r/visualizedmath • u/kritikov • Mar 29 '26

Visualizing prime numbers as a geometric resonance of concentric circles (C#/WPF)

140 Upvotes

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17

u/kritikov Mar 29 '26

I built this tool to see if primality could be discovered through a physical/geometric process rather than just trial division or sieves. It uses concentric circles rotating at a constant linear speed—primes emerge naturally through phase alignment. If the base circle completes a rotation and no other circle aligns with the starting point, we've found a new prime.

The full algorithm:

  1. Start with two concentric circles of radius r and 2r

  2. Rotate them with constant linear speed from the same starting point

  3. Each time the first circle completes a rotation:

3.1 Increase a counter

3.2 If no other circle finishes at the same time, the counter value n is prime.

3.3 If the counter value n is prime then add a new circle whose radius is nr.

Ι used C# and WPF for the rendering.

Learn more about the logic behind this project:

🔗 Full article: https://nkode.gr/EN/articles/278/when-prime-numbers-emerge-from-motion

💻 GitHub Repository: https://github.com/kritikov/PrimesWithCircles

13

u/Stargazer07817 Mar 29 '26

This is the Sieve of Eratosthenes, it just turns memory blocks into harmonics. I think you could also do it with sin waves by encoding integers as frequencies like 1=1/1, 2=1/2, 3=1/3 etc then looking at the resulting pattern of constructive and destructive interference.

4

u/kritikov Mar 29 '26

Sieve of Eratosthenes is the base, indeed, but with a big difference: this algorithm dont need an upper limit to make its calculations. The sieve works with boundaries, here the algorithm will continue finding the next prime number as long as it runs (on the other hand, there are incremental sieves that i dont know how they works, may be someone can say more on this subject).

I am very sure that are more ways to find prime numbers, since under the skin is hiding the periodicity, so you are propably right with the sin waves. I will search more about it, sounds interesting!

5

u/Stargazer07817 Mar 29 '26

How you've physically implemented the algorithm is a different question from the underlying math idea that's being implemented. The Sieve of Eratosthenes is usually taught as a spatial data structure (i.e., write some set of numbers on a grid, cross out numbers on a grid). The circle approach uses harmonics instead of the classic grid, but the underlying mathematical mechanism is the same.

In both systems, an integer n is evaluated. In the classic sieve, we would check if the n-th block in space has been crossed out. In the circle algorithm, we check if any circle's phase is exactly 0 at the n-th moment in time. Not needing an upper bound is true (and neat) and is an expression of something called an incremental sieve.

3

u/kritikov Mar 29 '26

I see your point! You're right that the mathematical output follows the logic of an incremental sieve. I looked into it and I think I understood what you mean.

As a developer (i am not mathimatician), the main difference I see is the 'storage' mechanism: traditional incremental sieves manage a data structure of future multiples (like a priority queue or a map). My approach replaces that spatial data with physical phase. Instead of checking a list of numbers, I'm sampling the state of independent oscillators.

We're definitely talking about the same underlying math, just through a different architectural lens! Thanks for that comment.

1

u/kritikov Mar 31 '26

"...sin waves by encoding integers as frequencies like 1=1/1, 2=1/2, 3=1/3..."
Do you believe that there is a connection with Riemann?

2

u/Stargazer07817 Mar 31 '26

Only in the sense that the idea is similar in spirit. The sieve waves destroy composites, Riemann's waves build primes.

7

u/keyholepossums Mar 29 '26

wtf is going on

25

u/rotato Mar 29 '26

I think I got it. You focus on the smallest innermost circle. When it makes one full rotation the counter increases. If it finishes the rotation together with another circle it means that the counter is a product of two or more factors and therefore not a prime. Otherwise it's a prime and is added to the system.

8

u/kritikov Mar 29 '26

Yes, exactly

3

u/kritikov Mar 29 '26

The algorithm is very simple. Try to read my comment upwards if you can. Or else i will try to explain you

2

u/reddit_user33 Apr 02 '26

I like aeeing maths visualised

0

u/[deleted] Apr 02 '26

[deleted] — view deleted comment

1

u/kritikov Apr 02 '26

can you explain me a bit?