Poincaré Disk Projection
Boundary: |z| < 1
Tessellation: None (Continuous Horocycle)
Angle Deficit: N/A
Isoperimetric Proof & Measurements
hyperbolic_linear_growth
GEOMETRIC MANIFOLD SHAPE
HYPERBOLIC TILING {p,q}
{5, 4}
0.65
-1.00
EUCLIDEAN BENCHMARK
Law: L² ≥ 4πA
Area: 1.3273
Perimeter: 4.0841
Ratio L/A: 3.0769
HYPERBOLIC HOROCYCLE
Law: L² ≥ 4πA + A²
Area: 3.8245
Perimeter: 4.1888
Asymptote L/A → 1.0000 (Linear!)
PERIMETER GROWTH COMPARISON: L(A)
Euclidean (√A) vs Hyperbolic (A)
Spatial Paradox: In flat space, expanding a circle makes its area grow quadratic in radius ($A \propto r^2$), while boundary perimeter grows linearly ($L \propto r$), causing the ratio $L/A \to 0$. In hyperbolic space, perimeter grows exponentially ($L \approx e^\rho$), keeping pace directly with enclosed area ($L/A \to \sqrt{-K} = 1$). A massive hyperbolic shape cannot isolate its interior from its border!