Mandelbrot and the Beauty of Shape

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” ~ Benoît Mandelbrot

Throughout human history, we have known about logarithmic growth spirals, planetary orbits, and the golden ratio, and many great mathematicians, architects, and artists have poured over how geometry presents rules which create or structure the very fabric of nature.

It’s not so much a belief, but a study of the geometric forms that make up our world. The natural world has long been the inspiration for engineers and artists, and by defining that fabric; intricately woven throughout every cartoid and sphere, there could be a rule. Too neuron-shattering for most, the mathematician Benoit Mandelbrot has been one of those brave enough to take on the challenge. After all, mathematics is an art in itself, and it is motivated by beauty.

The Mandelbrot Set

The Mandelbrot Set was actually named in honor of Benoit Mandelbrot, a well-known fractalist, who saw the order in seemingly chaotic forms, such as clouds and shorelines. The set can be generated by algorithms, see Mandelbrot Set, if you’re up for the challenge. It is an iteration, and can be simplified as follows:

A negative number will never have a square root because the answer will always be a positive number. Therefore in an equation like -n2, the squared part will be imaginary, or ‘i’. ‘i’ is a complex number, as when it’s squared it will equal -1.

If you were to draw this out with two axes, the pattern which would emerge, for example, -1, 0, -1, 0, would create a loop, or a fractal. The Mandelbrot Set is created with these repetitions and is present in river systems, lightning bolt pathways, and galaxies. The fractals demonstrated by the Mandelbrot Set are not typically self-similar and are non-linear systems which can be observed in many factions of nature.

If you’re feeling a bit boggled, this video explains it much better.

The Koch Snowflake


If you add one equilateral triangle to another, each side of the triangle will turn into four. The first repetition will have 3×4=12 sides, the second will have 3×42= 48 sides. The equation for this would be n=3x4n. The potential of sides becomes infinite, like that of a snowflake… a naturally occurring fractal.

The tessellation of the Koch snowflake is possible, but only with snowflakes of varying sizes. Another interesting note on the Koch Snowflake is that it has an infinite perimeter, but a finite area. Quite mind-blowing when you think about it. Here’s an animation to demonstrate:

Mandelbrot and the Times Tables

The Mandelbrot Set, or at least parts of it, can also be created using the timetables. Using a circle with 10 points plotted on it, the 2xs table, for example, will create the central motif of the fractal, being the cartoid shape, or heart.

The infinite properties of the shapes can be created by shifting around the points, 0 becoming 1, 1 becoming 2, and so on. The other timetables create varying shapes and can be a fantastic way for a fractal to be demonstrated on simple terms, and are shapes inherent in nature.

Check out this video for a demonstration:

“My life seemed to be a series of events and accidents. Yet when I look back I see a pattern.” ~ Mandelbrot

Fractals are Not Typically Self-Similar, and are Dimensional

The Koch Snowflake, plus Mandelbrot and the set named after him certainly give us the framework of a fractal, but what about the chaos? The chaos is definitely still in there. Beyond the pattern, fractal geometry could be described as a rebellion against calculus, in that they are not usually self-similar or uniform in their pattern.

Nature is not smooth or overly ordered, because the fractals that hold the fabric together aren’t. Fractal geometry upholds the chaos, in that it doesn’t idealize, but rather observes the inconsistencies of the pattern.

All non-fractal shapes are self-similar; a line (1 dimensional), a square (2 dimensional), a cube (3 dimensional), and a Sierpinski triangle. They can all be broken down into scaled-down versions of themselves, and can be clearly measured.

The Koch Snowflake IS self-similar, in that you can break it down into four identical parts of itself. This means it could be 1.2 dimensional, as the root number of the logarithm (the method early navigators used to use to plot where they were in the ocean), is: log3 (4) = 1.262.

In other words, what would you need to do to test if it was self-similar, is scale a curve on the Koch Snowflake down to 1/3rd (the 3 part of the equation), and chop it into 4 pieces (as there are 4 sections which are self-similar). If you did this quite simple logarithm, which is used to calculate the amount of times you multiply a number to get the answer – in this case, the answer is 1.262 – you would discover the root number is 4.

If you can get your head around it, you’ll find the way the dimensions of shapes could hypothetically be calculated, fascinating. Even more so are the fractals which aren’t self-similar, as they make up the chaotic part of nature, where the mathematics launches from ‘reasonable’, to ‘unreasonable’… Like with the Mandelbrot Set, the reasoning behind the ways the mass of a shape changes, based on the scale of a shape as it changes, relies on the original shape being self-similar.

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With non-self-similar shapes such as the coastline of Britain, (the image is of North India depicting fractals in nature), the same rules (surprisingly) apply. Fractals are rough, but still follow a dimensional pattern. When they’re scaled up, however, they behave more like a tube, or two dimensional, and then one dimensional as you zoom out from the coastline.

The point is, they change dimension as you change the scale. This gives mathematicians a quantitative way to describe roughness. The Mandelbrot Set demonstrates the complexity of the imaginary axis, and the trippy formations it creates.

And the moral of the mathematical story is? That math is imaginary, or purely hypothetical? Or there is an order, even to chaos? Much like Quantum Mechanics, it sure is fun learning about the fabric of the universe, and where we might fit in, to that impressive structure.

Image Sources:

Image by David Zydd
Smog over North India from NASA Earth Observatory

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