🌐 Poincare Disk (Hyperbolic Geometry)

🌐 What It Demonstrates

The Poincaré disk model represents the entire infinite hyperbolic plane inside a finite Euclidean unit disk. In this model, hyperbolic "straight lines" (called geodesics) are represented either as diameters of the disk, or as arcs of circles that meet the boundary circle at exactly 90° (perpendicular to it). Given two points inside the disk, the unique geodesic through them is found by constructing the circle through both points that crosses the unit circle at right angles — this simulator computes that circle directly using the standard inversion-geometry formula, then draws the correct arc, rather than approximating it with a straight line.

This model gives a concrete picture of Euclid's parallel postulate failing: given a geodesic and a point not on it, there are infinitely many distinct geodesics through that point that never meet the first one ("parallel" in the sense of never intersecting) — not just the single unique parallel line Euclidean geometry guarantees. Hyperbolic triangles built from three geodesic edges also have angle sums strictly less than 180°, with the "missing" amount (the angular defect) growing larger the bigger the triangle is, unlike Euclidean triangles where the angle sum is always exactly 180° regardless of size.

How to Use

• Click inside the disk to place up to 6 points; drag existing points to reposition them. The simulator connects them in order with true hyperbolic geodesic arcs, forming a hyperbolic polygon.
• Toggle Show Euclidean straight lines to overlay ordinary straight-line segments between the same points for comparison with the curved geodesics.
• Toggle Show hyperbolic tiling overlay to see a recursively reflected {7,3} (order-3 heptagonal) or {5,4} (order-4 pentagonal) tiling pattern for visual context — an approximate but geometrically plausible rendering of a true hyperbolic tessellation, clearly not an exact edge-to-edge Möbius-transformation tiling.
• Watch the Statistics panel for the polygon's angle sum, compared with the fixed Euclidean expectation for a polygon with that many sides.

Did You Know?

The Poincaré disk model was popularized by Henri Poincaré in the 1880s, but the underlying hyperbolic geometry dates to the independent 1820s–30s work of János Bolyai and Nikolai Lobachevsky, who showed that a fully self-consistent geometry exists in which Euclid's fifth postulate simply does not hold. The Dutch artist M. C. Escher used the Poincaré disk model (introduced to him by the mathematician H. S. M. Coxeter) as the basis for his famous "Circle Limit" woodcut series, tiling the disk with repeating, shrinking fish and angel/devil motifs that all represent congruent shapes in genuine hyperbolic space, despite looking smaller and smaller toward the disk's edge.

About this simulation

This simulator implements the Poincaré disk model of hyperbolic geometry. Given any two points you place inside the unit disk, it computes the real geodesic — the circular arc perpendicular to the boundary circle (or the diameter, in the degenerate case) — connecting them, using the standard construction from inversive geometry, and reports the resulting hyperbolic polygon's angle sum, which is always less than the Euclidean value for the same number of sides.

Frequently asked questions