Hyperbolic Lattice: Negatively Curved Space

Poincaré disk model of {p,q} hyperbolic tilings

Tiling {p, q}

Hyperbolic: 1/p + 1/q < 1/2
Euclidean: 1/p + 1/q = 1/2
Spherical: 1/p + 1/q > 1/2
Hyperbolic geometry (negative Gaussian curvature) is realized in the Poincaré disk model where the entire infinite hyperbolic plane is projected into the unit disk. Geodesics are circular arcs perpendicular to the boundary. A {p,q} tiling places regular p-gons so that q meet at each vertex. When (p−2)(q−2)>4, the tiling is hyperbolic: the number of tiles grows exponentially with radius, unlike the polynomial growth in flat Euclidean space. Hyperbolic lattices arise in AdS/CFT, holographic quantum error correction, and novel band theory with flat bands.