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The Spiral Story

Three years ago I played with Ruby and RMagick to generate pictures. I was always interessted in prime numbers, so I made pictures of the distribution of them. One day the Idea came up to order them in the form of a spiral (meanwhile, exactly since 4 weeks I know, that the mathematician Stanisław Ulam in 1963, did the same thing (You can start here). But, who cares? I lived 3 years in the illusion, that I have invented something very nice and becuse I didn’t know anyting and still do not know much of Ulam’s work, I had the chance to do other things then Ulam. (I hope this is the case and if not, I beg all Innovators pardon, that I haven’t had the time to read all the stuff about this field, to cite them here.)

Primes, Seconds, Thirds, ...

The natural numbers can be classified in a very natural way if we start by the class of prime numbers.

1 has only one divisor.


Primes have two

2, 3, 5, ...

Seconds have three

4, 9, 25, ... (Primes**2)

Thirds have 4

6, 8, 10, 14, 15, ...

Nths have n+1

Infinite many classes. Some of them are pictured here on this page (this was form the time I thought I found them first)

I think very nice structures become visable.

The City

I think this is obvious. The Ulam Spiral looks like a map of a town with sometimes complex buildings, streets and places. I think someone should build this town and give it to the mathematic people. (So not in reality, but for example an Island in a virtual reality like 2nd Life, would be a nice place.)

The houses (connected figures) in the center (there is one big building in the center, that should be the community center) should be dedicated to people like Eratosthenes, Pythagoras, ..., Euler, Gauß and so on. And there is place for everyone interessted in math. Here a picture (sorry wrong proportions):

Higher Dimensions

I came up with the idea of spirals (windings that are spirallike) in higher dimensions.

Something like this (3D-Ulam):

Ok I think there are 48 different ways to wind up in a spiral alike regular way in 3 dimensions and 96 with changing orientations. This leeds me to the Question:

Why did I do the primespiral only one way?

There are 8 ways to do the spiral in 2D. What happens if we overlay them?


Here are more Infos and a Zipfile with the single pictures

By the way, this 8 fold spiralisation stuff leeds to many many very nice pictures!

and with iterating only over the 8 fold overlapping Ulam Spiral:

Here are more Infos

Why did I do the spiral inside out?

Ok, because there are infinite many prime numbers! So we need the place to expand. But what about a iterative prozess, that shows us the outside in spirals step by step?

Here are more Infos

Spiral Symmetry


to bee continued …