Fractal Geometry

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Fractal Geometry

Fractal Geometry has fascinated me since I first discovered that such a thing even existed.

The term “fractal” was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning “broken” or “fractured”, and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.

Mandelbrot himself summarized fractals as being: “beautiful, damn hard, increasingly useful.” 1

I am not a mathematician, nor am I the daughter of a mathematician. Nor do I even have a solid grasp on the most basics principles of mathematics, let alone the infinite complexities of geometry, so as you probably already suspect, this article is more of a flight of fancy than and attempt at a scholarly treatise.

Fractal Geometry and Fractal Art are related, but separate and distinct topics.

Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The Julia set and Mandelbrot sets can be considered as icons of fractal art. Fractal art (especially in the western world) is not drawn or painted by hand. It is usually created indirectly with the assistance of fractal-generating software.”

My fascination is primarily with fractal geometry in that it ties in closely with my fascination with the patterns in nature which of course, ties in with my belief that the world was created by an infinitely creative Creator.

I am not qualified to write about fractal geometry or how God created the world. But I can share links to the sites, explanations and speculations of those who are considerably more informed than me.

LINKS TO ARTICLES ABOUT FRACTAL GEOMETRY

The Mandelbrot Set
The Fractal Pattern of the Torah, Bible
The Bible is a Fractal

One of the reasons fractal geometry fascinates me is that it expands my appreciation for the mathematical laws that govern the universe I inhabit.

I wonder what Mandelbrot’s discovery would have meant to Einstein, who diligently sought, but never found, the ‘unifying principal’ that holds the universe together. (The Mandelbrot set was discovered in 1980. Einstein died in 1955.)

It is my belief that this ‘unifying principal’ is Jesus.I base that belief on this passage from the letter Paul wrote to the early church in Colossae.

Colossians 1:16

“For in Him all things were created, things in heaven and on earth, visible and invisible, whether thrones or dominions or rulers or authorities. All things were created through Him and for Him.”

I also believe that this same Creator created all the laws that govern our universe.

Beliefs can be true or false. Believing something doesn’t make it true or false. Something is true or false because it can be proved or disproved.

For example, I believe that the mathematical equation 1 + 1 = 2 is true.

I believe it because I can see that when I have one apple, then add another apple, I have two apples. So I assume that what I believe is true. Now when it comes to God and the existence of God, I can’t prove that he exists or that he doesn’t exist.

But apparently, it can be proved that 1 + 1 does indeed = 2.

Here’s how…

The proof starts from the Peano Postulates, which define the natural
numbers N. N is the smallest set satisfying these postulates:

P1. 1 is in N.
P2. If x is in N, then its “successor” x’ is in N.
P3. There is no x such that x’ = 1.
P4. If x isn’t 1, then there is a y in N such that y’ = x.
P5. If S is a subset of N, 1 is in S, and the implication
(x in S => x’ in S) holds, then S = N.

Then you have to define addition recursively:
Def: Let a and b be in N. If b = 1, then define a + b = a’
(using P1 and P2). If b isn’t 1, then let c’ = b, with c in N
(using P4), and define a + b = (a + c)’.
Then you have to define 2:
Def: 2 = 1′
2 is in N by P1, P2, and the definition of 2.
Theorem: 1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.
Then 1 + 1 = 1′ = 2 Q.E.D.

Note: There is an alternate formulation of the Peano Postulates which
replaces 1 with 0 in P1, P3, P4, and P5.

Then you have to change the
definition of addition to this:
Def: Let a and b be in N. If b = 0, then define a + b = a.
If b isn’t 0, then let c’ = b, with c in N, and define
a + b = (a + c)’.

You also have to define 1 = 0′, and 2 = 1′.

Then the proof of the
 Theorem above is a little different:

Proof: Use the second part of the definition of + first:
1 + 1 = (1 + 0)’
Now use the first part of the definition of + on the sum in
parentheses: 1 + 1 = (1)’ = 1′ = 2 Q.E.D.

This begs the question: Who really cares? Besides scientists, mathematicians, philosophers and theologians, that is?

The Answer: I do!

Next Question: ‘Why?’

I care, because by believing the foundational truth of the equation 1+1=2 opens the way for me to believe in algebra, calculus and all of the other higher forms of mathematics on which the laws of our universe are based.

I care, because I believe that our universe did not happen by chance any more than (to quote a well worn cliche) the works of Shakespeare were created by the explosion of a printing press. I see Fractal Geometry as being one more example of the Creator’s progressive revelation to mankind at a specific point in history exposing yet another element of the infinite complexity of the world we inhabit.

“Any scientist who studies in the natural realm is always astonished when he comes to see the complexity of life, the marvellous symmetry of things, what lies behind all visible matter, the molecules, the atoms, the make-up of a flower or a star. The obvious order, design and symmetry of everything is astonishing; it is amazing.”
Ray Stedman

I’m going to jump from Fractal Geometry to a closer examination of the word Logos, because in my mind, the two seem to be intimately connected.

The Bible states that the world was created by “the Word” (Logos) and that “the Word” was Jesus himself.

That’s a very strange and staggering claim!

By the word (Logos) of the Lord the heavens were made, and by the breath (Pneuma) of his mouth. Psalm 33:6 (ESV)

‘Lord’ and ‘His’ imply that there was a rational, logical and powerful being who created the universe. The only other alternative I have to offer for the case of the non existence of a creator is that ‘nothing’ created the universe. As in the old stand-by joke: In the beginning there was nothing. And then it exploded. How an explosion of nothing brought in to being the existence of something is not ‘something’ I find easy to comprehend.

Even though I don’t fully understand the concepts that I am grappling with in this article, this study and the process of putting my thoughts about fractal geometry on paper has not been without merit. For me, the merit comes from a better understanding of the word faith. To quote one of the easiest verses in the Bible to remember, which is Hebrews 1:11

“Now faith is the substance of things hoped for, the evidence of things not seen.”

It is by faith that we come to God. Or refuse to come to God. The choice is ours. What fractal geometry gives me is substantial thoughts for examining the things I both believe and hope for.

NOTE: The links in this article are from common usage sources. They are not intended as scholastic references.