primarily ice, have rocks in their cores. Four years after this prediction was published in 2001 (In the Beginning, 7th edition, page 220), measurements of the largest asteroid, Ceres, found that it does indeed have a dense, rocky core and primarily a water-ice mantle.10
Question 2: Wasn’t asteroid Eros found to be primarily a large, solid rock?A pile of dry sand here on Earth cannot maintain a slope greater than about 30 degrees. If it were steeper, the sand grains would roll downhill. Likewise, a pile of dry pebbles or rocks on an asteroid cannot have a slope exceeding about 30 degrees. However, 4% of Eros’ surface exceeds this slope, so some scientists concluded that much of Eros must be a large, solid rock. This conclusion overlooks the possibility that ice is present between some rocks and acts as a weak glue—as predicted above. Ice in asteroids would also explain their low density. Endnote 8 gives another reason why asteroids are probably flying rock piles.
Question 3: Objects launched from Earth should travel in elliptical, cometlike orbits. How could rocky bodies launched from Earth become concentrated in almost circular orbits between Mars and Jupiter?Gases, such as water vapor and its components,11 were abundant in the inner solar system for many years after the flood. Hot gas molecules striking each asteroid’s hot side were repelled with great force. This jetting action was like air rapidly escaping from a balloon, applying a thrust in a direction opposite to the escaping gas.12 Cold molecules striking each asteroid’s cold side produced less jetting. This thrusting, efficiently powered by solar energy, pushed asteroids outward, away from the sun, concentrating them between the orbits of Mars and Jupiter.13 [See Figures 157 and 158.]
Figure 157: Thrust and Drag Acted on Asteroids (Sun, asteroid, gas molecules, and orbit are not to scale.) The fountains of the great deep launched rocks and muddy water from Earth. The larger rocks, assisted by water vapor and other gases within the spheres of influence of these rocks, captured other rocks and ice particles. Those growing bodies that were primarily rocks became asteroids.
The Sun heats an asteroid’s near side, while the far side radiates its heat into cold outer space. Therefore, large temperature differences exist on opposite sides of each rocky, orbiting body. The slower the body spins, the darker the body,14 and the closer it is to the Sun, the greater the temperature difference. (For example, temperatures on the sunny side of our Moon reach a searing 260°F, while on the dark side, temperatures can drop to a frigid -280°F.) Also, gas molecules (small blue circles) between the Sun and asteroid, especially those coming from very near the Sun, are hotter and faster than those on the far side of an asteroid. Hot gas molecules hitting the hot side of an asteroid bounce off with much higher velocity and momentum than cold gas molecules bouncing off the cold side. Those impacts slowly expanded asteroid orbits until too little gas remained in the inner solar system to provide much thrust. The closer an asteroid was to the Sun, the greater the outward thrust. Gas molecules, densely concentrated near Earth’s orbit, created a drag on asteroids. My computer simulations have shown how gas, throughout the inner solar system for years after the flood, herded asteroids into a tight region near Earth’s orbital plane—an asteroid belt.15 Thrust primarily expanded the orbits. Drag circularized orbits and reduced their angles of inclination.
Figure 158: The Radiometer Effect. This well-known novelty, called a radiometer, demonstrates the unusual thrust that pushed asteroids into their present orbits. Sunlight warms the dark side of each vane more than the light side. The partial vacuum inside the bulb approaches that found in outer space, so gas molecules travel relatively long distances before striking other molecules. Gas molecules bounce off the hotter, black side with greater velocity than off the colder, white side. This turns the vanes away from the dark side.
The black side also radiates heat faster when it is warmer than its surroundings. This can be demonstrated by briefly placing the radiometer in a freezer. There the black side cools faster, making the white side warmer than the black, so the vanes turn away from the white side. In summary, the black side gains heat faster when in a hot environment and loses heat faster when in a cold environment. Higher gas pressure always pushes on the warmer side.
Question 4: Could the radiometer effect push asteroids 1–2 astronomical units (AU) farther from the Sun?Each asteroid began as a swarm of particles (rocks, ice, and gas molecules) orbiting within a large sphere of influence. Because a swarm’s volume was quite large, its spin was much slower than it would be as it shrank to become an asteroid—perhaps orders of magnitude slower. The slow spin produced extreme temperature differences between the hot and cold sides. The cold side would have been so cold that gas molecules striking it would tend to stick, thereby adding “fuel” to the developing asteroid. Because the swarm’s volume was large, the radiometer pressure acted over a large area and produced a large thrust. The swarm’s large thrust and low density caused the swarm to rapidly accelerate—much like a feather placed in a gentle breeze. Also, the Sun’s gravity 93,000,000 miles from the Sun (the Earth-Sun distance) is 1,600 times weaker than Earth’s gravity here on Earth.17 So, pushing a swarm of rocks and debris farther from the Sun was surprisingly easy, because there is almost no resistance in outer space.
Question 5: Why are 4% of meteorites almost entirely iron and nickel? Also, why do meteorites rarely contain quartz, which constitutes about 27% of granite’s volume?Pillars were formed in the subterranean chamber when the thicker portions of the crust were squeezed downward onto the chamber floor. Twice daily, during the centuries before the flood, these pillars were stretched and compressed by tides in the subterranean water. This gigantic heating process steadily raised pillar temperatures. [See “What Triggered the Flood?” here.] As explained in Figure 159, temperatures in what are now iron-nickel meteorites once exceeded 1,300°F, enough to dissolve quartz and allow iron and nickel to settle downward and become concentrated in the pillar tips.18 (A similar gravitational settling process concentrated iron and nickel in the Earth’s core after the flood began. See “Melting the Inner Earth” here.)
Evolutionists have great difficulty explaining iron-nickel meteorites. First, everyone recognizes that a powerful heating mechanism must first melt at least some of the parent body from which the iron-nickel meteorites came, so iron and nickel can sink and be concentrated. How this could have occurred in the weak gravity of extremely cold asteroids has defied explanation.19 Second, the concentrated iron and nickel, which evolutionists visualize in the core of a large asteroid, must then be excavated and blasted into space. Available evidence shows that this has not happened.20
Figure 156: Asteroid Ida and Its Moon, Dactyl. Most iron-nickel meteorites display Widmanstätten patterns. That is, if an iron-nickel meteorite is cut and its face is polished and then etched with acid, the surface has the strange crisscross pattern shown above. This shows that temperatures throughout those meteorites exceeded 1,300°F.16 Why were so many meteoroids, drifting in cold space, at one time so uniformly hot? An impact would not produce such uniformity, nor would a blowtorch. The heating a meteor experiences in passing through the atmosphere is barely felt more than a fraction of an inch beneath the surface. If radioactive decay generated the heat, certain daughter products should be present; they are not. Question 5 explains how these high temperatures were probably reached.
Question 6: Aren’t meteoroids chips from asteroids?This commonly-taught idea is based on an error in logic. Asteroids and meteoroids have some similarities, but that does not mean that one came from the other. Maybe a common event produced both asteroids and meteoroids.
Also, three major discoveries suggest that meteoroids came not from asteroids, but from Earth.
1. In the mid-1970s, the Pioneer 10 and 11 spacecraft traveled out through the asteroid belt. NASA expected that the particle detection experiments on board would find 10 times more meteoroids in the belt than are present near Earth’s orbit.21 Surprisingly, the number of meteoroids diminished as the asteroid belt was approached.22 This showed that meteoroids are not coming from asteroids but from nearer the Earth’s orbit.
2. A faint glow of light, called the zodiacal light, extends from the orbit of Venus out to the asteroid belt. The light is reflected sunlight bouncing off dust-size particles. This lens-shaped swarm of particles orbits the Sun, near Earth’s orbital plane. (On dark, moonless nights, zodiacal light can be seen in the spring in the western sky after sunset and in the fall in the eastern sky before sunrise.) Debris chipped off asteroids would have a wide range of sizes and would not be as uniform and fine as the particles reflecting the zodiacal light. Debris expelled by the fountains of the great deep would place fine dust particles in the Earth's orbital plane.
3. Many meteorites have remanent magnetism, so they must have come from a larger magnetized body. Eros, the only asteroid on which a spacecraft has landed and taken magnetic measurements, has no net magnetic field. If this is true of other asteroids as well, meteorites probably did not come from asteroids.30 If asteroids are flying rock piles, as it now appears, any magnetic fields in the randomly oriented rocks would be largely self-canceling, so the asteroid would have no net magnetic field. Therefore, instead of coming from asteroids, meteorites likely came from a magnetized body such as a planet. Because Earth’s magnetic field is 2,000 times greater than that of all other rocky planets combined, meteorites probably came from Earth.
Remanent magnetism decays, so meteorites must have recently broken away from their parent magnetized body. Those who believe that meteorites were chipped off asteroids say this happened millions of years ago.
PREDICTION 34:Most rocks comprising asteroids will be found to be magnetized.
Two InterpretationsWith a transmission electron microscope, Japanese scientist Kazushige Tomeoka identified several major events in the life of one meteorite. Initially, this meteorite was part of a much larger parent body orbiting the Sun. The parent body had many thin cracks, through which mineral-rich water cycled. Extremely thin mineral layers were deposited on the walls of these cracks. These deposits, sometimes hundreds of layers thick, contained calcium, magnesium, carbonates, and other chemicals. Mild thermal metamorphism in this rock shows that temperatures increased before it experienced some final cracks and was blasted into space.31
Hydroplate Interpretation. Earth was the parent body of all meteorites, most of which came from pillars. [Pages 381–386 explain how, why, when, and where pillars formed.] Twice a day before the flood, tides in the subterranean water compressed and stretched these pillars. Compressive heating occurred and cracks developed. Just as water circulates through a submerged sponge that is squeezed and stretched, mineral-laden water circulated through cracks in pillars for years before they broke up. Pillar fragments, launched into space by the fountains of the great deep, became meteoroids. In summary, water did it.
Tomeoka’s (and Most Evolutionists’) Interpretation. Impacts on an asteroid cracked the rock that was to become this meteorite. Ice was deposited on the asteroid. Impacts melted the ice, allowing liquid water to circulate through the cracks and deposit hundreds of layers of magnesium, calcium, and carbonate bearing minerals. A final impact blasted rocks from this asteroid into space. In summary, impacts did it.
Figure 160: Shatter Cone. When a large, crater-forming meteorite strikes the Earth, a shock wave radiates outward from the impact point. The passing shock wave breaks the rock surrounding the crater into meteorite-size fragments having distinctive patterns called shatter cones. (Until shatter cones were associated with impact craters by Robert S. Dietz in 1969, impact craters were often difficult to identify.)
If large impacts on asteroids launched asteroid fragments toward Earth as meteorites, a few meteorites should have shatter cone patterns. None have ever been reported. Therefore, meteorites are probably not derived from asteroids. Likewise, impacts have not launched meteorites from Mars.
Question 7: Does other evidence support this hypothesis that asteroids and meteoroids came from Earth?Yes. Here are seventeen additional observations that either support the proposed explanation or are inconsistent with other current theories on the origin of asteroids and meteoroids:
1. The materials in meteorites and meteoroids are remarkably similar to those in the Earth’s crust.32 Some meteorites contain very dense elements, such as nickel and iron. Those heavy elements seem compatible only with the denser rocky planets: Mercury, Venus, and Earth—Earth being the densest.
A few asteroid densities have been calculated. They are generally low, ranging from 1.2 to 3.3 gm/cm3. The higher densities match those of the Earth’s crust. The lower densities imply the presence of empty space between loosely held rocks or something light such as water ice.33
PREDICTION 35:Rocks in asteroids are typical of the Earth’s crust. Expensive efforts to mine asteroids34 to recover strategic or precious metals will be a waste of money.
2. Meteorites contain different varieties (isotopes) of the chemical element molybdenum, each isotope having a slightly different atomic weight. If, as evolutionists teach, a swirling cloud of gas and dust mixed for millions of years and produced the Sun, its planets, and meteorites, then each meteorite should have about the same combination of these molybdenum isotopes. Because this is not the case,35 meteorites did not come from a swirling dust cloud or any source that mixed for millions of years.
3. Most meteorites36 and some asteroids37 contain metamorphosed minerals, showing that those bodies reached extremely high temperatures, despite a lifetime in the “deep freeze” of outer space. Radioactive decay within such relatively small bodies could not have produced the necessary heating, because too much heat would have escaped from their surfaces. Stranger still, liquid water altered some meteorites38 while they and their parent bodies were heated—sometimes heated multiple times.39
Impacts in space are often proposed to explain this mysterious heating throughout an asteroid or meteroite. However, an impact would raise the temperature only near the point of impact. Before gravel-size fragments from an impact could become uniformly hot, they would radiate their heat into outer space.40
For centuries before the flood, heat was steadily generated within pillars in the subterranean water chamber. As the flood began, the powerful jetting water launched rock fragments into space—fragments of hot, crushed pillars and fragments from the crumbling walls of the ruptured crust. Those rocks became meteoroids and asteroids.
4. Because asteroids came from Earth, they typically spin in the same direction as Earth (counterclockwise, as seen from the North). However, collisions have undoubtedly randomized the spins of many smaller asteroids in the last few thousand years.41
5. Some asteroids have captured one or more moons. [See Figure 156 at top of this page.] Sometimes the “moon” and asteroid are similar in size. Impacts would not create equal-size fragments that could capture each other.42 The only conceivable way for this to happen is if a potential moon enters an asteroid’s expanding sphere of influence while traveling about the same speed and direction as the asteroid. If even a thin gas surrounds the asteroid, the moon will be drawn closer to the asteroid, preventing the moon from being stripped away later. An “exploded planet” would disperse relatively little gas. The “failed planet explanation” meets none of the requirements. The hydroplate theory satisfies all the requirements.
Figure 161: Chondrules. The central chondrule above is 2.2 millimeters in diameter, the size of this circle: o. This picture was taken in reflected light. However, meteorites containing chondrules can be thinly sliced and polished, allowing light from below to pass through the thin slice and into the microscope. Such light becomes polarized as it passes through the minerals. The resulting colors identify minerals in and around the chondrules. [Meteorite from Hammada al Hamra Plateau, Libya.]
Chondrules (CON-drools) are strange, spherical, BB-size objects found in 86% of all meteorites. To understand the origin of meteorites we must also understand how chondrules formed.
Their spherical shape and texture show they were once molten, but to melt chondrules requires temperatures exceeding 3,000°F. How could chondrules get that hot without melting the surrounding rock, which usually has a lower melting temperature? Because chondrules contain volatile substances that would have bubbled out of melted rock, chondrules must have melted and cooled quite rapidly.23 By one estimate, melting occurred in about one-hundredth of a second.24
The standard explanation for chondrules is that small pieces of rock, moving in outer space billions of years ago, before the Sun and Earth formed, suddenly and mysteriously melted. These liquid droplets quickly cooled, solidified, and then were encased inside the rock that now surrounds them. Such vague conditions, hidden behind a veil of space and time, make it nearly impossible to test this explanation in a laboratory. Scientists recognize that this standard story does not explain the rapid melting and cooling of chondrules or how they were encased uniformly in rocks which are radiometrically older than the chondrules.25 As one scientist wrote, “The heat source of chondrule melting remains uncertain. We know from the petrological data that we are looking for a very rapid heating source, but what?”26
Frequently, minerals grade (gradually change) across the boundaries between chondrules and surrounding material.27 This suggests that chondrules melted while encased in rock. If so, the heating sources must have acted briefly and been localized near the center of what are now chondrules. But how could this have happened?
The most common mineral in chondrules is olivine.28 Deep rocks contain many BB-size pockets of olivine. Pillars within the subterranean water probably had similar pockets. As the subterranean water escaped from under the crust, pillars had to carry more of the crust’s weight. When olivine reaches a certain level of compression, it suddenly changes into another mineral, called spinel (spin-EL), and shrinks in volume by about 10%.29 (Material surrounding each pocket would not shrink.)
Tiny, collapsing pockets of olivine transforming into spinel would generate great heat, for two reasons. First, the transformation is exothermic; that is, it releases heat chemically. Second, it releases heat mechanically, by friction. Here’s why. At the atomic level, each pocket would collapse in many stages—much like falling dominos or the section-by-section crushing of a giant scaffolding holding up an overloaded roof. Within each pocket, as each microscopic crystal slid over adjacent crystals at these extreme pressures, melting would occur along sliding surfaces. The remaining solid structures in the olivine pocket would then carry the entire compressive load—quickly collapsing and melting other parts of the “scaffolding.”
The fountains of the great deep expelled pieces of crushed pillars into outer space where they rapidly cooled. Their tumbling action, especially in the weightlessness of space, would have prevented volatiles from bubbling out of the encased liquid pockets within each rock. In summary, chondrules are a by product of the mechanism that produced meteorites—a rapid process that started under the Earth’s crust as the flood began.
Also, tidal effects, as described on pages 425–428, limit the lifetime of the moons of asteroids to about 100,000 years.43 This fact and the problems in capturing a moon caused evolutionist astronomers to scoff at early reports that some asteroids have moons.
Figure 162: Peanut Asteroids. The fountains of the great deep expelled dirt, rocks, and considerable water from Earth. About half of that water quickly evaporated into the vacuum of space; the remainder froze. Each evaporated gas molecule became an orbiting body in the solar system. Asteroids then formed as explained on pages 298–302. Many are shaped like peanuts.
Gas molecules captured by asteroids or released by icy asteroids became their atmospheres. Asteroids with thick atmospheres sometimes captured smaller asteroids as moons. If an atmosphere remained long enough, the moon would lose altitude and gently merge with the low-gravity asteroid, forming a peanut-shaped asteroid. (We see merging when a satellite or spacecraft reenters Earth’s atmosphere, slowly loses altitude, and eventually falls to Earth.) Without an atmosphere, merging becomes almost impossible.
Japan’s Hayabusa spacecraft orbited asteroid Itokawa (shown above) for two months in 2005. Scientists studying Itokawa concluded that it consists of two smaller asteroids that merged. Donald Yeomans, a mission scientist and member of NASA’s Jet Propulsion Laboratory, admitted, “It’s a major mystery how two objects each the size of skyscrapers could collide without blowing each other to smithereens. This is especially puzzling in a region of the solar system where gravitational forces would normally involve collision speeds of 2 km/sec.”45 The mystery is easily solved when one understands the role that water played in the origin of comets and asteroids.
Notice, a myriad of rounded boulders, some 150 feet in diameter, litter Itokawa’s surface. High velocity water produces rounded boulders; an exploded planet or impacts on asteroids would produce angular rocks.
6. The smaller moons of the giant planets (Jupiter, Saturn, Uranus, and Neptune) are captured asteroids. Most astronomers probably accept this conclusion, but have no idea how these captures could occur.44
As explained earlier in this chapter, for decades to centuries after the flood the radiometer effect, powered by the Sun’s energy, spiraled asteroids outward from Earth’s orbit. Water vapor, around asteroids and in interplanetary space, temporarily thickened asteroid and planet atmospheres. This facilitated aerobraking which allowed massive planets to capture asteroids.
Recent discoveries indicate that Saturn’s 313-mile-wide moon, Enceladus (en-SELL-uh-duhs), is a captured asteroid. Geysers at Enceladus’ south pole are expelling water vapor and ice crystals which escape Enceladus and supply Saturn’s E ring.46 That water contains salts resembling Earth’s ocean waters.47 Because asteroids are icy and weak, they would experience strong tides if captured by a giant planet. Strong tides would have recently48 generated considerable internal heat, slowed the moon’s spin, melted ice, and boiled deep reservoirs of water. Enceladus’ spin has almost stopped, its internal water is being launched (some so hot that it becomes a plasma),49 and its surface near the geysers has buckled, probably due to the loss of internal water. Because the material for asteroids and their organic matter came recently from Earth, water is still jetting from cold Enceladus’ surprisingly warm south pole, and “dark green organic material”50 is on its surface.
7. A few asteroids suddenly develop comet tails, so they are considered both asteroid and comet. The hydroplate theory says that asteroids are weakly joined piles of rocks and ice. If such a pile cracked slightly, perhaps due to an impact by space debris, then internal ice, suddenly exposed to the vacuum of space, would violently vent water vapor and produce a comet tail. The hydroplate theory explains why comets are so similar to asteroids.
8. A few comets have nearly circular orbits within the asteroid belt. Their tails lengthen as they approach perihelion and recede as they approach aphelion. If comets formed beyond the planet Neptune, it is highly improbable that they could end up in nearly circular orbits in the asteroid belt.51 So, these comets almost certainly did not form in the outer solar system. Also, comet ice that near the Sun would evaporate relatively quickly. Only the hydroplate theory explains how comets (icy rock piles) recently entered the asteroid belt.
9. If asteroids passing near Earth came from the asteroid belt, too many of them have diameters less than 50 meters,52 and too many have circular orbits.53 However, we would expect this if the rocks that formed asteroids were launched from Earth.
10. Computer simulations, both forward and backward in time, show that asteroids traveling near Earth have a maximum expected lifetime of only about a million years. They “quickly” collide with the Sun.54 This raises doubts that all asteroids began 4,600,000,000 years ago as evolutionists claim—living 4,600 times longer than the expected lifetime of near-Earth asteroids.
11. Earth has one big moon and several small moons—up to 650 feet in diameter.55 The easiest explanation for the small moons is that they were launched from Earth with barely enough velocity to escape Earth’s gravity. (To understand why the largest of these small moons is about 650 feet in diameter, see Endnote 8.)
12. Asteroids 3753 Cruithne and 2000 AA29 are traveling companions of Earth.56 They delicately oscillate, in a horseshoe pattern, around two points that lie 60° (as viewed from the Sun) forward and 60° behind the Earth but on Earth’s nearly circular orbit. These points, predicted by Lagrange in 1764 and called Lagrange points, are stable places where an object would not move relative to the Earth and Sun if it could once occupy either point going at zero velocity relative to the Earth and Sun. But how could a slowly moving object ever reach, or get near, either point? Most likely, it barely escaped from Earth.
Also, Asteroid 3753 could not have been in its present orbit for long, because it is so easy for a passing gravitational body to perturb it out of its stable niche. Time permitting, Venus will pass near this asteroid 8,000 years from now and may dislodge it.57
13. Furthermore, Jupiter has two Lagrange points on its nearly circular orbit. The first, called L4, lies 60° (as seen from the Sun) in the direction of Jupiter’s motion. The second, called L5, lies 60° behind Jupiter.
Visualize planets and asteroids as large and small marbles rolling in orbitlike paths around the Sun on a large frictionless table. At each Lagrange point is a bowl-shaped depression that moves along with each planet. Because there is no friction, small marbles (asteroids) that roll down into a bowl normally pick up enough speed to roll back out. However, if a chance gravitational encounter slowed one marble right after it entered a bowl, it might not exit the bowl. Marbles trapped in a bowl would normally stay 60° ahead of or behind their planet, gently rolling around near the bottom of their moving bowl.
One might think an asteroid is just as likely to get trapped in Jupiter’s leading bowl as its trailing bowl—a 50–50 chance, as with the flip of a coin. Surprisingly, 1068 asteroids are in Jupiter’s leading (L4) bowl, but only 681 are in the trailing bowl.69 This shouldn’t happen in a trillion trials if an asteroid is just as likely to get trapped at L4 as L5. What concentrated so many asteroids near the L4 Lagrange point?
According to the hydroplate theory, asteroids formed near Earth’s orbit. Then, the radiometer effect spiraled them outward, toward the orbits of Mars and Jupiter. Some spiraled through Jupiter’s circular orbit and passed near both L4 and L5. Jupiter’s huge gravity would have slowed those asteroids that were moving away from Jupiter but toward L4. That braking action would have helped some asteroids settle into the L4 bowl. Conversely, asteroids that entered L5 were accelerated toward Jupiter, so they would quickly be pulled out of L5 by Jupiter’s gravity. The surprising excess of asteroids near Jupiter’s L4 is what we would expect based on the hydroplate theory.
Figure 163: Asteroid Belt and Jupiter’s L4 and L5. The size of the Sun, planets, and especially asteroids are magnified, but their relative positions are accurate. About 90% of the 30,000 precisely known asteroids lie between the orbits of Mars and Jupiter, a doughnut-shaped region called the asteroid belt. A few small asteroids cross Earth’s orbit.
Jupiter’s Lagrange points, L4 and L5, lie 60° ahead and 60° behind Jupiter, respectively. They move about the Sun at the same velocity as Jupiter, as if they were fixed at the corners of the two equilateral triangles shown. Items 12 and 13 explain why so many asteroids have settled near L4 and L5, and why significantly more oscillate around L4 than L5.
14. Without the hydroplate theory, one has difficulty imagining situations in which an asteroid would (a) settle into one of Jupiter’s Lagrange points, (b) capture a moon, especially a moon with about the same mass as the asteroid, or (c) have a circular orbit, along with its moon, about their common center of mass. If all three happened to an asteroid, astronomers would be shocked; no astronomer would have predicted that it could happen to a comet. Nevertheless, an “asteroid” discovered earlier, named 617 Patroclus, satisfies (a)–(c). Patroclus and its moon, Menoetius, have such low densities that they would float in water; therefore, both are probably comets70—dirty, fluffy snowballs. Paragraphs 5, 7, 8, and 13 (above) explain why these observations make perfect sense with the hydroplate theory.
15. As explained in “Shallow Meteorites,” meteorites are almost always found surprisingly near Earth’s surface. The one known exception is in southern Sweden, where 40 meteorites and thousands of grain-size fragments of one particular type of meteorite have been found at different depths in a few limestone quarries. The standard explanation is that all these meteorites somehow struck this same small area over a 1–2-million-year period about 480 million years ago.71
A more likely explanation is that some meteorites, not launched with enough velocity to escape Earth during the flood, fell back to Earth. One or more meteorites fragmented on reentering Earth’s atmosphere. The pieces landed in mushy, recently-deposited limestone layers in southern Sweden.
16. Light spectra (detailed color patterns, much like a long bar code) from certain asteroids in the outer asteroid belt imply the presence of organic compounds, especially kerogen, a coal-tar residue.72 No doubt the kerogen came from plant life. Life as we know it could not survive in such a cold region of space, but common organic matter launched from Earth could have been preserved.
17. Many asteroids are reddish and have light characteristics showing the presence of iron.73 On Earth, reddish rocks almost always imply iron oxidized (rusted) by oxygen gas. Today, oxygen is rare in outer space. If iron on asteroids is oxidized, what was the source of the oxygen? Answer: Water molecules, surrounding and impacting asteroids, dissociated (broke apart), releasing oxygen. That oxygen then combined chemically with iron on the asteroid’s surface, giving the reddish color.
Mars, often called the red planet, derives its red color from oxidized iron. Again, oxygen contained in water vapor launched from Earth during the flood, probably accounts for Mars’ red color.
Mars’ topsoil is richer in iron and magnesium than Martian rocks beneath the surface. The dusty surface of Mars also contains carbonates, such as limestone.74 Because meteorites and Earth’s subterranean water contained considerable iron, magnesium, and carbonates, it appears that Mars was heavily bombarded by meteorites and water launched from Earth’s subterranean chamber. [See “The Origin of Limestone” on pages 224–229.]
Those who believe that meteorites came from asteroids have wondered why meteorites do not have the red color of most asteroids.75 The answer is twofold: (a) as explained on page 301, meteorites did not come from asteroids but both came from Earth, and (b) asteroids contain oxidized iron, as explained above, but meteorites are too small to attract an atmosphere gravitationally.
Figure 164: Salt of the Earth. On 22 March 1998, this 2 3/4 pound meteorite landed 40 feet from boys playing basketball in Monahans, Texas. While the rock was still warm, police were called. Hours later, NASA scientists cracked the meteorite open in a clean-room laboratory, eliminating any possibility of contamination. Inside were salt (NaCl) crystals 0.1 inch (3 mm) in diameter and liquid water!58 Some of these salt crystals are shown in the blue circle, highly magnified and in true color. Bubble (B) is inside a liquid, which itself is inside a salt crystal. Eleven quivering bubbles were found in about 40 fluid pockets. Shown in the green circle is another bubble (V) inside a liquid (L). The length of the horizontal black bar represents 0.005 mm, about 1/25 the diameter of a human hair.
NASA scientists who investigated this meteorite believe that it came from an asteroid, but that is highly unlikely. Asteroids, having little gravity and being in the vacuum of space, cannot sustain liquid water, which is required to form salt crystals. (Earth is the only planet, indeed the only body in the solar system, that can sustain liquid water on its surface.) Nor could surface water (gas, liquid, or solid) on asteroids withstand high-velocity impacts. Even more perplexing for the evolutionist: What is the salt’s origin? Also, what accounts for the meteorite’s other contents: potassium, magnesium, iron, and calcium—elements abundant on Earth, but as far as we know, not beyond Earth?59 Dust-sized meteoroids often come from comets. Most larger meteoroids are rock fragments that never merged into a comet or asteroid.
Much evidence supports Earth as the origin of meteorites.
- Minerals and isotopes in meteorites are remarkably similar to those on Earth.32
- Some meteorites contain sugars,60 salt crystals containing liquid water,61 and possible cellulose.62
- Other meteorites contain limestone,63 which, on Earth, forms only in liquid water.
- Three meteorites contain excess amounts of left-handed amino acids64—a sign of once-living matter.
- A few meteorites show that “salt-rich fluids analogous to terrestrial brines” flowed through their veins.65
- Some meteorites have about twice the heavy hydrogen concentration as Earth’s water today.66 As explained in the preceding chapter and in “Energy in the Subterranean Water” here, this heavy hydrogen came from the subterranean chambers.
- About 86% of all meteorites contain chondrules, which are best explained by the hydroplate theory.
- Seventy-eight types of living bacteria have been found in two meteorites after extreme precautions were taken to avoid contamination.67 Bacteria need liquid water to live, grow, and reproduce. Obviously, liquid water does not exist inside meteoroids whose temperatures in outer space are near absolute zero (-460°F). Therefore, the bacteria must have been living in the presence of liquid water before being launched into space. Once in space, they quickly froze and became dormant. Had bacteria originated in outer space, what would they have eaten?
Water on MarsWater recently and briefly flowed at various locations on Mars.76 Photographic comparisons show that some water flowed within the last 2–5 years!77 Water is now stored as ice at Mars’ poles78 and in surface soil. Mars’ stream beds usually originate on crater walls rather than in ever smaller tributaries as on Earth.79 Rain formed other channels.80 Martian drainage channels and layered strata are found at almost isolated 200 locations.81 Most gullies are on crater slopes at high latitudes82—extremely cold slopes that receive little sunlight. One set of erosion gullies is on the central peak of an impact crater!83
Figure 165: Erosion Channels on Mars. These channels frequently originate in scooped-out regions, called amphitheaters, high on a crater wall. On Earth, where water falls as rain, erosion channels begin with narrow tributaries that merge with larger tributaries and finally, rivers. Could impacts of comets or icy asteroids have formed these craters, gouged out amphitheaters, and melted the ice—each within seconds? Mars, which is much colder than Antarctica in the winter, would need a heating source, such as impacts, to produce liquid water.
Today, Mars is cold, averaging -80°F (112 Fahrenheit degrees below freezing). Water on Mars should be ice, not liquid water. Mars’ low atmospheric pressures would hasten freezing even more.84
Water probably came from above. Soon after Earth’s global flood, the radiometer effect caused asteroids to spiral out to the asteroid belt, just beyond Mars. This gave asteroids frequent opportunities to collide with Mars. When crater-forming impacts occurred, large amounts of debris were thrown into Mars’ atmosphere. Mars’ thin atmosphere and low gravity allowed the debris to settle back to the surface in vast layers of thin sheets—strata.
PREDICTION 36Most sediments taken from layered strata on Mars and returned to Earth will show that they were deposited through Mars’ atmosphere, not through water. (Under a microscope, water deposited grains have nicks and gouges, showing that they received many blows as they tumbled along stream bottoms. Sediments deposited through an atmosphere receive few nicks.)
Impact energy (and heat) from icy asteroids and comets bombarding Mars released liquid water, which often pooled inside craters or flowed downhill and eroded the planet’s surface.87 (Most liquid water soaked into the soil and froze.) Each impact was like the bursting of a large dam here on Earth. Brief periods of intense, hot rain and localized flash floods followed.88 These Martian hydrodynamic cycles quickly “ran out of steam,” because Mars receives relatively little heat from the Sun. While the consequences were large for Mars, the total water was small by Earth’s standards—about twice the water in Lake Michigan.
Today, when meteorites strike icy soil on Mars, some of that ice melts. When this happens on a crater wall, liquid water flows down the crater wall, leaving the telltale gullies that have shocked the scientific community.77
PREDICTION 37As has been discovered on the Moon and apparently on Mercury, frost will be found within asteroids and in permanently shadowed craters on Mars. This frost will be rich in heavy hydrogen.
Are Some Meteorites from Mars?Widely publicized claims have been made that at least 30 meteorites from Mars have been found. With international media coverage in 1996, a few scientists also proposed that one of these meteorites, named ALH84001, contained fossils of primitive life. Later study rejected that claim.
The wormy-looking shapes discovered in a meteorite [supposedly] from Mars turned out to be purely mineralogical and never were alive.89
The 30 meteorites are presumed to have come from the same place, because they contain similar ratios of three types of oxygen: oxygen weighing 16, 17, and 18 atomic mass units. (That presumption is not necessarily true, is it?) A chemical argument then indirectly links one of those meteorites to Mars, but the link is more tenuous than most realize.90 That single meteorite had tiny glass nodules containing dissolved gases. A few of these gases (basically the noble gases: argon, krypton, neon, and xenon) had the same relative abundances as those found in Mars’ atmosphere in 1976. (Actually, a later discovery shows that the mineralogy of these meteorites differs from that of almost all Martian rock.91) Besides, if two things are similar, it does not mean that one came from the other. Similarity in the relative abundances of the noble gases in Mars’ atmosphere and in one meteorite may be because those gases originated in Earth’s preflood subterranean chamber. Rocks and water from the subterranean chamber may have transported those gases to Mars.
Could those 30 meteorites have come from Mars? To escape the gravity of Mars requires a launch velocity of 3 miles per second. Additional velocity is then needed to transfer to an orbit intersecting Earth, 34–236 million miles away. Supposedly, one or more asteroids slammed into Mars and blasted off millions of meteoroids. Millions are needed, because less than one in a million92 would ever hit Earth, be large enough to survive reentry, be found, be turned over to scientists, and be analyzed in detail. Besides, if meteorites can come to Earth from Mars, many more should have come from the Moon—but haven’t.93
For an impact suddenly to accelerate, in a fraction of a second, any solid from rest to a velocity of 3 miles per second requires such extreme shock pressures that much of the material would melt, if not vaporize.94 All 30 meteorites should at least show shock effects. Some do not. Also, Mars should have at least six giant craters if such powerful blasts occurred, because six different launch dates are needed to explain the six age groupings the meteorites fall into (based on evolutionary dating methods). Such craters are hard to find, and large, recent impacts on Mars should have been rare.
Then there are energy questions. Almost all impact energy is lost as shock waves and ultimately as heat. Little energy remains to lift rocks off Mars. Even with enough energy, the fragments must be large enough to pass through Mars’ atmosphere. To see the difficulty, imagine throwing a ball high into the air. Then visualize how hard it would be to throw a handful of dust that high. Atmospheric drag, even in Mars’ thin atmosphere, absorbs too much of the smaller particles’ kinetic energy. Finally, for large particles to escape Mars, the expelling forces must be focused, as occurs in a gun barrel or rocket nozzle. For best results, this should be aimed straight up, to minimize the path length through the atmosphere.
A desire to believe in life on Mars produced a type of “Martian mythology” that continues today. In 1877, Italian astronomer Giovanni Schiaparelli reported seeing grooves on Mars. The Italian word for groove is “canali”; therefore, many of us grew up hearing about “canals” on Mars—a mistranslation. Because canals are man-made structures, people started thinking about “little green men” on Mars.
In 1894, Percival Lowell, a wealthy, amateur astronomer with a vivid imagination, built Lowell Observatory primarily to study Mars. Lowell published a map showing and naming Martian canals, and wrote several books: Mars (1895), Mars and Its Canals (1906), and Mars As the Abode of Life (1908). Even into the 1960s, textbooks displayed his map, described vegetative cycles on Mars, and explained how Martians may use canals to convey water from the polar ice caps to their parched cities. Few scientists publicly disagreed with the myth, even after 1949 when excellent pictures from the 200-inch telescope on Mount Palomar were available. Those of us in school before 1960 were directly influenced by such myths; almost everyone has been indirectly influenced.
Artists, science fiction writers, and Hollywood helped fuel this “Martian mania.” In 1898, H. G. Wells wrote The War of the Worlds telling of strange-looking Martians invading Earth. In 1938, Orson Welles, in a famous radio broadcast, panicked many Americans into thinking New Jersey was being invaded by Martians. In 1975, two Viking spacecraft were sent to Mars to look for life. Carl Sagan announced, shortly before the tests were completed, that he was certain life would be discovered—a reasonable conclusion, if life evolved. The prediction failed. In 1996, United States President Clinton read to a global television audience, “More than 4 billion years ago this piece of rock [ALH84001] was formed as a part of the original crust of Mars. After billions of years, it broke from the surface and began a 16-million-year journey through space that would end here on Earth.” “... broke from the surface ...”? The myth is still alive.
Final ThoughtsAs with the 24 other major features listed on page 106 [of the book, In the Beginning], we have examined the origin of asteroids and meteoroids from two directions: “cause-to-effect” and “effect-to-cause.”
Cause-to-Effect. We saw that given the assumption listed on page 115 [of the book, In the Beginning], consequences naturally followed: subterranean water became supercritical, the fountains of the great deep erupted; large rocks, muddy water, and water vapor were launched into space; gas and gravity assembled asteroids; and gas pressure powered by the Sun’s energy (the radiometer effect) herded asteroids into the asteroid belt. Isolated rocks still moving in the solar system are meteoroids.
Effect-to-Cause. We considered seventeen effects (pages 302–306)[of the book, In the Beginning], each incompatible with present theories on the origin of asteroids and meteoroids. Each effect was evidence that many rocks and large volumes of water vapor were launched from Earth.
Portions of Part III will examine this global flood from a third direction: historical records from claimed eyewitnesses. All three perspectives reinforce each other, illuminating in different ways this catastrophic event.
To access the footnotes for this article, click here. | http://4thdayalliance.com/articles/solar-system/origin-of-asteroids/ | 13 |
71 | 2.1 The Strength of Gravity and Electric Forces
Gravity is a relatively very weak force. The electric Coulomb force between a proton and an electron is of the order of 1039 (that’s 1 with 39 zeros after it) times stronger than the gravitational force between them.
We can get a hint of the relative strength of electromagnetic forces when we use a small magnet to pick up an iron object, say, a ball bearing. Even though the whole of Earth’s gravitation attraction is acting upon the ball bearing, the magnet overcomes this easily when close enough to the ball bearing. In space, gravity only becomes significant in those places where the electromagnetic forces are shielded or neutralized.
For spherical masses and charges, both the gravity force and the electric Coulomb force vary inversely with the square of the distance and so decrease rapidly with distance. For other geometries/configurations, the forces decrease more slowly with distance. For example, the force between two relatively long and thin electric currents moving parallel to each other varies inversely with the first power of the distance between them.
Electric currents can transport energy over huge distances before using that energy to create some detectable result, just like we use energy from a distant power station to boil a kettle in our kitchen. This means that, over longer distances, electromagnetic forces and electric currents together can be much more effective than either the puny force of gravity or even the stronger electrostatic Coulomb force.
Remember that, just in order to explain the behavior of the matter we can detect, the Gravity Model needs to imagine twenty-four times more matter than we can see, in special locations, and of a special invisible type. It seems much more reasonable to investigate whether the known physics of electromagnetic forces and electric currents can bring about the observed effects instead of having to invent what may not exist.
2.2 The “Vacuum” of Space
Until about 100 years ago, space was thought to be empty. The words “vacuum” and “emptiness” were interchangeable. But probes have found that space contains atoms, dust, ions, and electrons. Although the density of matter in space is very low, it is not zero. Therefore, space is not a vacuum in the conventional sense of there being “nothing there at all”. For example, the Solar “wind” is known to be a flow of charged particles coming from the Sun and sweeping round the Earth, ultimately causing visible effects like the Northern (and Southern) Lights.
The dust particles in space are thought to be 2 to 200 nanometers in size, and many of them are also electrically charged, along with the ions and electrons. This mixture of neutral and charged matter is called plasma, and it is suffused with electromagnetic fields. We will discuss plasma and its unique interactions with electromagnetic fields in more detail in Chapter 3. The “empty” spaces between planets or stars or galaxies are very different from what astronomers assumed in the earlier part of the 20th century.
(Note about terminology in links: astronomers often refer to matter in the plasma state as “gas,” “winds,” “hot, ionized gas,” “clouds,” etc. This fails to distinguish between the two differently-behaving states of matter in space, the first of which is electrically-charged plasma and the other of which may be neutral gas which is just widely-dispersed, non-ionized molecules or atoms.)
The existence of charged particles and electromagnetic fields in space is accepted in both the Gravity Model and the Electric Model. But the emphasis placed on them and their behavior is one distinctive difference between the models. We will therefore discuss magnetic fields next.
2.3 Introduction to Magnetic Fields
What do we mean by the terms “magnetic field” and “magnetic field lines”? In order to understand the concept of a field, let’s start with a more familiar example: gravity.
We know that gravity is a force of attraction between bodies or particles having mass. We say that the Earth’s gravity is all around us here on the surface of the Earth and that the Earth’s gravity extends out into space. We can express the same idea more economically by saying that the Earth has a gravitational field which extends into space in all directions. In other words, a gravitational field is a region where a gravitational force of attraction will be exerted between bodies with mass.
Similarly, a magnetic field is a region in which a magnetic force would act on a magnetized or charged body. (We will look at the origin of magnetic fields later). The effect of the magnetic force is most obvious on ferromagnetic materials. For example, iron filings placed on a surface in a magnetic field align themselves in the direction of the field like compass needles.
Because the iron filings tend to align themselves south pole to north pole, the pattern they make could be drawn as a series of concentric lines, which would indicate the direction and, indirectly, strength of the field at any point.
Therefore magnetic field lines are one convenient way to represent the direction of the field, and serve as guiding centers for trajectories of charged particles moving in the field (ref. Fundamentals of Plasma Physics, Cambridge University Press, 2006, Paul Bellan, Ph.D.).
It is important to remember that field lines do not exist as physical objects. Each iron filing in a magnetic field is acting like a compass: you could move it over a bit and it would still point magnetic north-south from its new position. Similarly, a plumb bob (a string with a weight at one end) will indicate the local direction of the gravitational field. Lines drawn longitudinally through a series of plumb bobs would make a set of gravitational field lines. Such lines do not really exist; they are just a convenient, imaginary means of visualizing or depicting the direction of force applied by the field. See Appendix I for more discussion of this subject, or here, at Fizzics Fizzle.
A field line does not necessarily indicate the direction of the force exerted by whatever is causing the field. Field lines may be drawn to indicate direction or polarity of a force, or may be drawn as contours of equal intensities of a force, in the same way as contour lines on a map connect points of equal elevation above, say, sea level. Often, around 3-dimensional bodies with magnetic fields, imaginary surfaces are used to represent the area of equal force, instead of lines.
By consensus, the definition of the direction of a magnetic field at some point is from the north to the south pole.
In a gravitational field, one could choose to draw contour lines of equal gravitational force instead of the lines of the direction of the force. These lines of equal gravitational force would vary with height (that is, with distance from the center of the body), rather like contour lines on a map. To find the direction of the force using these elevation contour lines, one would have to work out which way a body would move. Placed on the side of a hill, a stone rolls downhill, across the contours. In other words the gravitational force is perpendicular to the field lines of equal gravitational force.
Magnetic fields are more complicated than gravity in that they can either attract or repel. Two permanent bar magnets with their opposite ends (opposite “poles”, or N-S) facing each other will attract each other along the direction indicated by the field lines of the combined field from them both (see image above). Magnets with the same polarity (N-N or S-S) repel one other along the same direction.
Magnetic fields also exert forces on charged particles that are in motion. Because the force that the charged particle experiences is at right angles to both the magnetic field line and the particle’s direction, a charged particle moving across a magnetic field is made to change direction (i.e. to accelerate) by the action of the field. Its speed remains unchanged to conserve kinetic energy. The following image shows what happens to an electron beam in a vacuum tube before and after a magnetic field is applied, in a lab demonstration.
The magnetic force on a charged particle in motion is analogous to the gyroscopic force. A charged particle moving directly along or “with” a magnetic field line won’t experience a force trying to change its direction, just as pushing on a spinning gyroscope directly along its axis of rotation will not cause it to turn or “precess”.
Even though the force on different charged particles varies, the concept of visualizing the direction of the magnetic field as a set of imaginary field lines is useful because the direction of the force on any one material, such as a moving charged particle, can be worked out from the field direction.
2.4 The Origin of Magnetic Fields
There is only one way that magnetic fields can be generated: by moving electric charges. In permanent magnets, the fields are generated by electrons spinning around the nuclei of the atoms. A strong magnet is created when all the electrons orbiting the nuclei have spins that are aligned, creating a powerful combined force. If the magnet is heated to its Curie temperature, the thermal motion of the atoms breaks down the orderly spin alignments, greatly reducing the net magnetic field. In a metal wire carrying a current, the magnetic field is generated by electrons moving down the length of the wire. A more detailed introduction to the complex subject of exchange coupling and ferromagnetism can be found here.
Either way, any time electric charges move, they generate magnetic fields. Without moving electric charges, magnetic fields cannot exist. Ampère’s Law states that a moving charge generates a magnetic field with circular lines of force, on a plane that is perpendicular to the movement of the charge.
Since electric currents made up of moving electric charges can be invisible and difficult to detect at a distance, detecting a magnetic field at a location in space (by well-known methods in astronomy, see below) is a sure sign that it is accompanied by an electric current.
If a current flows in a conductor, such as a long straight wire or a plasma filament, then each charged particle in the current will have a small magnetic field around it. When all the individual small magnetic fields are added together, the result is a continuous magnetic field around the whole length of the conductor. The regions in space around the wire where the field strength is equal (called “equipotential surfaces”) are cylinders concentric with the wire.
Time-varying electric and magnetic fields are considered later. (See Chapter IV and Appendix III)
The question of the origin of magnetic fields in space is one of the key differences between the Gravity Model and the Electric Model.
The Gravity Model allows for the existence of magnetic fields in space because they are routinely observed, but they are said to be caused by dynamos inside stars. For most researchers today, neither electric fields nor electric currents in space play any significant part in generating magnetic fields.
In contrast, the Electric Model, as we shall see in more detail later, argues that magnetic fields must be generated by the movement of charged particles in space in the same way that magnetic fields are generated by moving charged particles here on Earth. Of course, the Electric Model accepts that stars and planets have magnetic fields, too, evidenced by magnetospheres and other observations. The new insight has been to explain a different origin for these magnetic fields in space if they are not created by dynamos in stars.
2.5 Detecting Magnetic Fields in Space
Since the start of the space age, spacecraft have been able to measure magnetic fields in the solar system using instruments on board the spacecraft. We can “see” magnetic fields beyond the range of spacecraft because of the effect that the fields have on light and other radiation passing through them. We can even estimate the strength of the magnetic fields by measuring the amount of that effect.
Optical image Magnetic field intensity, direction
We have known about the Earth’s magnetic field for centuries. We can now detect such fields in space, so the concept of magnetic fields in space is intuitively easy to understand, although astronomers have difficulty in explaining the origination of these magnetic fields.
Magnetic fields can be detected at many wavelengths by observing the amount of symmetrical spectrographic emission line or absorption line splitting that the magnetic field induces. This is known as the Zeeman effect, after Dutch physicist and 1902 Nobel laureate, Pieter Zeeman, (1865—1943). Note in the right image above how closely the field direction aligns with the galactic arms visible in the optical image, left.
Another indicator of the presence of magnetic fields is the polarization of synchrotron emission radiated by electrons in magnetic fields, useful at galactic scales. See Beck’s article on Galactic Magnetic Fields, in Scholarpedia, plus Beck and Sherwood’s Atlas of Magnetic Fields in Nearby Galaxies. Measurement of the degree of polarization makes use of the Faraday effect. The Faraday rotation in turn leads to the derivation of the strength of the magnetic field through which the polarized light is passing.
The highly instructional paper by Phillip Kronberg et al, Measurement of the Electric Current in a Kpc-Scale Jet, provides a compelling insight into the direct link between the measured Faraday rotation in the powerful “knots” in a large galactic jet, the resultant magnetic field strength, and the electric current present in the jet.
Magnetic fields are included in both the Gravity Model and the Electric Model of the Universe. The essential difference is that the Electric Model recognizes that magnetic fields in space always accompany electric currents. We will take up electric fields and currents next.
2.6 Introduction to Electric Fields
An electric charge has polarity. That is, it is either positive or negative. By agreement, the elementary (smallest) unit of charge is equal to that of an electron (-e) or a proton (+e). Electric charge is quantized; it is always an integer multiple of e.
The fundamental unit of charge is the coulomb (C), where e = 1.60×10-19 coulomb. By taking the inverse of the latter tiny value, one coulomb is 6.25×1018 singly-charged particles. One ampere (A) of electric current is one coulomb per second. A 20A current thus would be 20 C of charge per second, or the passage of 1.25×1020 electrons per second past a fixed point.
Every charge has an electric field associated with it. An electric field is similar to a magnetic field in that it is caused by the fundamental force of electromagnetic interaction and its “range” or extent of influence is infinite, or indefinitely large. The electric field surrounding a single charged particle is spherical, like the gravitational acceleration field around a small point mass or a large spherical mass.
The strength of an electric field at a point is defined as the force in newtons (N) that would be exerted on a positive test charge of 1 coulomb placed at that point. Like gravity, the force from one charge is inversely proportional to the square of the distance to the test (or any other) charge.
The point in defining a test charge as positive is to consistently define the direction of the force due to one charge acting upon another charge. Since like charges repel and opposites attract, just like magnetic poles, the imaginary electric field lines tend to point away from positive charges and toward negative charges. See a short YouTube video on the electric field here.
Here is a user-controlled demonstration of 2 charges and their associated lines of force in this Mathematica application.
You may need to download Mathematica Player (just once, and it’s free) from the linked web site to play with the demo. Click on “Download Live Demo”after you install Mathematica Player. You can adjust strength and polarity of charge (+ or -) with the sliders, and drag the charged particles around the screen. Give the field lines time to smooth out between changes.
Electromagnetic forces are commonly stronger than gravitational forces on plasma in space. Electromagnetism can be shielded, while gravity can not, so far as is known. The common argument in the standard model is that most of the electrons in one region or body are paired with protons in the nuclei of atoms and molecules, so the net forces of the positive charges and negative charges cancel out so perfectly that “for large bodies gravity can dominate” (link: Wikipedia, Fundamental Interactions, look under the Electromagnetism sub-heading).
What is overlooked above is that, with the occasional exception of relatively cool, stable and near-neutral planetary environments like those found here on Earth, most other matter in the Universe consists of plasma; i.e., charged particles and neutral particles moving in a complex symphony of charge separation and the electric and magnetic fields of their own making. Gravity, while always present, is not typically the dominant force.
Far from consisting of mostly neutralized charge and weak magnetic and electric fields and their associated weak currents, electric fields and currents in plasma can and often do become very large and powerful in space. The Electric Model holds that phenomena in space such as magnetospheres, Birkeland currents, stars, pulsars, galaxies, galactic and stellar jets, planetary nebulas, “black holes”, energetic particles such as gamma rays and X-rays and more, are fundamentally electric events in plasma physics. Even the rocky bodies – planets, asteroids, moons and comets, and the gas bodies in a solar system – exist in the heliospheres of their stars, and are not exempt from electromagnetic forces and their effects.
Each separate charged particle contributes to the total electric field. The net force at any point in a complex electromagnetic field can be calculated using vectors, if the charges are assumed stationary. If charged particles are moving (and they always are), however, they “create” – are accompanied by – magnetic fields, too, and this changes the magnetic configuration. Changes in a magnetic field in turn create electric fields and thereby affect currents themselves, so fields that start with moving particles represent very complex interactions, feedback loops and messy mathematics.
Charges in space may be distributed spatially in any configuration. If, instead of a point or a sphere, the charges are distributed in a linear fashion so that the length of a charged area is much longer than its width or diameter, it can be shown that the electric field surrounds the linear shape like cylinders of equal force potential, and that the field from this configuration decreases with distance from the configuration as the inverse of the distance (not the inverse square of the distance) from the centerline. This is important in studying the effects of electric and magnetic fields in filamentary currents such as lightning strokes, a plasma focus, or large Birkeland currents in space.
Remember that the direction of applied force on a positive charge starts from positive charge and terminates on negative charge, or failing a negative charge, extends indefinitely far. Even a small charge imbalance with, say, more positively-charged particles here and more negatively-charged particles a distance away leads to a region of force or electric field between the areas of separated dissimilar charges. The importance of this arrangement will become more clear in the discussion of double layers in plasma, further on.
Think of an electrical capacitor where there are two separated, oppositely charged plates or layers, similar to the two charged plates “B” in the diagram above. There will be an electric field between the layers. Any charged particle moving or placed between the layers will be accelerated towards the oppositely charged layer. Electrons (which are negatively charged) accelerate toward the positively charged layer, and positive ions and protons toward the negatively charged layer.
According to Newton’s Laws, force results in acceleration. Therefore electric fields will result in charged particles’ acquiring velocity. Oppositely charged particles will move in opposite directions. An electric current is, by definition, movement of charge past a point. Electric fields therefore cause electric currents by giving charged particles a velocity.
If an electric field is strong enough, then charged particles will be accelerated to very high velocities by the field. For a little further reading on electric fields see this.
2.7 Detecting Electric Fields and Currents in Space
Electric fields and currents are more difficult to detect without putting a measuring instrument directly into the field, but we have detected currents in the solar system using spacecraft. One of the first was the low-altitude polar orbit TRIAD satellite in the 1970s, which found currents interacting with the Earth’s upper atmosphere. In 1981 Hannes Alfvén described a heliospheric current model in his book, Cosmic Plasma.
Since then, a region of electric current called the heliospheric current sheet (HCS) has been found that separates the positive and negative regions of the Sun’s magnetic field. It is tilted approximately 15 degrees to the solar equator. During one half of a solar cycle, outward-pointing magnetic fields lie above the HCS and inward-pointing fields below it. This is reversed when the Sun’s magnetic field reverses its polarity halfway through the solar cycle. As the Sun rotates, the HCS rotates with it, “dragging” its undulations into what NASA terms “the standard Parker spiral”.
Spacecraft have measured changes over time in the current sheet at various locations since the 1980s. They have detected near-Earth and solar currents as well. The Gravity Model accepts that these currents exist in space but assumes they are a result of the magnetic field. We will return to this point later.
Electric fields outside the reach of spacecraft are not detectable in precisely the same way as magnetic fields. Line-splitting or broadening in electric fields occurs, but it is asymmetrical line splitting that indicates the presence of an electric field, in contrast to the symmetric line splitting in magnetic fields. Further, electric field line broadening is sensitive to the mass of the elements emitting light (the lighter elements being readily broadened or split, and heavier elements less so affected), while Zeeman (magnetic field) broadening is indifferent to mass. Asymmetric bright-line splitting or broadening is called the Stark effect, after Johannes Stark (1874–1957).
Another way in which we can detect electric fields is by inference from the behavior of charged particles, especially those that are accelerated to high velocities, and the existence of electromagnetic radiation such as X-rays in space, which we have long known from Earth-bound experience are generated by strong electric fields.
Electric currents in low density plasmas in space operate like fluorescent lights or evacuated Crookes Tubes. In a weak current state, the plasma is dark and radiates little visible light (although cold, thin plasma can radiate a lot in the radio and far infrared wavelengths). As current increases, plasma enters a glow mode, radiating a modest amount of electromagnetic energy in the visible spectrum. This is visible in the image at the end of this chapter. When electrical current becomes very intense in a plasma, the plasma radiates in the arc mode. Other than scale, there is little significant difference between lightning and the radiating surface of a star’s photosphere.
This means, of course, that alternative explanations for these effects are also possible, at least in theory. The Gravity Model often assumes that the weak force of gravity multiplied by supernatural densities that are hypothesized to make up black holes or neutron stars creates these types of effect. Or maybe particles are accelerated to near-light-speed by supernovae explosions. The question is whether “multiplied gravity” or lab-testable electromagnetism is more consistent with observations that the Universe is composed of plasma.
The Electric Model argues that electrical effects are not just limited to those parts of the solar system that spacecraft have been able to reach. The Electric Model supposes that similar electrical effects also occur outside the solar system. After all, it would be odd if the solar system was the only place in the Universe where electrical effects do occur in space.
End of Chapter 2 | http://www.thunderbolts.info/wp/2011/10/17/essential-guide-to-the-eu-chapter-2/ | 13 |
158 | An Introduction to MATLAB: Basic Operations
MATLAB is a programming language that is very useful for numerical simulation and data analysis. The following tutorials are intended to give you an introduction to scientific computing in MATLAB.
Lots of MATLAB demos are available online at
You can work through these at your leisure, if you want. Everything you need for EOS 225 should be included in the following tutorials.
At its simplest, we can use MATLAB as a calculator. Type
What do you get?
ans = 5
What do you get?
ans = 21
Can also do more complicated operations, like taking exponents: for "3 squared" type
ans = 9
For "two to the fourth power" type
ans = 16
"Scientific notation" is expressed with "10^" replaced by "e" - that is, 10^7 is written 1e7 and 2.15x10^-3 is written 2.15e-3. For example:
ans = 0.0150
2e-3 * 1000
ans = 2
MATLAB has all of the basic arithmetic operations built in:
+ addition - subtraction * multiplication \ division ^ exponentiation
as well as many more complicated functions (e.g. trigonometric, exponential):
sin(x) sine of x (in radians) cos(x) cosine of x (in radians) exp(x) exponential of x log(x) base e logarithm of x (normally written ln)
The above are just a sample - MATLAB has lots of built-in functions.
When working with arithmetic operations, it's important to be clear about the order in which they are to be carried out. This can be specified by the use of brackets. For example, if you want to multiply 5 by 2 then add 3, we can type
ans = 13
and we get the correct value. If we want to multiply 5 by the sum of 2 and 3, we type
ans = 25
and this gives us the correct value. Carefully note the placement of the brackets. If you don't put brackets, Matlab has its own built in order of operations: multiplication/division first, then addition/subtraction. For example:
ans = 13
gives the same answer as (5*2)+3. As another example, if we want to divide 8 by 2 and then subtract 3, we type
ans = 1
and get the right answer. To divide 8 by the difference between 2 and 3, we type
ans = -8
and again get the right answer. If we type
ans = 1
we get the first answer - the order of operations was division first, then subtraction.
In general, it's good to use brackets - they invovle more typing, and may make a computation look more cumbersome, but they help reduce ambiguity regarding what you want the computation to do.
This is a good point to make a general comment about computing. Computers are actually quite stupid - they do what you tell them to, not what you want them to do. When you type any commands into a computer program like MATLAB, you need to be very careful that these two things match exactly.
You can always get help in MATLAB by typing "help". Type this alone and you'll get a big list of directories you can get more information about - which is not always too useful. It's more useful to type "help" with some other command that you'd like to know more about. E.g.:
SIN Sine of argument in radians. SIN(X) is the sine of the elements of X. See also ASIN, SIND. Reference page in Help browser doc sin
ATAN Inverse tangent, result in radians. ATAN(X) is the arctangent of the elements of X. See also ATAN2, TAN, ATAND. Reference page in Help browser doc atan
You can get a list of all the built-in functions by typing
Elementary math functions. Trigonometric. sin - Sine. sind - Sine of argument in degrees. sinh - Hyperbolic sine. asin - Inverse sine. asind - Inverse sine, result in degrees. asinh - Inverse hyperbolic sine. cos - Cosine. cosd - Cosine of argument in degrees. cosh - Hyperbolic cosine. acos - Inverse cosine. acosd - Inverse cosine, result in degrees. acosh - Inverse hyperbolic cosine. tan - Tangent. tand - Tangent of argument in degrees. tanh - Hyperbolic tangent. atan - Inverse tangent. atand - Inverse tangent, result in degrees. atan2 - Four quadrant inverse tangent. atanh - Inverse hyperbolic tangent. sec - Secant. secd - Secant of argument in degrees. sech - Hyperbolic secant. asec - Inverse secant. asecd - Inverse secant, result in degrees. asech - Inverse hyperbolic secant. csc - Cosecant. cscd - Cosecant of argument in degrees. csch - Hyperbolic cosecant. acsc - Inverse cosecant. acscd - Inverse cosecant, result in degrees. acsch - Inverse hyperbolic cosecant. cot - Cotangent. cotd - Cotangent of argument in degrees. coth - Hyperbolic cotangent. acot - Inverse cotangent. acotd - Inverse cotangent, result in degrees. acoth - Inverse hyperbolic cotangent. hypot - Square root of sum of squares. Exponential. exp - Exponential. expm1 - Compute exp(x)-1 accurately. log - Natural logarithm. log1p - Compute log(1+x) accurately. log10 - Common (base 10) logarithm. log2 - Base 2 logarithm and dissect floating point number. pow2 - Base 2 power and scale floating point number. realpow - Power that will error out on complex result. reallog - Natural logarithm of real number. realsqrt - Square root of number greater than or equal to zero. sqrt - Square root. nthroot - Real n-th root of real numbers. nextpow2 - Next higher power of 2. Complex. abs - Absolute value. angle - Phase angle. complex - Construct complex data from real and imaginary parts. conj - Complex conjugate. imag - Complex imaginary part. real - Complex real part. unwrap - Unwrap phase angle. isreal - True for real array. cplxpair - Sort numbers into complex conjugate pairs. Rounding and remainder. fix - Round towards zero. floor - Round towards minus infinity. ceil - Round towards plus infinity. round - Round towards nearest integer. mod - Modulus (signed remainder after division). rem - Remainder after division. sign - Signum.
MATLAB can be used like a calculator - but it's much more. It's also a programming language, with all of the basic components of any such language.
The first and most basic of these components is one that we use all the time in math - the variable. Like in math, variables are generally denoted symbolically by individual characters (like "a" or "x") or by strings of characters (like "var1" or "new_value").
In class we've distinguished between variables and parameters - but denoted both of these by characters. MATLAB doesn't make this distinction - any numerical quantity given a symbolic "name" is a variable".
How do we assign a value to a variable? Easy - just use the equality sign. For example
a = 3
a = 3
sets the value 3 to the variable a. As another example
b = 2
b = 2
sets the value 2 to the variable b. We can carry out mathematical operations with these variables: e.g.
ans = 5
ans = 6
ans = 9
ans = 20.0855
Although operation of setting a value to a variable looks like an algebraic equality like we use all the time in math, in fact it's something quite different. The statement
a = 3
should not be interpreted as "a is equal to 3". It should be interpreted as "take the value 3 and assign it to the variable a". This difference in interpretation has important consequences. In algebra, we can write a = 3 or 3 = a -- these are equivalent. The = symbol in MATLAB is not symmetric - the command
a = b
should be interpreted as "take the value of b and assign it to the variable a" - there's a single directionality. And so, for example, we can type
a = 3
a = 3
with no problem, if we type
we get an error message. The value 3 is fixed - we can't assign another number to it. It is what it is.
Another consequence of the way that the = operator works is that a statement like
a = a+1
makes perfect sense. In algebra, this would imply that 0 = 1, which is of course nonsense. In MATLAB, it means "take the value that a has, add one to it, then assign that value to a". This changes the value of a, but that's allowed. For example, type:
a = 3 a = a+1
a = 3 a = 4
First a is assigned the value 3, then (by adding one) it becomes 4.
There are some built in variables; one of the most useful is pi:
ans = 3.1416
We can also assign the output of a mathematical operation to a new variable: e.g.
b = a*exp(a)
b = 218.3926
b = 218.3926
If you want MATLAB to just assign the value of a calculation to a variable without telling you the answer right away, all you have to do is put a semicolon after the calculation:
b = a*exp(a);
Being able to use variables is very convenient, particularly when you're doing a multi-step calculation with the same quantity and want to be able to change the value. For example:
a = 1; b = 3*a; c = a*b^2; d = c*b-a;
d = 26
Now say I want to do the same calculation with a = 3; all I need to do is make one change
a = 3; b = 3*a; c = a*b^2; d = c*b-a;
How does this make things any easier? Well, it didn't really here - we still had to type out the equations for b, c, and d all over again. But we'll seee that in a stand-alone computer program it's very useful to be able to do this.
In fact, the sequence of operations above is an example of a computer program. Operations are carried out in a particular order, with the results of earlier computations being fed into later ones.
It is very important to understand this sequential structure of programming. In a program, things happen in a very particular order: the order you tell them to have. It's very important to make sure you get this order right. This is pretty straightforward in the above example, but can be much more complicated in more complicated programs.
Any time a variable is created, it's kept in memory until you purposefully get rid of it (or quit the program). This can be useful - you can always use the variable again later. It can also make things harder - for example, in a long program you may try using a variable name that you've already used for another variable earlier in the program, leading to confusion.
It can therefore be useful sometimes make MATLAB forget about a variable; for this the "clear" command is used. For example, define
b = 3;
Now if we ask what b is, we'll get back that it's 3
b = 3
Using the clear command to remove b from memory
now if we ask about b
we get the error message that it's not a variable in memory - we've succeeded in getting rid of it. To get rid of everything in memory, just type
An important idea in programming is that of an array (or matrix). This is just an ordered sequence of numbers (known as elements): e.g.
M = [1, 22, -0.4]
is a 3-element array in which the first element is 1, the second element is 22, and the third element is -0.4. These are ordered - in this particular array, these numbers always occur in this sequence - but this doesn't mean that there's any particular structure ordering in general. That is - in an array, numbers don't have to increase or decrease or anything like that. The elements can be in any order - but that order partly defines the array. Also note that the numbers can be integers or rational numbers, or positive or negative.
While the elements of the array can be any kind of number, their positions are identified by integers: there is a first, a second, a third, a fourth, etc. up until the end of the array. It's standard to indicate the position of the array using bracket notation: in the above example, the first element is
M(1) = 1
the second element is
M(2) = 22
and the third element is
M(3) = -0.4.
These integers counting off position in the array are known as "indices" (singular "index").
All programming languages use arrays, but MATLAB is designed to make them particularly easy to work with (the MAT is for "matrix"). To make the array above in MATLAB all you need to do is type
M = [1 22 -0.4]
M = 1.0000 22.0000 -0.4000
Then to look at individual elements of the array, just ask for them by index number:
ans = 1
ans = 22
ans = -0.4000
We can also ask for certain ranges of an array, using the "colon" operator. For an array M we can ask for element i through element j by typing
ans = 1 22
ans = 22.0000 -0.4000
If we want all elements of the array, we can type the colon on its own
ans = 1.0000 22.0000 -0.4000
We can also use this notation to make arrays with a particular structure. Typing
M = a:b:c
makes an array that starts with first element
and increases with increment b:
M(2) = a+b M(3) = a+2b M(4) = a+3b
The array stops at the largest value of N for which M(N) <= c.
M = 1:1:3
M = 1 2 3
The array starts with 1, increases by 1, and ends at 3
M = 1:.5:3
M = 1.0000 1.5000 2.0000 2.5000 3.0000
The array starts at 1, increases by 0.5, and ends at 3
M = 1:.6:3
M = 1.0000 1.6000 2.2000 2.8000
Here the array starts at 1, increases by 0.6, and ends at 2.8 - because making one more step in the array would make the last element bigger than 3.
M = 3:-.5:1
M = 3.0000 2.5000 2.0000 1.5000 1.0000
This kind of array can also be decreasing.
If the increment size b isn't specified, a default value of 1 is used:
M = 1:5
M = 1 2 3 4 5
That is, the array a:c is the same as the array a:1:c
It is important to note that while the elements of an array can be any kind of number, the indices must be positive integers (1 and bigger). Trying non-positive or fractional integers will result in an error message:
Each of the elements of an array is a variable on its own, which can be used in a mathematical operation. E.g.:
ans = 4
ans = 6
The array itself is also a kind of variable - an array variable. You need to be careful with arithmetic operations (addition, subtraction, multiplication, division, exponentiaion) when it comes to arrays - these things can be defined, but they have to be defined correctly. We'll look at this later.
In MATLAB, when most functions are fed an array as an argument they give back an array of the function acting on each element. That is, for the function f and the array M, g=f(M) is an array such that
g(i) = f(M(i)).
a = 0:4; b = exp(a)
b = 1.0000 2.7183 7.3891 20.0855 54.5982
Let's define two arrays of the same size
a = 1:5; b = exp(a);
and what we get is a plot of the array a versus the array b - in this case, a discrete version of the exponential function exp(x) over the range x=1 to x=5.
We can plot all sorts of things: the program
a = 0:.01:5; b = cos(2*pi*a); plot(a,b)
sets the variable a as a fine discretisation of the range from x=0 to x=5, defines b as the cosine of 2 pi x over that range, and plots a agaist b - showing us the familiar sinusoidal waves.
We can also do all sorts of things with plots - stretch them vertically and horizontally, flip them upside down, give them titles and label the axes, have multiple subplots in a single plot ... but we'll come to these as we need them.
Arithmetic operations (addition, subtraction, multiplication, division) between an array and a scalar (a single number) are straightforward. If we add an array and a scalar, every element in the array is added to that scalar: the ith element of the sum of the array M and the scalar a is M(i)+a.
M = [1 3 -.5 7]; M2 = M+1
M2 = 2.0000 4.0000 0.5000 8.0000
Similarly, we can subtract, multiply by, and divide by a scalar.
M3 = 3*M
M3 = 3.0000 9.0000 -1.5000 21.0000
M4 = M/10
M4 = 0.1000 0.3000 -0.0500 0.7000
It's even possible to add, subtract, multiply and divide arrays with other arrays - but we have to be careful doing this. In particular, we can only do these things between arrays of the same size: that is, we can't add a 5-element array to a 10-element array.
If the arrays are the same size, these arithmetic operations are straightforward. For example, the sum of the N-element array a and the N-element array b is an N-element array c whose ith element is
c(i) = a(i)+b(i)
a = [1 2 3]; b = [2 -1 4]; c = a+b; c
c = 3 1 7
That is, addition is element-wise. It's just the same with subtraction.
d = a-b; d
d = -1 3 -1
With multiplication we use a somewhat different notation. Mathematics defines a special kind of multiplication between arrays - matrix multiplication - which is not what we're doing here. However, it's what MATLAB thinks you're doing if you use the * sign between arrays. To multiply arrays element-wise (like with addition), we need to use the .* notation (note the "." before the "*"):
e = a.*b; e
e = 2 -2 12
Similarly, to divide, we don't use /, but rather ./
f = a./b; f
f = 0.5000 -2.0000 0.7500
(once again, note the dot). As we'll see over and over again, it's very useful to be able to carry out arithmetic operations between arrays.
For example, say we want to make a plot of x versus 1/x between x = 2 and x = 4. Then we can type in the program
x = 2:.1:4; y = 1./x; plot(x,y) xlabel('x'); ylabel('y');
Note how we put the labels on the axes - using the commands xlabel and ylabel, with the arguments 'x' and 'y'. Because the arguments are character strings - not numbers - they need to be in single quotes. The axis labels can be more complicated, e.g.
xlabel('x (between 1 and 5)') ylabel('y = 1/x')
We haven't talked yet about how to exponentiate an array. To take the array M to the power b element-wise, we type M.^b Note again the "." before the "^" in the exponentiation. As an example
x = [1 2 3 4]; y = x.^2
y = 1 4 9 16
As another example, we can redo the earlier program:
x = 2:.1:4; y = x.^(-1); plot(x,y) xlabel('x'); ylabel('y');
Note that we put the "-1" in brackets - this makes sure that the minus sign associated with making the exponent negative is applied before the "^" of the exponentiation. In this case, we don't have to do this - but when programming it doesn't hurt to be as specific as possible.
These are the basic tools that we'll need to use MATLAB. Subsequent tutorials will cover other aspects of writing a program - but what we've talked about above forms the core. Everything that follows will build upon the material in this tutorial.
The following exercises will use the tools we've learned above and are designed to get you thinking about programming.
In writing your programs, you'll need to be very careful to think through:
(1) what is the goal of the program (what do I need it to do?)
(2) what do I need to tell MATLAB?
(3) what order do I need to tell it in?
It might be useful to sketch the program out first, before typing anything into MATLAB. It can even be useful to write the program out on paper first and walk through it step by step, seeing if it will do what you think it should.
Plot the following functions:
(a) y = 3x+2 with x = 0, 0.25, 0.5, ...., 7.75, 8
(b) y = exp(-x^2) with x = 0, 0.1, 0.2, ..., 2
(c) y = ln(exp(x^-1)) with x = 1, 1.5, 2, ..., 4
(d) y = (ln(exp(x)))^-1 with x = 1, 1.5, 2, ..., 4
A mountain range has a tectonic uplift rate of 1 mm/yr and erosional timescale of 1 million years. If the mountain range starts with a height h(0) = 0 at time t = 0, write a program that predicts and plots the height h(t) at t=0, t=1 million years, t=2 million years, t=3 million years, t=4 million years, and t=5 million years (neglecting isostatic effects). Label the axes of this plot, including units.
Repeat Exercise 2 in the case that the erosional timescale is 500,000 years.
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