Algebraic numbers glow dust
Using WebGL for realtime approximation. See it live: www.glslsandbox.com/e#42524.0
An aggregation of the complex roots of
a₈z^8 + a₇z^7 + a₆z^6 + a₅z^5 + a₄z^4 ... +a₁z +a₀ = 0
where a₀ = 1 and a₁...a₈ = 1 or 0.
Rather than computing the roots, this visualization takes advantage of GPU parallelism--it evaluates the polynomials at each pixel and sets brightness according to the proximity to zero. Brightness is additive, so nearby roots create brighter, larger regions.
The brightest points lie on the unit circle, representing the various roots of unity: -1, ±i, (1±i√3) / 2, etc. Note that 1 does not appear, as no positive real roots exist for this set of polynomials.
Different colors are assigned, arbitrarily, according to the coefficients of the first and second powers of z.