<commit_before>---
title: Coastline Paradox
layout: article
ontology: terminology
category: what
---

The coastline paradox is the counterintuitive observation that the coastline of
a landmass does not have a well-defined length. This results from the fractal-like
properties of coastlines. The first recorded observation of this phenomenon was
by <NAME>.

More concretely, the length of the coastline depends on the method used to measure
it. Since a landmass has features at all scales, from hundreds of kilometers in
size to tiny fractions of a millimeter and below, there is no obvious limit to
the size of the smallest feature that should not be measured around, and hence no
single well-defined perimeter to the landmass. Various approximations exist when
specific assumptions are made about minimum feature size.

### Mathematical Aspects

The basic concept of length originates from Euclidean distance. In the familiar
Euclidean geometry, a straight line represents the shortest distance between two
points; this line has only one length. The geodesic length on the surface of a
sphere, called the great circle length, is measured along the surface curve which
exists in the plane containing both end points of the path and the center of the
sphere. The length of basic curves is more complicated but can also be calculated.
Measuring with rulers, a person can approximate the length of a curve by adding
the sum of the straight lines which connect the points:

![illustration1](illustration1.png)

Using a few straight lines to approximate the length of a curve will produce a
low estimate. Using shorter and shorter lines will produce sums that approach the
curve's true length. A precise value for this length can be established using
calculus, a branch of mathematics which enables calculation of infinitely small
distances. The following animation illustrates how a smooth curve can be meaningfully
assigned a precise length:

![illustration2](illustration2.gif)

However, not all curves can be measured in this way. A fractal is by definition
a curve whose complexity changes with measurement scale. Whereas approximations
of a smooth curve get closer and closer to a single value as measurement precision
increases, the measured value of fractals may change wildly.

The length of a "true fractal" always diverges to infinity. However, this
figure relies on the idea that space can be subdivided indefinitely. This
fiction—which underlies Euclidean geometry and serves as a useful model in everyday
measurement—almost certainly does not reflect the changing realities of 'space'
and 'distance' on the atomic level.

Coastlines differ from mathematical fractals because they are formed by numerous
small events, which create patterns only statistically.

### Practical

For practical considerations, an appropriate choice of minimum feature size is on
the order of the units being used to measure. If a coastline is measured in kilometers,
then small variations much smaller than one kilometer are easily ignored. To measure
the coastline in centimeters, tiny variations the size of centimeters must be
considered. However, at scales on the order of centimeters various arbitrary and
non-fractal assumptions must be made, such as where an estuary joins the sea, or
where in a broad tidal flat the coastline measurements ought to be taken. Using
different measurement methodologies for different units also destroys the usual
certainty that units can be converted by a simple multiplication.

Extreme cases of the coastline paradox include the fjord-heavy coastlines of Norway,
Chile and the Pacific Northwest of North America. From the southern tip of Vancouver
Island northwards to the southern tip of the Alaska Panhandle, the convolutions
of the coastline of the Canadian province of British Columbia make it over 10% of
the entire Canadian coastline—25,725 km (15,985 mi) vs 243,042 km (151,019 mi)
over a linear distance of only 965 km (600 mi), including the maze of islands of
the Arctic archipelago.
<commit_msg>Modified content<commit_after>---
title: Coastline Paradox
layout: article
category: what
menu:
  - link: "#1"
    title: Mathematical Aspects
  - link: "#2"
    title: Practical

---

The coastline paradox is the counterintuitive observation that the coastline of
a landmass does not have a well-defined length. This results from the fractal-like
properties of coastlines. The first recorded observation of this phenomenon was
by <NAME>.

More concretely, the length of the coastline depends on the method used to measure
it. Since a landmass has features at all scales, from hundreds of kilometers in
size to tiny fractions of a millimeter and below, there is no obvious limit to
the size of the smallest feature that should not be measured around, and hence no
single well-defined perimeter to the landmass. Various approximations exist when
specific assumptions are made about minimum feature size.

### Mathematical Aspects

The basic concept of length originates from Euclidean distance. In the familiar
Euclidean geometry, a straight line represents the shortest distance between two
points; this line has only one length. The geodesic length on the surface of a
sphere, called the great circle length, is measured along the surface curve which
exists in the plane containing both end points of the path and the center of the
sphere. The length of basic curves is more complicated but can also be calculated.
Measuring with rulers, a person can approximate the length of a curve by adding
the sum of the straight lines which connect the points:

![illustration1](illustration1.png)

Using a few straight lines to approximate the length of a curve will produce a
low estimate. Using shorter and shorter lines will produce sums that approach the
curve's true length. A precise value for this length can be established using
calculus, a branch of mathematics which enables calculation of infinitely small
distances. The following animation illustrates how a smooth curve can be meaningfully
assigned a precise length:

![illustration2](illustration2.gif)

However, not all curves can be measured in this way. A fractal is by definition
a curve whose complexity changes with measurement scale. Whereas approximations
of a smooth curve get closer and closer to a single value as measurement precision
increases, the measured value of fractals may change wildly.

The length of a "true fractal" always diverges to infinity. However, this
figure relies on the idea that space can be subdivided indefinitely. This
fiction—which underlies Euclidean geometry and serves as a useful model in everyday
measurement—almost certainly does not reflect the changing realities of 'space'
and 'distance' on the atomic level.

Coastlines differ from mathematical fractals because they are formed by numerous
small events, which create patterns only statistically.

### Practical

For practical considerations, an appropriate choice of minimum feature size is on
the order of the units being used to measure. If a coastline is measured in kilometers,
then small variations much smaller than one kilometer are easily ignored. To measure
the coastline in centimeters, tiny variations the size of centimeters must be
considered. However, at scales on the order of centimeters various arbitrary and
non-fractal assumptions must be made, such as where an estuary joins the sea, or
where in a broad tidal flat the coastline measurements ought to be taken. Using
different measurement methodologies for different units also destroys the usual
certainty that units can be converted by a simple multiplication.

Extreme cases of the coastline paradox include the fjord-heavy coastlines of Norway,
Chile and the Pacific Northwest of North America. From the southern tip of Vancouver
Island northwards to the southern tip of the Alaska Panhandle, the convolutions
of the coastline of the Canadian province of British Columbia make it over 10% of
the entire Canadian coastline—25,725 km (15,985 mi) vs 243,042 km (151,019 mi)
over a linear distance of only 965 km (600 mi), including the maze of islands of
the Arctic archipelago.

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