Flowing Brush Strokes

The math, plainly

Every painting starts as a field of arrows: at each point of the canvas, a direction. The arrows come from two sources added together. Vortices — a few points the sky turns around, spinning hardest at a core radius and fading smoothly beyond it. And turbulence — a lattice of random values smoothed between grid corners, stacked at several zoom levels so small eddies ride on big ones. That sum is the whole physics — each brush just shapes the vortex falloff its own way. There is no simulation, just geometry.

v(p) ≈ Σᵢ sᵢ · (−Δyᵢ, Δxᵢ) · r₀ᵢ / (r² + r₀ᵢ²)  +  c · curl N(p)        Night,  r = |p − cᵢ|
N(p) = Σₒ aᵒ · n(λᵒ p),  a ≈ ½,  λ ≈ 2        n = lattice noise, corners blended by t²(3 − 2t)

Strokes are not scattered at random; pure scatter reads as static, not brushwork. A new stroke may only begin a set distance from the ones already painted — one brush lays its candidates on an evenly-spread sequence, another spawns each stroke beside the last — and a coarse grid of already-painted cells enforces the spacing. Each start point is then traced along the arrows, so neighbors run parallel and comb around the eddies the way brush hair lies.

pₙ = ( {½ + n·0.754877…},  {½ + n·0.569840…} )        R2 low-discrepancy sequence
grow while dist(other strokes) > 0.52·d_sep        spawn neighbors at ±1.05·d_sep

A stroke is not a line but one filled shape: a belly that tapers to a point at both ends, laid at 55–96% opacity. Many carry a thin lighter ridge along their lit side — paint catching light. That, not blur, is what makes flat pixels read as raised paint.

w(t) = w_max · (0.30 + 0.70·sin(πt)^0.70),  α = 0.60 + 0.32·ξ        Night
w(t) = w_max · (0.22 + 0.78·sin(πt)^0.62),  ridge + underside        Impasto

The three brushes are three answers to where the strokes should go. Night currents composes a picture first: a moon, sometimes a second star, halo rings, a combed dark ground. Impasto cares about the paint itself: fatter ribbons with bristle streaks, an underside shadow, seams of canvas left showing. Cypress lets the line loose: long licks that overshoot around the vortices, and dark flame-shapes climbing the sky.

And none of it is random at the moment you look. A seed — one number, 0 to 9,999 — feeds a small generator (mulberry32) that deals out every "random" decision in a fixed order, like a shuffled deck dealt the same way twice. Same seed, same screen, same brush, same sliders: the same painting, stroke for stroke. Ten thousand seeds, three brushes, three palettes — ninety thousand paintings in this one file, before you touch a slider.

mulberry32:  s ← s + 0x6D2B79F5 (mod 2³²)  →  xor-shift mix  →  uniform [0, 1)
painting = F(seed, brush, palette, sliders, W, H)        bit-for-bit reproducible

The constants above are lifted straight from the source. The rest is one View Source away.