Ulam Spiral

A simple Ulam spiral generator written in Crystal.

Place integers on a grid in a spiral pattern starting from the center, then highlight the primes. What you get is unexpected: the primes line up along diagonal lines. Nobody fully understands why.

Stanisław Ulam discovered this in 1963 while doodling during a conference talk. It has been a curiosity in number theory ever since.

Ulam spiral 800x800

Build

shards install
crystal build ulam_spiral.cr --release

Usage

# White primes on black, 800x800
./ulam_spiral -w 800 -h 800

# Pick your own color
./ulam_spiral -w 1000 -h 1000 -c "#00FF88"

# Quick run without compiling first
crystal run ulam_spiral.cr -- -w 400 -h 400

The center pixel is always yellow. Everything else is black (background) or your chosen prime color.

Green variant

How it works

Start at the center of the image
Walk outward in a spiral (right, up, left, down, ...)
At each position, check if the current number is prime
If it is, paint that pixel

Here's a 7x7 example. First, lay out the numbers in a spiral from the center:

 37  36  35  34  33  32  31
 38  17  16  15  14  13  30
 39  18   5   4   3  12  29
 40  19   6  [1]  2  11  28
 41  20   7   8   9  10  27
 42  21  22  23  24  25  26
 43  44  45  46  47  48  49

Now keep only the primes. Notice the diagonals:

 37   .   .   .   .   .  31
  .  17   .   .   .  13   .
  .   .   5   .   3   .  29
  .  19   .   .   2  11   .
 41   .   7   .   .   .   .
  .   .   .  23   .   .   .
 43   .   .   .  47   .   .

Scale that up to thousands of pixels and the diagonals become unmistakable.

There's also a 4K example in the repo if you want to zoom in.