Demos
Geodesic Ripples
Solvin the geodesic equations on a surface that looks like ripples in a pond.
Equirectangular Map
Equirectangular projection of the earth, but not always from the north pole.
Stereographic Map
Stereographic projection from the sphere to the plane, conformal and circle-preserving
Archimedes Map
Archimedes equal-area projection from sphere to cylinder, preserving area while distorting shape
Hemisphere Map
Hemisphere projection mapping the sphere vertically onto a flat disk
Featured Mercator
Mercator projection: conformal map from sphere to plane preserving compass bearings
Five Body
Five gravitationally interacting bodies with chaotic trajectories and close encounters
Geodesic Circles
Geodesic circles of varying radii on a curved surface, distorted by Gaussian curvature
Featured Double Pendulum
Double pendulum exhibiting chaotic sensitivity to initial conditions
Pendulum
Simple pendulum with full nonlinear dynamics, showing anharmonic motion at large amplitudes
Featured Geodesic Billiards
Billiard trajectories following geodesics on a curved surface, reflecting off boundaries
Norton Dome
Norton's dome: a Newtonian surface where determinism fails at the Lipschitz-singular apex
Physical Billiards
Classical billiards with reflections off boundary walls, integrable or chaotic depending on table shape
Gravity Star
A cartoon (not realistic) of light rays under gravity, for a talk.
Geodesics Graphing Calc
Interactive graphing calculator for geodesics on arbitrary user-defined surfaces
Geodesics Cosr
Geodesics on a $z = \cos(r)$ surface, traversing regions of alternating curvature
Geodesic Paraboloid
Geodesics on a paraboloid, wrapping around the positively curved vertex
Geodesics Gaussian
Geodesics on a Gaussian bump, deflected by curvature in analogy to gravitational lensing
Geodesics Sinusoid
Geodesics on a $\sin(x)\sin(y)$ egg-carton surface, weaving between peaks and saddle points
Bh Slice
Flamm paraboloid: isometric embedding of a constant-time spatial slice of the Schwarzschild geometry
Gravity Curvature Sphere
Geodesic deviation near a gravitating sphere, illustrating curvature as the origin of gravitational attraction
Tangent to Graph
Tangent line to $y = f(x)$ at a movable point, visualizing the derivative
Baumslag-Solitar
Cayley graph of a Baumslag-Solitar group, with tree-like branching from the defining relation
Graph3d
Interactive 3D graphing calculator for user-defined surfaces $z = f(x,y)
Schrodinger_barrier
Quantum tunneling: wave packet penetrating a classically forbidden potential barrier
Featured Quantum Elliptical Billiards
Quantum wave packet in an elliptical billiard domain, developing self-interference patterns
Particle in a Box
Quantum particle in a 2D box: probability density $|\psi|^2$ evolving as a superposition of eigenstates
Diffraction through Periodic Potential
Quantum diffraction of a wave packet through a periodic array of potential barriers
Featured The Double Slit Experiment
Double slit experiment: quantum wave packet producing interference fringes
Exp Branch Cut
Branch cut of the complex logarithm, showing the Riemann surface sheets near the negative real axis
Zn Branch Cut
Riemann surface of $z^{1/n}$: $n$ sheets cyclically connected along the branch cut
Complex Exponential Product
Modulus of a product of complex exponentials, revealing zeros and growth behavior
Complex Modulus
Modulus $|f(z)|$ of a complex function plotted as a height field over the complex plane
Featured Iterated Integral Cartesian
Fubini's theorem: double integral evaluated as iterated 1D integrals in Cartesian coordinates
Iterated Integral Variable Bounds
Double integral with variable bounds, illustrating Fubini's theorem on non-rectangular domains
Iterated Integral Polar
Double integral in polar coordinates, with wedge-shaped cross sections sweeping the domain
Sudanese Mobius
Sudanese Möbius band: a minimal non-orientable surface in $S^3$ projected to $\mathbb{R}^3
Kleinbottle
Klein bottle immersed in $\mathbb{R}^3$, a closed non-orientable surface with a circle of self-intersection
Featured Boys Surface
Boy's surface: an immersion of the real projective plane in $\mathbb{R}^3$ without singular points
Kleinbottle Fig8
Figure-eight immersion of the Klein bottle, with self-intersection along a line
Complex Spiral
Spiral in the complex plane lifted to 3D via the modulus of a complex function
Tangent Plane
Tangent plane to a surface at a movable point, the best linear approximation
Lorenz Attractor
The Lorenz attractor: butterfly-shaped strange attractor of a three-variable ODE system
Exp Taylor Series
Taylor polynomial approximations to $e^z$ converging on the complex plane
Frenet Frame
Frenet-Serret frame (tangent, normal, binormal) moving along a user-defined space curve
Parametric Curve Tangent
Tangent vector $\mathbf{r}''(t)$ moving along a parametric space curve
Featured Partial Derivatives
Partial derivatives as slopes of cross-sectional curves on a surface $z = f(x,y)
Parametric Plane
Parametric plane $\mathbf{r}(s,t) = \mathbf{p} + s\mathbf{u} + t\mathbf{v}$ in 3-space
Parametric Curve Animation
Animated tracing of a parametric curve, emphasizing the role of the parameter as time
Parametric Curve
User-defined parametric curve $\mathbf{r}(t)$ rendered as a tube in 3-space
Parametric Surface Animation
Time-dependent parametric surface evolving to show deformations and wave-like motion
Levelsets
Level sets of $f(x,y)$ displayed on the 3D graph and projected to the $xy$-plane
Featured Contour Slicing
Horizontal slicing plane intersecting a surface to reveal its level curves
Morse Function
Morse function on a surface: sublevel set topology changing at each critical point
Parametric Surfaces
Parametric surface $\mathbf{r}(u,v)$ rendered in 3-space with normal-based lighting
Cross Product
Cross product of two vectors, showing the perpendicular result and parallelogram area
Parameterized Line
Parametric line $\mathbf{r}(t) = \mathbf{p} + t\mathbf{v}$ in 3-space with its direction vector
Vector Components
Vector in $\mathbb{R}^3$ decomposed into its $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ components