# 8. Self-similarity

#### — Variety Makes All the Difference —

## So, how does nature do it?

How does nature direct molecules into snowflakes, or crystals or any other regular form? Chaos Theory has an answer: *Self Similarity*, a fundamental principle that allows building blocks to mimic their own shape in the building they make.

Remember this one, from chapter 2?

##### Hold that thought (2b)

A large number of particles will display a pattern that is near equal to the initial possibilities of a single particle.

A similar thing is going on here: A large number of elements may form a shape that is derived from the shape of one element. And because no element can be coerced to follow a certain path, no large number of elements will display the exact same pattern as another group. Patterns caused by large numbers of elements are alike but never the same. Hence all snowflakes look alike but no two are exactly identical.

Self-similarity is a really big deal. It occurs all over nature and many have argued that self-similarity is one of the key natural principles that shape our world the way it is. Self-similarity has been observed in all fields of research: physics but also biology and even psychology and sociology.

It also happens all over Scriptures and has been studied extensively, most often referred to as type-Theology. In the Book of Exodus for instance, Moses constructs the Tabernacle according to heavenly patterns he observed (Exodus 25:40, Hebrews 8:5), making the tabernacle a self-similarity of something that exists on a different level of complexity, namely heaven.

No wonder that at a certain point Jesus said, "I will open My mouth in parables; I will say things hidden from the foundation of the world" (Matthew 13:35).

##### Hold that thought (8a)

Self-similarity is the repetition of a shape, form or behavior on different levels of complexity. Not as an identical copy, but as a variation of the same basic shape.

## And how do mathematicians do it?

An image that displays self-similarity is usually called a fractal. One of the first works of Fractal Art was made in 1904 by a Swedish mathematician named Helge von Koch, and his piece was the so-called Koch Snow Flake. He took a triangle and added a similar but smaller triangle to each of the sides of the first one. Then he added smaller triangles to the sides of the second ones, again to the sides of the third ones, and so on at infinitum.

Every textbook reporting on the Koch Snow Flake will demand that if indeed we continue this process at infinitum we will add length to the outline at infinitum, hence producing an infinitely long line. And yes, theoretically this is true (Click here for the details). But with every step the added triangle gets smaller and smaller, and in the real world there is no such thing as infinitely small. After a great many steps the sides of the smallest triangle will be one quantum long, and no smaller triangle can be added.

And that is more important than it seems. The big difference between God's snowflakes and Koch's snowflakes is that God's are all different while Koch's are all the same. Koch's Snow Flake is sterile while with God's flakes, variety makes all the difference. This is why no two trees are the same, no two mountains are the same and no two humans are the same. But there's more:

Koch's tiniest triangles are identical to the big triangle, but way down the line this concoction proves to be an impossible structure. It's an un-fractal since the tiniest triangles will lose their form and fuzz up. A sleek triangle like that will only occur in the minds of Koch and perhaps Plato, but not in nature. Nature produces snowflakes that are never the same because the large-scale phenomenon doesn't mimic the *shape* of the small-scale phenomenon, but the *behavior* : unpredictability, randomness and sovereignty. And Math has no way of generating randomness.

## Randomness

Generating random numbers is somewhat of a sport among mathematicians. Because how does one *program* randomness? There's no way! And so they cheat. They take some kind of infinite number sequence (like the number pi) and tell a computer to select decimals at certain intervals (like every fourth digit, or seventh, or whatever). Pi is a number without inner structure and its digits are random (for as far as we know now), so out comes a random number sequence. But! this is not real randomness because any other smart computer could analyze the result and blow the whistle: This is not *random* random, this is *pi* random. And as the secret is out, the rest of the pseudo-random sequence can be predicted. If the second computer divides every outcome of the first by the way it generates the pseudo-random sequence, it would spew out 1 every cycle. That violates the prime definition of randomness, and the sequence is not random. True randomness cannot be divided by something other than itself.

And besides that, the decimals of the number pi go on forever. That means that if we try to express the relationship between diameter and circumference of a circle in numbers we need infinite detail to stay truthful. But infinite detail does not exist, and so we must yield to the rather shocking conclusion that the before mentioned relationship cannot be expressed in numbers, and that *pi* is not a number at all!

The same goes for that other famous 'number' *e*, and all so-called transcendental numbers (numbers that go on forever after the dot; in other words, numbers that represent infinite detail). And to make matters worse, transcendental numbers out-number real numbers with an infinite factor.

It may seem a bit paradoxical but since mathematics cannot release its detailed accuracy, it loses connection with the real world around quantum level.

##### Hold that thought (8b)

Since reality is fuzzy, accuracy is approximation.

Some say that the number sequence displays true randomness in its prime-number distribution. If you would like to see that idea debunked go to a series of 4 supplements, starting with: Who Framed the Number Sequence? →

For more on the number *pi*, and specifically the troublesome occurrence of *pi* in the Bible, see our article on
*Pi* in the Bible →

If you don't care much about numbers, skip ahead to a formal decapitation of all logical systems:
Children of the Primes →

## Summary 8: Self-Similarity

- Self-similarity is a structure repeated on a different level of complexity or at a different scale.
- Numbers cannot fully represent reality because (1) they are too accurate, and (2) they can't mimic the randomness that comes from the freedom which is the most fundamental principle of nature.