Math, Infinity, and Beautiful Design

In his book The Blind Watchmaker, Dawkins discusses the appearance of design.

Biology is the study of complicated things that give the appearance of having been designed for a purpose.

In the context of evolution, the appearance of design makes sense. But I want to explore an area of geometry and mathematics that relates simple algorithms to beautiful appearances of design.

Welcome to the world of fractals. This is a world where I can draw a bounded snowflake that has a boundary of infinite length. It’s a world of paradox, a world where it doesn’t matter how big you are, because you won’t notice the difference when you zoom in. It’s a world of infinity, a world of endless iteration. More importantly, it’s a world of simplicity in terms of beginnings, and stunning beauty and complexity in terms of ends.

The Koch snowflake shows how simple the rules are for fractals to be constructed. One starts with an equilateral triangle. The next iteration involves the union with another triangle, which forms triangles on the periphery. Now we union the smaller triangles with rotated versions of themselves, which forms even smaller triangles on the periphery, and so on…

The resulting snowflake is not only aestheticallly pleasing, but it can also be shown to have infinite length.


The most iconic figure of fractal geometry, however, is the Mandelbrot Set.

Few people know or understand how simple the rules are that define this set. Formally, the M-set consists of all the elements C in the complex plane such that the iteration Z_n+1 = (Z_n)^2 + C is bounded, starting with Z_o = (0,0).

Computers can calculate by repeating the iteration over and over for different values of C to see if the norm of Z_n goes toward infinity.

The resulting graph of the M-set is mesmerizing. It’s a set that features very complicated arrangements of symmetry and self-similarity.

The more beautiful renditions of the M-set are created by assigning colors to values that allow Z_n to diverge, depending on the speed of divergence. As we can see below, there are all kinds of intricate “designs” in these renditions. There are seahorses and islands, shells and spirals, peninsulas and antennas.

So what can computer generated graphics show about nature, evolution, and biology life? Do fractals have any useful function besides being “beautiful”?

1) Fractals allow things that work on a large scale to be reproduced on a smaller scale (or maybe vice versa). The heart pumps blood to larger blood vessels, which branch out to smaller vessels, to smaller capillaries, and so on, and each branching pattern mimics the level above it. No designer had to use his intelligence to think about each step along the way. No God had to draw a blueprint of the location of every tiny blood vessel. Fractals allow simple patterns that work to be expanded and repeated into smaller, more intricate settings without any need to think about them, without any design.

2) Fractals are nearly ubiquitous in nature; they are found in places ranging from snowflakes and seashells to leaves and mountains.

3) Fractals are optimal for many purposes. For example, it was proven mathematically that the self-similarity of fractals allowed them to serve as the one and only optimal solution for antenna design. The fact that so many living things have fractal qualities also suggests that natural selection favored the advantages that fractals gave to certain species.

Of course, the creationists are really looking for trouble here. They look at the word “infinity” and not only do they think “intelligent design”, they think Yahweh, Jesus, and one version of one book in particular.

And they trip all over their watchmaker-design argument. After all, everything that is complex and specified must have a clear designer, right? And what do they say when we can’t find a designer for such things as fractals?

Their conjectured unknowable and untouchable god was, in fact, the very knowable Judeo-Christian God of Creation. And His methods actually express themselves as geometry. But it is a geometry that the Greeks could scarcely have imagined.

Just assert and assert and assert the superiority of your God and your One True Religion over and over again.

Let’s give them a round of applause.


Posted on June 17, 2011, in Math, Religion and tagged , , , , , , , , , , , , . Bookmark the permalink. Leave a comment.

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