## Fractals and God

### June 13, 2012

I’ve been taken up with fractals the past few months. I’ve started seeing them everywhere. They’ve also given me some ideas.

If you are not familiar with fractals, I suggest you research what they are, what kinds there are, how they are used, and how they are generated. I’ve provided a few resources below for reference. A fractal can be roughly defined as s a geometric figure or curve marked by self-similarity: the characteristic where the parts of a figure look like the whole. Self-similarity occurs when a figure forms a pattern that repeats itself in iterations. See below.

These iterations show a simple example of a fractal tree’s development. With each iteration of the “motif”–the pattern repeated–the figure becomes more complex.

Example of a fractal tree.

There could be a number of different fractal trees depending on the number of iterations, regularity, how many branches per iteration, how long each branch, what angle each branch stems off at, and other factors. The tree is one example of a kind of fractal. However, other shapes and designs produce different patterns. Because fractals can form complex and irregular shapes unlike Classical Euclidean geometry there are things you can model or measure using fractals that cannot be done by a triangle or circle or trapezoid. Fractals are used in computer modeling, metallurgy, and other fields particularly to predict seemingly unpredictable data.

Fractals appear throughout nature in trees, coastlines, snowflakes, plants, and a number of other entities. In the physical world they have a limited number of iterations, but in theory, a fractal can produce an infinite number of iterations. The iterations are not necessarily as symmetrical and regular as the above example. Certain mathematical algorithms produce fractals like the Mandelbrot set discovered in the late 1970’s.

Mandelbrot Set.

With online videos of fractal zooms you can see the infinite detail in a fractal set. As you zoom in on the detail of one area more detail can be seen, without end. The experience is uncanny.

When I started looking at fractals I was struck by them. Two paradoxes came to me. First, they can be seemingly simple and yet complex. By “simple” I do not necessarily mean uncomplicated, but not made of composite parts. This is easiest seen in a Koch Island or Gosper Island fractal. However, the infinite iteration of a simple pattern results in infinite complexity. This brings to mind the difference between classic Christian theological doctrine, which states that God is divinely simple, not made up of parts but whole, and Hinduism, which worships thousands of gods as different facets of one God. As fractals can be seen to be both whole and made of composite parts they exhibit both unity and complexity. This is not the same as the Hindu depiction, which sees God as divided into several different facets. Rather, I am suggesting how, as a fractal demonstrates, God could possibly be both undivided like any 2-dimensional shape and simultaneously infinitely complex, thereby resolving the contradiction between simplicity and complexity.Because of this a fractal can have a shape and still be infinite.

Random Koch Island or Snowflake Fractal.

Closely related to this first paradox is a second paradox. I am a sort of Deist admittedly leaning towards Theism; that is to say, I am closer than I used to be to believing God is involved in human life. But I used to have difficulty seeing how God could possibly be infinite, as most theists believe, and have a personality or will. For to have a personality is to be finite; to be a certain kind of person is for there to be other kinds of people you are not. To have a will is to want certain things done and not others. Infinity by contrast implies having no boundaries like an infinite landscape would, and personality and will therefore seem antithetical to infinity. Who can define the space of infinity? Who can put borders around it? You can imagine an infinite shape would have no boundaries. It could not be represented by a perimeter because that’s represented by a number, which is finite. Therefore there seemingly can’t exist an infinite shape. It would take up an entire plane–yet fractals have a recognizable shape; the shape of a Mandelbrot is distinct from a Koch Island or Box Fractal. However, a complete fractal set can have an infinite number of iterations so that the deeper you zoom into a fractal you will continue to see self-similarity. In a complete fractal (bad adjective for something infinite) there is infinite detail and therefore an infinite perimeter.

These are a few of the thoughts about fractals that have stood out to me.  They could be a source of inspiration for many ideas.

Other Resources:

Elementary explanation of a Fractal: http://mathworld.wolfram.com/Fractal.html

A Slightly less elementary explanation: http://www.math.umass.edu/~mconnors/fractal/fractal.html

Fractals Unleashed (more interesting): http://library.thinkquest.org/26242/full/

Fractal Geometry (much more extensive): http://classes.yale.edu/fractals/