Demos

Solving the schrodinger equation inside the Koch Fractal, starting from a gaussian initial condition.

Wave Eqn Snowflake

The wave equation in a snowflake domain.

Wave String

One-dimensional wave equation $\partial_t^2 u = c^2 \partial_x^2 u$ solved on a vibrating string

Orbital Density

Volumetric rendering of hydrogen orbital densities $|\psi_{n,\ell,m}|^2$ via GPU ray marching

Sphere and Cylinder Deform

Animated deformation between a sphere and its circumscribing cylinder, preserving horizontal slice areas

Cylinder Eigenfunctions

Laplacian eigenfunctions on a cylinder, displayed as standing wave patterns

Orbital Isosurface

GPU-raymarched isosurfaces of hydrogen wavefunctions $\psi_{n,\ell,m}$, colored by sign

Sphere and Cylinder

Archimedes' theorem that horizontal slices of a sphere and its circumscribing cylinder have equal area

Spherical Harmonics

Spherical harmonics $Y_{\ell m}$ displayed as vibrating spheres, modulus plots, and polar plots

Torus Eigenfunctions

Laplacian eigenfunctions on a torus of revolution, color-coded and deformed to show standing wave patterns

Cylindrical Shells

Interactive calculator for the method of cylindrical shells, showing volumes of revolution

Hydrogen 2d Slice

Cross-sectional slice through the probability density $|\psi_{n,\ell,m}|^2$ of hydrogen atom orbitals

Square Drum

Standing wave modes of a vibrating square drum, given by products of sine eigenfunctions

Circular Drum

Vibration modes of a circular drum, given by Bessel function eigenmodes $J_m(\alpha_{mn}r)

Area Function

Interactive visualization of the accumulation function $A(x) = \int_a^x f(t)\,dt

Ftc Plotter

Interactive illustration of the Fundamental Theorem of Calculus, showing $F(x) = \int_a^x f(t)\,dt$ alongside $f

Volume by Slicing

Interactive calculator for computing volumes by the method of cross-sectional slicing

DisksAndWashers

Interactive calculator for the disk and washer methods of computing volumes of revolution

Genus2 Teichmuller

Right-angled hexagon tiling of the hyperbolic plane, with moduli tracing a path through genus-2 Teichmüller space

Newton Fractal

Newton fractal for a cubic polynomial in the complex plane, rendered in real time on the GPU

Projective Triangle Group

Triangle group tiling of the real projective plane with a continuously varying parameter

Right Angled Hexagons

Tiling of the hyperbolic plane by right-angled hexagons in the Poincaré disk model, with oscillating side lengths

Right Angled Pentagons

Tiling of the hyperbolic plane by right-angled pentagons in the Poincaré disk model, deforming through Teichmüller space

Schwarzschild Checkerboard

relativistic pathtracer

GPU ray tracer showing the visual distortion of a checkerboard near a Schwarzschild black hole

Accretion Disk Intrinsic

Intrinsic geometry of an accretion disk around a Schwarzschild black hole, embedded in three-dimensional space

Spherical Dinosaur

Spherical harmonic decomposition of a dinosaur mesh, reconstructed from a truncated expansion

Newtons Method

Interactive calculator for Newton's method, showing tangent-line iterations converging to a root

Fubini

Interactive visualization of Fubini's theorem, showing iterated double integrals as cross-sectional slices

Gaussian Geodesics

Geodesics on a Gaussian bump surface, showing convergence and divergence due to curvature

Gradient

Interactive calculator showing the gradient $\nabla f$ of a function $f(x,y)$ as a vector field

Riemann Sum 2D

Riemann Sum Sequence

Animated convergence of Riemann sums to a definite integral as the partition refines

Schwarzschild Geodesics

Geodesics of the Schwarzschild metric around a black hole, integrated numerically and displayed in three dimensions

Slope Field

Interactive slope field calculator for first-order differential equations $y' = f(x,y)

Vector Field 2D

GaussCurvature

Interactive Gaussian curvature calculator for surfaces $z = f(x,y)$, color-coded by sign

Riemann Sum

Interactive Riemann sum calculator with left, right, and midpoint sampling

Three Body

Gravitational three-body problem integrated numerically in real time, exhibiting chaotic orbits

Code: Connect Sum Torus Euclidean Hd

riemannian raytracer

High-resolution version of the Euclidean connect-sum-of-tori raymarcher

Code: Connect Sum Torus Euclidean

riemannian raytracer

Inside a connect sum of tori with Euclidean geometry, rendered by ray marching

SpringCloth

Cloth simulation via a grid of spring-connected vertices with gravity and drag

Spring Cube

Jello-like deformable cube simulated as a $16^3$ lattice of spring-connected vertices

Curvature & Torsion

Space curves reconstructed from prescribed curvature $\kappa(s)$ and torsion $\tau(s)$ via the Frenet–Serret equations

Hopf Tori

Flat tori in $S^3$ arising as Hopf preimages of closed curves on $S^2$, projected stereographically to $\mathbb{R}^3

Integral Curves

Integral curves of the Aizawa attractor, a chaotic three-dimensional dynamical system

Knot Complements

Boundary torus of a tubular neighborhood of torus knots in $S^3$, under stereographic projection

Dynamics & Strange Attractors

Six strange attractors (Aizawa, Chen, Dadras, Rössler, Sprott, Thomas) rendered as trajectories in three-dimensional space

The Shadow of a Cube

Stereographic projection of a cubical tiling of the sphere, with animated interpolation to the plane

Curvature of Surfaces

Gaussian and mean curvature on surfaces, color-coded to show elliptic, hyperbolic, and parabolic regions

The Fundamental Group of the Torus

Closed curves on a torus representing elements $(p,q) \in \mathbb{Z} \oplus \mathbb{Z}$ of the fundamental group

Stereographic Nullhomotopy

Nullhomotopy of a closed curve on $S^2$ and its image under stereographic projection to the plane

Torus Geodesics