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POINCARÉ HYPERBOLIC DISK

Negative Gaussian curvature, Möbius transformations, and M.C. Escher hyperbolic tessellations.

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POINCARÉ DISK H²

// REGULAR {p, q} SCHLÄFLI TILINGS

POLYGON p 7

VERTEX q 3

RECURSIVE TILING DEPTH 4

MOUSE DRAG: Hyperbolic Möbius Translation | 1/p + 1/q < 1/2: Hyperbolic Invariant | BOUNDARY: Infinite Horizon

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RESEARCH & LEARNING VAULT // Poincaré Hyperbolic Disk & Non-Euclidean Tessellations

Infinite hyperbolic geometry H², Möbius automorphisms, and regular {p, q} Schläfli tilings.

ACADEMIC & ALGORITHMIC REFERENCE

// HISTORICAL ORIGINS

Henri Poincaré introduced the conformal disk model in 1882. Geometers and artists like M.C. Escher (Circle Limit I–IV) used hyperbolic geometry to depict infinity within finite circular boundaries.

// GOVERNING EQUATIONS

d(z_1, z_2) = 2 \operatorname{artanh} \left| \frac{z_1 - z_2}{1 - z_1 \bar{z}_2} \right|, \quad z \mapsto \frac{z - w}{1 - \bar{w} z}, \quad \frac{1}{p} + \frac{1}{q} < \frac{1}{2}

// BROWSER IMPLEMENTATION

Generates regular {p, q} hyperbolic polygons by constructing orthogonal Euclidean circle arcs (geodesics) and translates the infinite plane in real time via complex Möbius transformations.

// GUIDED EXPERIMENTS TO TRY IN THIS SANDBOXNORMAL

1Möbius Warp & Recenter

🎯 Action:Click and drag near the disk boundary to pull infinitely small polygons into the center.

✨ Observe:Notice how shapes expand naturally into full Euclidean proportions at the origin without angle distortion (conformal property).

// CURATED PAPERS, RFCS & RESOURCES

Theory of Fuchsian Groupspaper

By Henri Poincaré (1882)

Poincaré's breakthrough work establishing conformal representations of non-Euclidean hyperbolic space.

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POINCARÉ HYPERBOLIC DISK

RESEARCH & LEARNING VAULT // Poincaré Hyperbolic Disk & Non-Euclidean Tessellations