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Measuring things with fractional dimensions

Using fractional dimensions is incredibly useful for measuring things. Traditionally we measure spatial things either by counting, measuring length, area or volume. These correspond to m0, m1, m2 and m3 using metres as units. But how do you measure the length of the coastline of Great Britain? The figures given vary wildly, because the coastline has essentially a fractional dimension, roughly 1.25, so an accurate measure is only possible using the correct units: m1.25 as visualised here:

When we think of a m or a m2 we have a picture of what one of these looks like (as shown above). For a m1.25 curve we can use a Koch curve, in fact the Koch curve is a great choice because the bend angle in the fractal allows the curve to represent any fractional dimension from 1 (a straight line) to 2 (an area):

This is great for coastlines, but if you want to define the length of a river in the map, you have a similar problem but now the geometry is not a curve but a tree. Here's a simple fractal that gives a tree of the specified dimension:

If you next want to find the total circumference of lakes and puddles in the map, below a certain size this is also approximated with a fractional dimension. But is neither a curve or a tree, the following fractal could be used:

Extending these line fractals to 3d is easy for the above case and can be used for cluster structures such as the surface area of the asteroid belt or saturn's rings. The Koch curve can be extended to a 3d curve, and the tree fractal can be extended to 3d by using the long diagonal of a cube (rather than the diagonal of a square) and therefore three times as many branches.

These shapes are great references, but we still have a problem with measuring the surface area of Great Britain, below a certain scale (about 100m) the surface itself has a fractional dimension, let's say it is about 2.2, so the apparent area is greater when we measure it in more detail. What shape represents one m2.2? Ideally we could use a simple fractal that is parameterised by dimension (like the Koch curve)... there isn't an obvious candidate, but here is a fractal surface that I created which can take you from a m2 up to about a m2.3 (before it self-intersects), it is called the volcanic surface fractal:

There is a choice of style of surface, the above has negative curvature on the left (the surface of a shell fractal) and positive curvature on the right three (the surface of a tree fractal). 

You can also have both positive and negative curvature so it is the same either way up, as in this fractal I built which uses spherical transforms, and can be mapped onto a sphere as in the sequence on the right:

Both of the above assume object is made of bumps or dents or both. A third class of surface fractal are those with zero Gaussian curvature, so they are developable surfaces meaning that you can make them by crumpling paper. But, since they are fractal, you would need an infinitely large piece of paper to make a finite size surface. A family of such fractals are possible by repeated concertina, yaw and scale operations:
Here we see it at multiple bend angles. 
A nice property of this family is that the yaw rotation parameter generalises a Koch curve, the above fractal, a hexagonal version and also a very random looking version when taken at a random point on the surface using a yaw value of the pi/phi, which gives a crumpled paper appearance:
These yaw angles are pi, pi/2, pi/3 and pi/phi (at random location) respectively. However, the pi/2 yaw angle is the only one that has a 'simple' fold pattern.

It is likely that this fractal is a tree of ridges intertwined with a tree of valleys, however this is hard to see. The concept is more easily demonstrated on height-field fractals, where construction is less constrained:
The simplest such construction is probably like so:
This is a fractal height field which has a tree of valleys between its tree of ridges, which are blancmange curves along their length. Such height fields are fractal in a different sense, if the height is modeled as an infinitesimal (with a corresponding dimension that is 2 + an infinitesimal). These can be useful as reference shapes for natural objects that are strongly influenced by gravity, just like hills and valleys, and they have trivially variable dimension by scaling vertically. This shape is formed very like the 2d Koch curve, instead of replacing a line by two upside down lines with a bend, it replaces one triangle with two upside down triangles in a v.

By comparison, a surface made of both hills and dents can be constructed by replacing an equilateral triangle with 4 upside down equilateral triangles with equal gradient change:
See that is is composed of hollows and hills rather than ridges and valleys. So is a tree-shell rather than shell tree in my classification.

All these above fractals makes useful archetypes, simple shapes that represent a particular fractal dimension and the basic different qualitative measures such as whether ti is tree-like (fractal bumps) or ridge-like etc. For each unique archetype, stochastic versions are usually feasible, these give a more natural reference, just as a stochastic Koch curve is more natural (and higher entropy) than the single Koch curve. An example is the pi/phi crumpled surface fractal above, which has a good resemblance to crumpled paper or a rock surface. 

There are many other structures we can measure which have fractional dimension that aren't covered above, the borders and surfaces of the many classes of fractal as defined here:

The above are summarised in this article