Demos
Featured Cubic Lines Live
Blowing up the plane at six points to give a cubic surface
Featured black hole embedding
Isometric embedding of the optical geometry of a Schwarzschild black hole into $\mathbb{R}^3$
Featured hyperbolic disk embedding
Isometric embedding of a hyperbolic disk in Euclidean space with adjustable mesh parameters
Featured hyperbolic disk embedding gradient descent
Hyperbolic disk embedding via gradient descent with switchable optimization methods
Featured geodesic board: nyc II
Interactive geodesic board with six surface presets showing how curvature governs geodesic deviation
Featured geodesic board: nyc I
Interactive geodesic board: drag to aim geodesics across interchangeable curved surfaces
Featured Torus Mcg
Mapping class group of the torus acting on texture coordinates via Dehn twists and the Anosov map
Featured black hole loop
Light ray tuned to orbit a Schwarzschild black hole one or more times before escaping
Featured Pathtrace Sim Single
Pedagogical GPU pathtracer: single ray bounce per pixel
Featured Gradient Descent Domain
Gradient vector field over the domain of a surface, revealing critical points and basins of attraction
Featured Code: Wormhole
Raymarched traversable connect sum "wormhole" connecting two SF and Paris
Featured Mercator
Mercator projection: conformal map from sphere to plane preserving compass bearings
Featured Double Pendulum
Double pendulum exhibiting chaotic sensitivity to initial conditions
Featured Geodesic Billiards
Billiard trajectories following geodesics on a curved surface, reflecting off boundaries
Featured Quantum Elliptical Billiards
Quantum wave packet in an elliptical billiard domain, developing self-interference patterns
Featured The Double Slit Experiment
Double slit experiment: quantum wave packet producing interference fringes
Featured Iterated Integral Cartesian
Fubini's theorem: double integral evaluated as iterated 1D integrals in Cartesian coordinates
Featured Boys Surface
Boy's surface: an immersion of the real projective plane in $\mathbb{R}^3$ without singular points
Featured Partial Derivatives
Partial derivatives as slopes of cross-sectional curves on a surface $z = f(x,y)
Featured Contour Slicing
Horizontal slicing plane intersecting a surface to reveal its level curves
Featured Orbital Density
Volumetric rendering of hydrogen orbital densities $|\psi_{n,\ell,m}|^2$ via GPU ray marching
Featured Sphere and Cylinder
Archimedes' theorem that horizontal slices of a sphere and its circumscribing cylinder have equal area
Featured Spherical Harmonics
Spherical harmonics $Y_{\ell m}$ displayed as vibrating spheres, modulus plots, and polar plots
Featured Fubini
Interactive visualization of Fubini's theorem, showing iterated double integrals as cross-sectional slices
Featured Code: Connect Sum Torus Euclidean
Inside a connect sum of tori with Euclidean geometry, rendered by ray marching
Featured SpringCloth
Cloth simulation via a grid of spring-connected vertices with gravity and drag
Featured Knot Complements
Boundary torus of a tubular neighborhood of torus knots in $S^3$, under stereographic projection
Featured Dynamics & Strange Attractors
Six strange attractors (Aizawa, Chen, Dadras, Rössler, Sprott, Thomas) rendered as trajectories in three-dimensional space
Featured Curvature of Surfaces
Gaussian and mean curvature on surfaces, color-coded to show elliptic, hyperbolic, and parabolic regions
Featured Wave Eqn Flash
2D wave equation with a localized initial pulse, demonstrating the failure of Huygens' principle in two dimensions
Featured Wave Eqn Refraction
2D wave equation with spatially varying speed, demonstrating refraction and Snell's law
Featured Complex Zn
Graphs of complex power functions $z^n$ projected from $\mathbb{C}^2$ to $\mathbb{R}^3$ with domain coloring
Featured Nil Geodesics Earth Animate
Expanding geodesic spheres in Nil geometry, animated to show increasing distortion at larger radii
Featured Sol Geodesic Spheres
Geodesic spheres in Sol geometry, stretched by the exponential expansion and contraction
Featured Fourier Series
Fourier series partial sums converging to target functions, showing trigonometric decomposition
Featured Hopf Complement
Geodesics in the complement of a Hopf fiber in $S^3$, stereographically projected to $\mathbb{R}^3
Featured Punctured Disk Geometrization
Hyperbolic metric on a punctured disk, showing the cusp that forms at the removed point
Featured Code: 3torus Double
Inside a 3-torus manifold double connected by a wormhole, with earth and moon orbiting through the periodic space
Featured Code: Schwarzschild Accretion Disk
Schwarzschild black hole with a luminous accretion disk, rendered by geodesic ray tracing
Featured Code: Air Lens
Light bending through a varying refractive index gradient in Euclidean space, producing a lens effect
Featured Square Orbit
Nested tube-shaped orbits interpolating between circles and squares
Featured Code: Schwarzschild Earth
Gravitational lensing of an earth-textured sphere by a Schwarzschild black hole, producing multiple images
Featured Code: Sol Hyperbolic Planes
Totally geodesic hyperbolic planes in Sol, forming two orthogonal foliations
Featured Code: Nil Earth Moon
Earth and moon in Nil geometry, with viewpoint-dependent distortion from non-commutative translations
Featured Code: Hyp Coxeter Dodecahedron
Coxeter group tiling of $\mathbb{H}^3$ by right-angled regular dodecahedra
Featured Code: Hyp Coxeter Cubes
Coxeter group tiling of $\mathbb{H}^3$ by right-angled hyperbolic cubes
Featured Code: S2e Earth S2xs1
Earth sphere in $S^2 \times S^1$, the compact product of a 2-sphere with a circle
Featured Code: Sph Earth
Earth-textured sphere in $S^3$, with the antipodal image visible from behind
Featured Hyp Lattice
Horosphere lattice in hyperbolic 3-space with cusp acceleration for efficient raymarching
Featured Nil Lattice Fast
Performance-optimized Nil geometry lattice using approximate distance functions