Where Parallel Lines Meet (and Don't)
For 2,000 years, Euclid's fifth postulate—that parallel lines never meet—seemed self-evident. Then mathematicians discovered entire universes where it fails. In hyperbolic space, infinitely many parallels pass through a single point. On a sphere, every pair of "straight lines" eventually crosses. These aren't abstract curiosities: Einstein showed our actual universe has non-Euclidean curvature. Explore 20 interactive visualizations that bend your intuition about the shape of space itself.
Venture into the Poincaré disk, where space expands infinitely toward the boundary, triangles have too-small angles, and beautiful tessellations tile the hyperbolic plane.
Draw geodesics—the "straight lines" of hyperbolic space—that appear as beautiful arcs in the Poincaré disk model.
Generate {p,q} tessellations with regular polygons that only tile in hyperbolic space—impossible patterns in flat geometry.
Recreation of Escher's Circle Limit III with animated hyperbolic tilings of interlocking figures shrinking to infinity.
Draw triangles whose angles sum to less than 180°. Watch the angle defect grow as triangles get larger.
Euclid's parallel postulate fails: through any point, infinitely many lines are parallel to a given line.
Hypnotic spiraling geodesics, horocycles, and equidistant curves creating mesmerizing patterns in hyperbolic space.
Draw in one wedge and watch infinite hyperbolic reflections create stunning kaleidoscopic patterns.
Animate Möbius transformations: rotations, translations, and reflections that preserve hyperbolic distance.
On a sphere, all "straight lines" are great circles that wrap around and intersect. Triangles have too-large angles, and you can't comb a hairy ball flat.
Draw triangles on a sphere where angles sum to more than 180°—the larger the triangle, the greater the excess.
Why flights from NYC to Tokyo go over Alaska: the shortest path on a sphere is never a straight line on a flat map.
Platonic solids inflated to spherical surfaces, morphing between icosahedra, dodecahedra, and geodesic domes.
Try to comb a vector field flat on a sphere—topology guarantees you'll always create at least one cowlick.
Every flat map of Earth is a lie. These demos reveal exactly how and why, showing the beautiful mathematics of projection and distortion.
Smoothly morph between Mercator, Gall-Peters, and other projections, revealing how each distorts the world.
Equal circles on a globe stretched into ellipses by map projections—the classic tool for measuring distortion.
Project a sphere onto a plane from its pole—the unique projection that preserves all angles perfectly.
Beyond flat and curved surfaces lie stranger topologies: one-sided bottles, four-dimensional cubes, and spaces that smoothly shift between geometries.
A surface with no inside or outside, visualized as a 3D immersion with an animated ant crawling its impossible surface.
A 4D hypercube rotating through the fourth dimension, its shadow dancing in our 3D world projected to your 2D screen.
Pac-Man's world is a torus: straight lines that exit one side re-enter the other, creating surprising periodic patterns.
First-person navigation through a hyperbolic maze—infinite corridors that defy Euclidean expectations.
Watch space continuously transform from spherical to flat to hyperbolic, morphing tessellations and geodesics in real time.