The following essay is going to attempt to show a relationship that doesn’t appear to be a complete description. However if you follow along you, the reader, may find within its contents that the idea has some merit.
You, the reader, may be familiarized with the theory of finding a series using a logarithmic spiral from this website, Constructors Corner. It proposes that any series (a pattern among numbers) can be described by a logarithmic spiral.
But the question arises: Is the theory true? Can every pattern be described by a logarithmic or approximate logarithmic spiral?
In order to answer those questions we, the mathematicians, must find a case in which the logarithmic spiral can not find the pattern. My guess is you, the reader, is thinking Prime numbers. Well that guess may or may not be true. That is, Prime numbers might be able to be described by a logarithmic spiral.
Back to the original question: Can we, the mathematicians, find a series that cannot be described by a logarithmic spiral? And the answer is probably “Yes”.
Think of 3 dice that are each a 6 sided cube. Each of the sixth sides corresponds to numbers 1 through 6. Using these dice a three digit number is chosen. The interesting part is that this number appears to be random with no pattern, however, the dice had to be turned a certain direction to get this number. So if another 3 digit number is chosen there would appear to be no relation to the first number, again, however, there is a geometric process or direction that the dice must turn to end up with the second, 3 digit number.
So even though the numbers appear random there is a pattern because the position on the dice is known. This is a complicated pattern. Imagine it increasing exponentially with numbers in the millions or more turns on dice of a greater number of sides! There would appear to be no pattern at all. Or at least not one which could be solved.
Does this describe Primes? I believe that all the processing power in the World would not solve a Prime number pattern. It would have to be done graphically. Does a logarithmic spiral or a logarithmic spiral shifted by a function solve it? It is probably wishful thinking or estimated concept. However it is a start and may lead to better ideas.
By the way, the dice analogy came from looking at “Christmas Day” days till Christmas blocks. The numbers of the day move in a pattern according to the placement of number of the day on the dice.
Also if you could solve a pattern or series in this way you, the cryptographer, could solve many types of encryption by finding a pattern where none seems present. Imagine if we took the original 3 dice and rolled them. That would be comparable to encrypting them. It may appear we have no way to decrypt the numbers. However if we know the geometry of the dice, all the numbers and turns will produce that encrypted number a certain number of ways. In other words we can eliminate many of the options if the “numbers and position” that fill the dice don’t line up. The pattern only fits so many ways. It is not meant to be implied that this is an easy task. But a geometric model (maybe a logarithmic spiral) may help.
But until then...
May the Creative Force be with You
Hopefully it is clear what is being attempted to be solved here. I will post updates to better explain and hopefully solve this problem. This is a good group project. If you have read this and want to work on a problem email: email@example.com . Also more math can be found in the math_hunches section of Constructor’s Corner.