Jason Lisle. Fractals: The Secret Code of Creation. Master Books, 2021.
In recent years people have become more familiar with the concept of fractals. Fractals are geometric figures that are traced by repeating a formula over and over in a smaller scale each time. Fractals: The Secret Code of Creation gives us some of the formulas and many lovely and colorful illustrations of fractals.
Because fractals are created by repeating an algebraic formula, they could not really be described or even discovered until the invention of the computer. The man who first discovered and wrote about them, Benoit Mandelbrot, worked for IBM.
I first learned about fractals when I was attempting to teach Tom Stoppard’s Arcadia. Fractals and other iterating algorithms figure into that tale. A mathematician is trying to explain what an iterating algorithm is to a non-scientist and simply says—alas, typical of some experts in many fields who lack communication skills—“It’s an algorithm that iterates.” In other words, if you are plotting the solution to an algebraic formula containing x, let the solution become the starting point and your new x using the same formula. Try repeating that (iterating) many times and see what you get.
This book illustrates examples of a number of fractals, giving us the formulas and illustrating the results. Though softcover, it is the kind of book that one associates with coffee tables because it is full of color illustrations and photographic quality paper. We begin to see the pattern of fractals in many forms, going into great degrees of magnification.
What became fascinating to people about fractals is that they can be used to describe natural features and phenomena. Around 1980, for example, computer programmers were trying to create a graphical interface for a flight simulator program. They realized that the most realistic way to create graphics of mountains that looked real was with fractals.
We see that fractals can be used to describe or illustrate many other things we see in nature such as lightning, shorelines, river deltas, tree branches, nautilus shells, snowflakes, or fern leaves. This is fascinating, especially illustrated the way Lisle’s book does.
But the book has another purpose other than to introduce us to and illustrate fractals. Fractals are graphed by using what we call imaginary numbers. A little over a hundred years ago, imaginary numbers were simply a mathematical exercise. On an ordinary number line, negative numbers do not have square roots. Both negative and positive numbers when multiplied once by themselves yield a positive number. So 2 × 2 = 4, but also -2 × -2 = 4.
Mathematicians hypothesized imaginary numbers. For the sake of argument, let us say negative one has a square root. It is not on the number line, but let us call it i for imaginary. So i × i = -1. That would mean, then, that 2i × 2i = -4. Now negative numbers have square roots. The math is not too complicated for anyone who has had a year of algebra.
We can plot these imaginary numbers on a plane, rather than a number line, and it looks like the typical x and y axes of algebra. Soon, though, people realized that this kind of algebra helps us graph and solve problems involving waves, especially when describing electricity. These so-called imaginary numbers had a real purpose.
Graphs of fractals also use imaginary numbers. Lisle declares these fractal designs as “the secret code of creation.” He draws a conclusion similar to the one Newton concludes his Principia with. Just as Newton had to use a new mathematical system to help describe gravity and planetary motions, so the discovery (we really cannot say invention) of fractals helps describe many natural phenomena. We see that the universe hangs together using very sophisticated mathematical models. Yet these models are intellectual and thought initially to be abstract. The fact that there is deep math underlying these things demonstrates what Newton affirmed (and as we have quoted elsewhere) that:
This most beautiful system…could only proceed from the counsel and dominion of an intelligent and powerful Being…This Being governs all things, not as the soul of the world, but as Lord over all; and on account of His dominion he is wont to be called Lord God pantokrator, Universal Ruler. (Newton 369-370)
In Newton’s day, it was the calculus. Now we have seen how many other abstractions have been used to describe natural phenomena, notably relativity, subatomic wave-particles, probability of habitability, and the apparent eleven dimensions needed for unifying the four known forces. It really appears that behind our existence is a very sophisticated mind.
The more sophisticated the math behind nature, the greater the evidence of the intelligence needed to create it. Random? Hardly. Check it out.
Newton, Sir Isaac. Mathematical Principles of Natural Philosophy. 1687. Translated by Andrew Motte and Florian Cajori, 1939; edited by Robert Maynard Hutchins, Encyclopedia Britannica, 1952. Great Books of the Western World.