A geometry where parallel lines diverge, triangles lose their angles, and an infinite universe compresses into a disk. Experience it before you try to understand it.
For two thousand years, Euclid's geometry was the only geometry. Parallel lines stayed parallel. Triangles summed to 180°. Then in the 1820s, Gauss, Bolyai, and Lobachevsky proved another geometry existed — internally consistent, equally valid, and deeply strange.
Drag the slider. Watch flat space bend into hyperbolic space. Parallel lines stop behaving the way Euclid promised.
The entire infinite hyperbolic plane can be represented inside a finite disk — the Poincaré disk model. The boundary of the disk represents infinity: it is infinitely far from every interior point.
Every fish below is the same size in hyperbolic geometry. Near the boundary they appear smaller only because we are projecting through Euclidean eyes. Each fish swims outward forever — approaching the boundary but never arriving.
Click anywhere in the disk to release a fish.
In Euclidean geometry, the angles of any triangle always sum to exactly 180°. This is not a fact about triangles — it is a fact about flat space. In hyperbolic space, triangle angle sums are always less than 180°. The farther the vertices are from the center, the more the angles shrink.
Watch what happens as the triangle's vertices approach the boundary of the disk.
This identity has no Euclidean analogue. In hyperbolic geometry, the area of a triangle is literally determined by its angles alone — larger triangles have smaller angle sums. Geometry and measurement are intertwined in a way Euclid never imagined.
In hyperbolic geometry, the shortest path between two points — a geodesic — appears curved to our Euclidean eyes. It is an arc of a circle that meets the boundary of the Poincaré disk at right angles. From inside the space, it is perfectly straight.
Can you guess where the geodesic goes? Your Euclidean intuition will mislead you. Try to click where you think the shortest path passes through before the answer is revealed.
Every geodesic in the Poincaré disk is an arc of a circle meeting the boundary at 90°. Click multiple points to build a web of geodesics. The resulting patterns are the fingerprints of hyperbolic geometry — the same symmetries that appear in the holonomy groups of hyperbolic 3-manifolds.
Click points to build · right-click to clear.
In Euclidean geometry, only three regular polygons tile the plane: triangles, squares, hexagons. In hyperbolic geometry, infinitely many do — because hyperbolic space has exponentially more room. Regular pentagons tile the hyperbolic plane. So do heptagons, octagons, and beyond.
M.C. Escher spent years studying these patterns. His "Circle Limit" series are hyperbolic tessellations — infinite patterns compressed into a finite disk, each tile shrinking toward a boundary it can never reach.
The most important quantitative difference between Euclidean and hyperbolic geometry: how much space is inside a circle of radius r.
This is where the abstract geometry becomes physically significant. Every cusped hyperbolic 3-manifold in the SnapPy census with cusp field discriminant −283 has volume that is an exact rational multiple of v₀ = 0.9814... — the volume of the Meyerhoff manifold, the minimum-volume closed hyperbolic 3-space.
Six manifolds. Six exact rational multiples. All verified to machine precision. Click any bar to reveal the details.
The following chain is the core claim of the Hyperbolic Flavor Geometry programme. Scroll slowly — each stage lights up as you read. The chain is astonishing not because any individual step is mysterious, but because all the steps connect.
Each arrow in this chain is either a proved mathematical theorem or a verified numerical result. The connection between hyperbolic 3-manifolds and particle physics is not an analogy or a metaphor. It is a precise mathematical claim, subject to verification and falsification.
Marvin L. Gentry · Independent Researcher · Seattle WA
SSRN ·
GitHub ·
hyperbolicflavorgeometry.org