The Ball and the Quilt

Begin with the eight spherical triangles of the octahedron. Refine each into four. The counts on both sides of the fold grow in lockstep:

Triangles grow by 4^k; pixels grow by 4^k. A bijection between spherical triangles and square pixels therefore exists at every level. The discretizations do not merely resemble each other — they match exactly.

Let R be refinement (triangle → four triangles) and π the sphere→square projection. Refinement uses midpoints and straight edges; projection preserves midpoints and edges; sub-triangles map to sub-triangles. Therefore:

The mapping is structurally exact, not approximately faithful. Below, the refined spherical mesh is pushed through π and lands on a clean subdivided square — the commuting diagram, drawn.

Angular spacing between samples behaves as

Doubling k multiplies triangles by four and halves angular spacing: linear gain in precision, exponential cost in geometry. This is the complexity loss, and it is predictable rather than chaotic, because refinement is symmetric and uniform.

Angular resolution does not depend on radius; physical patch size does. Two hundred fifty-six squares look like glass on a marble and like a quilt on a planet. That is why the renderer above shades with Lambert diffusion, a tight specular lobe and a rim term: the eye needs a light model, not more triangles, to accept a sphere as a solid body in three dimensions. Nothing here is a higher-dimensional diagram pretending to be a ball.

The metrics panel in the studio is computed from the actual polyhedron, not from a formula for a smooth sphere. Volume comes from summing signed tetrahedra from the centre; area from triangle cross products; the inradius from the minimum face distance; the solid angle per face from 4π / T_k. As k rises, the deficits against 4πr³/3 and 4πr² fall in a clean geometric ladder — you can watch the polyhedron become the sphere.

Volume alone is a poor description of a ball. Edge length says how the mesh feels to walk; inradius says how deeply the discretization cuts; arc Δs says how large a patch is in the world; solid angle says how much sky a single pixel of the quilt owns. These together are the disposition of the ball at octave k.

The wider laGEOsis repository carries a second eight-triangle system projected by another method. The two are relatable exactly: both start from the octahedral face set of eight, both refine by the same operator R, and both admit a fold of the lower hemisphere into the corners of a square. They differ only in the metric of the fold — one preserves the L1 norm (|x|+|y|+|z| = 1, the mapping used here), the other distributes area differently across the seam.

Because R commutes with both projections, any statement proved at octave k in one system transfers to the other by a per-face reparametrisation that is smooth away from the seam. The seam — the equator fold, glowing gold in the quilt above — is where the two systems disagree, and it is exactly the locus that Laegna Physics reads as the membrane between interior and exterior.