Every integer from 1 to N is mapped to a screen position with real coordinate math, not decoration: the Ulam spiral walks outward ring by ring on a square grid, while the Sacks spiral places n at polar radius r = √n and angle θ = 2π√n, so perfect squares fall on a single ray.
Primality is decided by a genuine Sieve of Eratosthenes computed up to N on every change — no lookup table, no faked colouring. The square and modulus-residue modes recompute their own membership test per n the same way.
Hover any plotted point to see its actual value and, for primes mode, its factorization computed on demand by trial division.
Ulam: ring k, side 2k, four straight segments
Sacks: r = sqrt(n), theta = 2*pi*sqrt(n)
Sieve: mark multiples of every i from 2..sqrt(N)