Can you provide examples of fractal coordinates for points? On a 1D line there is one real number needed to uniquely locate a point by coordinates.
On a 2D plane, a pair of real numbers are needed.
But what if a fractal has dimension d, between 1 and 2 ($1&lt;d&lt;2$), how can points on the fractal be specified with more than one, but less than 2 coordinates?
Can you provide examples of 1.xD coordinates for points?
Edit: I'm asking for a numerical example, so this is not the same question tan this one.
A point in the plane can be specified by a single number in base 4
if each one of the symbols represents one quadrant
O=0
→=1
↑=i
↗=1+i
. is a decimal separator

a point with coordinates 1.5+2.5i can be written ↑→.↗
meaning (the dot is a decimal separator)
$\uparrow \rightarrow . \nearrow = i*2^1+1*2^0+(1+i)*2^{-1} $
That representation system can be rewritten as pair of numbers in decimal system.
On the same way, a point on sierpinski triangle can be written in base 3 using a symbol for each of the 3 subtriangles
Any point on the fractal can be specified with arbitrary precision with a string like
←→↑←↑↑←←→.←↑←→←←↑
where each position is multiplied by a power of 2 and
←=0
→=1
↑=0.5+(3/4)^½i
. is a decimal separator

So, I expect that number to be decomposed into 1.D coordinates as a real number, and something else.
On information theory, a character may represent a non integer number of bits, and that is solved by combining multiple characters, so I expected a similar solution for expressing coordinates.
Note that on the real line, if base N is used, the characters are multiplied by power of 4, but on other dimensions, it looks like the dimension depend on the difference between the base N and the power used as positional weight.
When we talk of &quot;dimensions&quot; of fractals we mean something entirely different that the number of independant spatial coordinates to determine a point.
So the question of how to indicate a point doesn't apply or even make sense.
So what does the dimensions of a fractal mean?
In euclidean geometry, if we increase the radius of an $n$dimensional object in $\mathbb R^n$ by a factor of $k$, then the hyper-volume of the object increase by a factor of $k^n$.  And it is the linear increase causes an exponential growth in volume that we mean by &quot;dimension&quot; and that has nothing to do with coordinates.
So the area of a fraction increases in comparison to an increase in linear size by a non-integer factor.
So in that definition it is fractional dimensional.  But it has nothing to do with coordinates.
https://medium.com/@cosinekitty/understanding-fractional-dimensions-f2ed2e4e1600
(Sometimes we teach by analogy.  But then analogy breaks.  The classic example if this is $b^{n} = \underbrace{b\times b\times b......\times b}_{n\text{ times}}$.  And as just by counting we must have $b^{n+m} = b^nb^m$.  But what if $x\not \in \mathbb Z$? What does $b^x$ mean?  Well, knowing that $b^{x+y} = b^xb^y$ we come up with another definition based on... other stuff.  So $2^\pi = 8.824977827076287623856429604208$......  Punch it into a calculator! That's what you get.  But does that mean $2^\pi = \underbrace{2\times2\times 2\times ....}_{\pi \text{ times}} =  8.824977827076287623856429604208$????  How the heck do you multiply a number by itself $\pi$ times?  Does that make any sense?  The answer is &quot;no&quot; it doesn't make sense but that is not what $b^x$ means any more.  It means something else.)
You're confusing the topological dimension of an embedding with the intrinsic dimension.  You need as many (integer) coordinates as the topological dimension.