What are Fractals?
By TomWatch Frax in action
The natural world around us is defined by irregular surfaces and shapes with uneven edges and rough corners. However, since Euclid classical geometry has only described the smooth ideal shapes - the circle, sphere, square, cube… - rarely, if ever, found in nature.
Fractals are the geometry of the natural world, they describe the texture of reality!
This insight was introduced by the Polish born French/American mathematician, Benoit Mandelbrot.
In 1975 he coined the word ‘fractal’ as a way to describe shapes that are detailed at all levels
of scale. What started as an investigation into an obscure area of mathematics culminated
in Mandelbrot defining the new field of fractal geometry.
“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…”
as described by Mandelbrot in his introduction to The Fractal Geometry of Nature.
Fractal geometry is an extension to classical geometry, which with the aid of
computers, can model and describe structures from sea-shells to galaxies!
The most striking characteristic of fractals is their self-similarity, the way the whole resembles smaller parts of itself at different scales. This property reveals natural fractal structures all around us - the way a branch with small twigs can look like a larger branch, which looks similar to the entire tree. How the jagged surface of a rock can look similar to an entire mountain, or how a river network resembles the smaller streams and tributaries that feed it.
Once you are aware of fractal patterns you will start seeing them everywhere!
Fancy a fractal sight-seeing tour? Here are some Google Earth locations to get you started!
Another key characteristic of a fractal is its dimension. We are all familiar with topological dimension; the single dimension of a straight line, the two dimensions of a shape or the three dimensions of objects and the space around us. However, things aren't so straight forward with shapes more complicated than the normal Euclidean forms.
It was Felix Hausdorff who proposed a different way of looking at the measure of dimension by introducing the notion of a fractional, or fractal dimension (also called the Hausdorff dimension).
This allows the intriguing notion of one-and-a-half-dimensional objects, i.e. fractals!
To properly understand fractional dimensions we first need a general concept of measuring dimension with more complicated objects. A cube, which has three dimensions, can be cut into eight (23) half-sized cubes. This is generalised as an n-dimensional shape being composed of mn copies of itself that are scaled to 1/m size.
Divide each side by 2
- 1D line
- 2 copies
mn = 21 = 1
- 2D square
- 4 copies
mn = 22 = 4
- 3D cube
- 8 copies
mn = 23 = 8
Divide each side by 3
- 1D line
- 3 copies
mn = 31 = 1
- 2D square
- 9 copies
mn = 32 = 9
- 3D cube
- 27 copies
mn = 33 = 27
Now if we can count the number of copies that an object contains of itself, and we know how they were scaled, we can determine its dimension using:
dimension = log(no. copies) / -log(scale)
So for the cube split into 8 copies each at 1/2 scale gives a dimension result of 3, which of course we already knew… however, let's see how this applies to a fractal shape.
The classic example of a self-similar shape with a fractional dimension is the Koch Curve. Devised by Helge von Koch in 1904, this snowflake curve is part of a family of shapes affectionately referred to by mathematicians as a ‘pathological’ - it is defined as being infinitely long and yet contained within a finite area!
For every step each line segment is replaced with the base shape from step one.
This effectively adds four copies of the previous shape at a third of the scale.
Plugging this into our formula gives the resulting dimension: log(4) / -log(1/3) ≈ 1.26
A comparison of dimensions for many other fractals can be seen in this list.
How Long is a Coastline?
It was when Mandelbrot was scouring through obscure and forgotten journals like an 18th century naturalist, that he came across an eccentric and unremembered mathematician called, Lewis Fry Richardson. Richardson's paper, "How long is the coastline of Britain?", caught Mandelbrot's attention because such a seemingly simple question of geography actually exposes essential features of fractal geometry.
The measured length of the coastline actually depends on the size of your measuring stick. Using a map you could estimate a rough distance. Driving around the coast you would record a more accurate, and longer result. If you were then to walk around measuring every nook and cranny your answer would be greater still. Ultimately as the measurements of the coastline become more and more accurate the resulting length approaches infinity!
Obviously the practicalities of such a measurement make it infeasible, so a technique called box counting is often used to to estimate fractal dimension.
Mandelbrot's paper that discussed the Hausdorrf dimension in relation to this so-called coastline paradox was a turning point that set his thinking along a fractal path.
Incidentally, the fractal dimension of Britain's coastline is about the same as the Koch curve above, ≈ 1.26
The First Fractal Explorations
The discovery of the first mathematical fractal is attributed to, Karl Weierstrauss, in 1861. He found a continuous function that is nowhere differentiable, i.e. a curve made entirely of corners that is impossible to determine its rate of change at any point.
Weierstrauss's function was a shock to mathematicians at the time and it was deemed to be an aberration resembling nothing found in nature.
In his pursuit to bring a new rigour to the field of logical calculus, pioneered by Augustin-Louis Cauchy, Weierstrauss is now known as the father of modern analysis where precise notions of number and continuity are sought.
Soon after Weierstrauss, Georg Cantor (the inventor of set theory), was on a quest to understand continuity and infinity. This led him, in 1883, to the fractal now known as the Cantor set (an unfortunate misattribution as it was actually discovered by, Henry Smith, in 1875).
Starting with a line, remove the middle third, leaving two equal lines. Then remove the middle third from each of these and so on ad infinitum, and you are left with the Cantor set:
At the time, Cantor, wouldn't have referred to this as a fractal, he used it as an example of a "nowhere dense" set that has "zero measure". Paradoxically a randomly thrown dart would have an infinitesimally small chance to hit it as every part consists almost entirely of holes!
Complex numbers are the building blocks of mathematical fractals, but to understand them we should first clarify the different groups (sets) of numbers:
Naturals: 1, 2, 3 etc…
Integers: 0 and negative whole numbers
Rationals: all numbers that can be written as a fraction of two integers
Irrationals: numbers such as π, φ, √2 that can't be written as a fraction
Reals: all of the above and represent all points on an infinitely long number line.
Finally we get to complex numbers, which encompass all real numbers as well as combinations that use the square root of negative numbers! Complex numbers were originally conceived as tools for solving cubic equations but are now used in many scientific and engineering fields.
Leonhard Euler denoted the symbol i, to mean the square root of -1 and called it the imaginary unit. Complex numbers are made up of two parts, real and imaginary, and are usually expressed in the form:
z = x + yi where x & y are real numbers and i = √-1
In 1685, John Wallis, realised that complex numbers could be represented in a diagram.
The horizontal x axis is the real number line with the vertical y axis for the imaginary numbers.
This representation helps lead to an intuitive understanding of complex arithmetic.
Incidentally, Wallis also introduced the symbol ∞ for infinity.
To dig deeper we highly recommend A Visual, Intuitive Guide to Imaginary Numbers.
Julia and Fatou
Our flyby through the field of fractals now brings us to two French mathematicians, Gaston Julia, and, Pierre Fatou. Around the time of the First World War they were both independently studying transformations in the complex plane.
A transformation is a rule that for any given point on a plane provides another point. It can be thought of acting simultaneously on the entire plane, picking it up, moving, spinning, stretching, folding, twisting then laying it back down in a new configuration.
Julia and Fatou were both particularly interested in the process of iteration - that is using the result of one transformation as the input for the next transformation.
This was of course at a time before modern computer graphics, so their calculations were carried out by hand and sketched manually. Even so they found attractors, points in space that pulled in the surrounding points towards them, and repellors that pushed the points away.
This can be seen below when iteratively squaring a complex number - depending on the starting location the result will either orbit around an attractor spiralling down to a point, or the orbit will 'escape to infinity' as the point gets pushed further and further away.
Julia and Fatou realised that the boundary between the areas of attraction and repulsion was very complicated, but without computers to automate the calculating process they were never to see the beauty and detail of their ideas. The area defined by this boundary is now known as the Julia set.
The work of Julia and Fatou remained largely unrecognised by mathematicians until Mandelbrot shone new light on it in the late 1970s. At the time, Mandelbrot, was working at IBM but it was as a visiting professor at Harvard University that he began to study Julia sets.
Mandelbrot started with the simplest possible transformation:
z → z2 + c
z and c are complex numbers where c is a constant and z starts as the coordinates of a point on the complex plane. The result of the equation then becomes the new value of z and the process is repeated for a set number of iterations.
For every point in the complex plane, Mandelbrot, plotted a black dot for the points that were captured (attracted to a point) and left blank those where the magnitude of z grew very large, 'escaped to infinity'. Below are Julia set fractals plotted for different values of c:
Mandelbrot was astonished by their complexity and called them "self-squared dragons"!
A Map of Julia Sets - The Mandelbrot Map!
Looking at the example Julia set fractals above you will notice that some are single connected shapes while others are disconnected, almost dust-like. For a while, Mandelbrot, wondered what the relationship between the connected and disconnected forms was, then he stuck on the idea of making a map of the behaviour.
This time, Mandelbrot, created Julia sets for each coordinate on the complex plane and plotted a point for those that were connected. He realised the disconnected Julia sets were those where the orbit of the starting point grew to infinity, which greatly simplified the generation of the map.
Using the same formula as before, but with an initial value of z as [0, 0] (the origin) and c as the coordinate of the point on the plane, z is calculated iteratively until its magnitude exceeds a 'bailout' threshold. If this threshold is reached, the point is considered disconnected and outside the Mandelbrot set. If, after a fixed number of iterations, the magnitude of the point is still under the threshold then it is within the Mandelbrot set (marked in orange below):
However, the truly amazing aspect of this seemingly trivial quadratic formula, z2 + c, is that as you zoom into the boundary more and more detail is resolved - in fact in mathematical terms it is infinitely detailed!
Dressing the Fractal
In their naked black and white skeletal forms, the Mandelbrot and Julia set fractals have intriguing structures, but we can elevate these to a level of real beauty.
In the fractal world there are hundreds of different schemes that have been devised to add interesting color and form to the raw fractal shape. They all work as a by-product of the iteration process. The simplest is iteration based coloring - the iteration number, at which the magnitude of z exceeds the bailout threshold, is used as an index to choose a color from a lookup table.
In Frax, if you use the spread gesture in Lights mode to flatten the 3D height, the result is an extended form of this iteration based coloring. Of course we don't stop there as we add 3D height, two independent light sources, surface marbling, two procedural texture layers that can twist, ripple and swirl, and combine together in unique ways to add an additional surface profile…
This has been a very top-level introduction to the world of fractals and we hope it has aided your understanding of this fascinating subject. The field is itself almost fractal in nature, with many different orbits and branches to follow!
These are some good books and websites worth investigating if you want to dig deeper:
- The Fractal Geometry of Nature Mandelbrot
The original introduction to fractals from the master himself
- Chaos and Fractals: New Frontiers of Science Peitgen, Jürgens, Saupe
A very comprehensive study into all areas of fractal science
- Introducing Fractals: A Graphic Guide Rood
A very approachable introduction to the subject even for those without a maths background
- Fractal Geometry course from Yale University
A very comprehensive website covering many different types of fractals
- Fractals on Mathworld
Mathworld is a very deep rabbit hole to go exploring down - you have been warned ;)
- Fractal Foundation
Inspiring interest in science, math and art
- Fractal Forums
A very active discussion forum that continues to push the boundaries of fractal exploration