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Geometry and Dimensions. - Part 4

Updated: Apr 17, 2021

Fractals, shorelines, lenses and negative dimensions (?): They're all here!

A doodle I made on the train

Keywords: Self-similarity, Seed Fractal, non-integer dimensions, Hausdorff dimension, coastline fractals, negative dimensions.

Welcome to this blog post. Here we will enter the realms of the uncanny: quirks of dimensions!


“A fractal is a way of seeing infinity.” – Benoit Mandelbrot

First of all, what is a fractal? Some would say that a fractal is a self-similar shape; a shape whose component parts are similar to the shape itself. In some cases, they would not be wrong. At the end of this post, however, we’ll discuss whether this is the best definition.

First of all, let’s see some common fractals:

Sierpinski’s Triangle from 3D Warehouse - Sketchup

Hexaflake from

Minkowski Sausage or Seed Fractal

Koch snowflake/Island

Now, let’s face this question: What dimensionality is a fractal?

At first, I would say 2-D. But after careful consideration, I would say that it depends on the fractal (Sierpinski Triangle: log[2](3); Seed Fractal: 1.5 etc.).

Wait, back up. How are these dimensionalities non-integers?

First, let’s confirm how we calculate dimensionalities. Imagine a 1-D, 2-D and 3-D shape (line, square and cube).

Photo from 3Blue1Brown.

Now imagine, as shown above, that each of these shapes are divided into shapes similar to the original shape but with half the side length. How many of those mini shapes would fit to make the original shape? For a line the number is 2. For a square the number is 4. For a cube the number is 8. For a point the number is 1. For an N-D cube, the number is 2^N.

We can say that officially N = S^D, where N is the number of times the “mini-shapes” fit in the original shape, S is the ratio of the original shape’s side length to the “mini shape’s” side length and D is the dimension of the shape.

Now consider the Sierpinski Triangle:

When S = 2, N = 3 (this is because one of the triangles is empty space).

By the formula: N = S^D

3 = 2^D

D = log[2](3) = 1.58 (2 d.p.) (!)

This explains the non-integer dimensions we saw earlier.

For the Minkowski Sausage, N = 8 and S = 4 (shown below) so D = log[S](N) = log[4](8) = 1.5(!).

For the hexaflake, the dimensionality would be log[3](7), or 1.77 (2 d.p.). This is because seven mini hexaflakes are fitted into a large hexaflake with a scaled side length of 3 times the mini hexaflake.

This method of calculating fractal dimension results in the Hausdorff dimension of fractals.

Now I’ll give you a moment to try to calculate the dimensionality of the Koch snowflake:

The answer is 1.26 (2 d.p.) or log[3](4).

These analogies also work with higher dimensions: Sierpinski’s cube has a dimension of 2.73 (2 d.p.) or log[3](20).

Now we will state something about fractals which seems obvious but is useful later.

Note that a fractal is rough at arbitrarily small levels. This means that even if you zoom in at large magnitudes, it will not conform to a line or a square or any integer dimensional shape. It will stay as it was, as if you did not zoom in at all in the first place.

That means that we can draw a distinction between Fractals and Smooth shapes.

Special Case: which of these do country coastlines fit in?

Well, country coastlines are not smooth at all levels of magnification (as there are different shapes of the borderline atoms; the fluctuations of the electrons on the orbits of atoms). So, they are fractals! Bear with me here, we will see why it is useful later.

But then, what is their dimensionality?

We can measure it by placing these shorelines on a grid and counting how many boxes the shoreline goes through. It would be a very inaccurate measure at first. Then we increase the width of the coastline and count how many boxes the coastline goes through then.

If you keep on doing this, the number of boxes would increase as the width increases, but almost in a power graph, where the power is the dimensionality of the shoreline.

If you do this, England’s shoreline is approximately 1.21 dimensional and Norway’s shoreline is almost 1.52 dimensional.

Why is this useful?

As we said before, fractals are rough. The fractal dimension is a way of quantifying roughness (the Seed fractal seems to be more "rough" than the Koch Snowflake). In the case of shorelines, roughness could indicate more coastline and therefore, more opportunity for harbours and shipping termini and hence trade opportunities. Of course there are other factors to consider such as approachability etc. but a more rough coastline is open to more possibilities.

So, fractals are not always self-similar, but they have non-integer dimensions.

Negative Dimensions?

First of all, what do we mean by negative dimensions?

Well, maybe this initial concept can be cleared by considering what the units of a negative D length would be.

Well, for an N-D measure, the units are m^N.

For the -1D world, the units are therefore m ^ -1, or 1/m.

In physics, this relates to at least two things: the cyclic frequency of a wave and the power of a lens (in Dioptres).

But how can we relate either of these to a world handicapped of space and devoid of even the existence of a point?

Let’s try a different point of view.

What actually is a point?

A point is a figure that is defined with a fixed coordinate for every dimension (e.g. (a, b, c, d, …) where a, b, c, d and so on are constants).

A line, on the other hand, is a figure that is defined with a fixed coordinate for every dimension except the first (e.g. (x, b, c, d, …) where b, c, d and so on are constants and x is a variable).

A plane is a figure that is defined with a fixed coordinate for every dimension except the first and second (e.g. (x,y,c,d,…) where c, d and so on are constants and x and y are variables).

With every higher dimension, more and more constants are replaced with variables, leading to higher uncertainty of where a specific point in that figure might be. To go the opposite direction, we need to increase certainty of where a specific point may be. But how can you be more certain than literally pointing out where the point is (as we defined above)?

I don’t know the answer to this yet, but one day we will figure it out! Any thoughts in this line are welcome.


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1 Comment

Imre Matajz
Imre Matajz
Dec 23, 2023

I interpret a Negative dimension = density or capacity. This would be analogous to frequency as a fraction of a wave, or power of lens i.e. its capacity to discern pixels of a picture.

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