The {p, 3} Tilings
A {p, 3} tiling places regular p-gons (p-sided polygons) with exactly
3 meeting at each vertex. The interior angle of a regular p-gon is (p-2)·180°/p.
For 3 polygons to fit perfectly around a vertex, we need their angles to sum to exactly 360°:
This is why hexagons tile the flat plane. For p < 6, the angles sum to less than 360° — there's angular deficit, and the surface must curve positively (sphere). For p > 6, the angles exceed 360° — there's angular excess, forcing negative curvature (hyperbolic space).
As p → ∞, the polygons become "apeirogons" (infinite-sided). Three apeirogons meeting at a vertex gives the ideal triangle tiling of hyperbolic space — exactly the structure of Ford circles, where each rational number is an ideal vertex.