Hyperbolic Lattice: Poincaré Disk Tessellation
{p,q} tilings • Gaussian curvature K = −1 • hyperbolic geodesics
Tessellation {p, q}
p (polygon sides)
4
q (vertex valence)
5
Depth (layers)
4
Color mode
Depth gradient
Rainbow
B&W wireframe
Hyperbolic plane H²
Gaussian K = −1/R²
{p,q} valid iff:
1/p + 1/q < 1/2
Angle deficit per vertex:
π − 2π/p − 2π/q > 0
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Show Geodesics
The Poincaré disk model maps the infinite hyperbolic plane into the unit disk. Straight lines (geodesics) appear as circular arcs meeting the boundary at right angles. Infinite regular tilings {p,q} with 1/p+1/q < 1/2 exist only in hyperbolic space.