Prime Number Sieve and the Ulam Spiral

The Ulam Spril was publicized by Stanislaw Ulam in 1963. The spiral can be generated using the sieve of Eratosthenes to etch out the prime numbers. This animation depicts 123 iterations of the sieve. Initially, the Ulam Spiral is drawn and the primes are colored dark red. Those primes that have a rank, the order that the prime appears in the prime number sequence, that is also prime are colored yellow, and those that have a rank that is a prime with a rank that is also prime are colored blue. The sieve begins with 2 and 'evaporates' every other cell. When a cell is evaporated, the cells ahead of it in the spiral shift back to fill the vacant space but they retain their original value for future division tests. The next iteration evaporates those cells divisible by 3, and then 5, 7 and so on. Red cells(primes) change to black when they are less than p^2. Thus, the area of the inner square is π(x)^2 (the prime counting function). What is left after all composites have been evaporated and the newly created black cells are interperted as composites is an exact replica of what we started with, only shifted in the hierarchy of primeness so to speak. The path of a couple of prime points is traced as they move to their final destination. Music by: cloudtalk, Austin Texas.

Category:

Science & Technology

Tags:

• prime
• number
• numbers
• ulam
• spiral
• fractal
• sieve
• Eratosthenes
• math
• mathematics
• beauty
• shawn
• paske
• cloudtalk
• animation

[Werelion2367]

5 stars, this is different than the other "prime spiral" formed by crossing out primes, like the one of the mathworld site right?

[spaske2528]

Thanks! This animation is similar to the mathworld demonstration in that it uses the sieve of Eratosthenes. However, when a composite is 'scratched out', those numbers above the composite, shift back to fill the empty space. When all composites are scratched out what is left is only primes. However, the original coloring arrangement of the primes leads to the Ulam Spiral.

[spiritfulness]

Would love to see you animate the Croft Spiral Sieve!

[pipolopulus]

spiral?

[llladelll]

what program can i use to paint this points, im trying with mathlab but i´d like to know another software

[Buzzlybonk]

The noice looked like waterdrops hitting a watersurface for awhile, anyway it looks also like a language of some sort

[kaleidostar88]

I more or less understood all of that until you started explaining why the cells turn black then you lost me

[luked82]

Any Calc fans out there? I have a problem I want to see if anyone can solve it. I did. You can find the problem at the end of my video called String Art is Calculus.

[W4d5Y]

I don't know what's going on here but I actually recognized the last images to be those of the prime numbers of I think up to ten k something, when all integers are arranged in a spiraling fashion.

[forgotmypassword3]

waves?

[cobaltdan9]

this is awesome