So, how does nature do it?
Hold that thought (2b)
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 snow flakes 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 (Ex 25:40, Heb 8:5), making the tabernacle a self-similarity of something that exists on a different level of complexity, namely heaven.
No wonder that it is written:
I will open My mouth in parables; I will say things hidden from the foundation of the world.
- Matt 13:35.
Hold that thought (8a)
And how do mathematicians do it?
And that is more important than it seems. The big difference between God's snow flakes and Koch's snow flakes 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 tree is alike, no mountain is alike, no human is alike. 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 snow flakes that are never the same because the large-scale phenomenon doesn't mimick the shape of the small-scale phenomenon, but the behavior : unpredictability, randomness and sovereignty. And Math has no way of generating randomness.
RandomnessGenerating 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 can not 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 can not 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 can not release its detailed accuracy, it loses connection with the real world around quantum level.
Hold that thought (8b):
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:
If you don't care much about numbers, skip ahead to a formal decapitation of all logical systems:
Helge von Koch