Demos
hopf torus list
Gallery of flat tori realized as Hopf surfaces in the 3-sphere, one for each lattice type
hyerbolic embedding flows
Comparison of optimization methods for embedding a hyperbolic disk in 3-space
weierstrass hex
Hexagonal-lattice elliptic curve embeded in R4 by the Weierstrass functions.
weierstrass square
Square-lattice elliptic curve embedded in R4 via the Weierstrass functions.
hex curve
An elliptic curve over a finite field embedded on the characteristic zero curve with hexagonal symmetry.
hex curve
An elliptic curve over a finite field embedded on the characteristic zero curve with square symmetry.
elliptic 11 4
The elliptic curve $y^2=x^3+x+1$ mod 5, over the field with 625 elements.
elliptic 11 5
The elliptic curve $y^2=x^3+x+1$ mod 5, over the field with 3125 elements.
elliptic 3 4
The elliptic curve $y^2=x^3+3$ mod 7, over the field with 2401 elements, embedded in a characteristic zero curve with hexagonal symmetry.
elliptic 3 3
The elliptic curve $y^2=x^3+3$ mod 7, over the field with 343 elements, embedded in a characteristic zero curve with hexagonal symmetry.
elliptic 4 3
The elliptic curve $y^2=x^3+3x$ mod 5, over the field with 625 elements, embedded in a characteristic zero curve with square symmetry.
elliptic 4 4
The elliptic curve $y^2=x^3+3x$ mod 5, over the field with 3125 elements, embedded in a characteristic zero curve with square symmetry.
elliptic 7 2
The elliptic curve $y^2=x^3+5x+7$ mod 11, over the field with 121 elements.
elliptic 7 3
The elliptic curve $y^2=x^3+5x+7$ mod 11, over the field with 1331 elements.
elliptic 8 2
The elliptic curve $y^2=x^3+x+3$ mod 11, over the field with 121 elements, embedded in a characteristic zero rectangular torus.
elliptic 8 3
The elliptic curve $y^2=x^3+x+3$ mod 11, over the field with 1331 elements, embedded in a characteristic zero rectangular torus.
pathtracer touch controls
Path-traced Menger sponge in a Cornell box with mobile touch controls
Brouwer Fixed Point Test
Interactive linking-number proof of the Brouwer fixed point theorem
Featured black hole embedding
Isometric embedding of the optical geometry of a Schwarzschild black hole into $\mathbb{R}^3$
rhombic tori in S3
Family of rhombic flat tori isometrically embedded in the 3-sphere, parameterized by vertex angle
rich's minimal paper torus
Rich Schwartz's eight-vertex paper torus: a flat polyhedral torus embedded in 3-space
hex tori in R3
Attempting to embed the flat hexagonal torus in $\mathbb{R}^3$ — impossible smoothly, but revealing to try
hex tori in S3
Isometric embedding of the flat hexagonal torus in the 3-sphere with adjustable parameters
charged bunny
Spring and charge dynamics reshaping the Stanford bunny mesh
square tori in R3
Attempting to embed the flat square torus in $\mathbb{R}^3$ — a necessarily imperfect compromise
square tori in S^3
Square tori of varying size in the 3-sphere — only one embeds isometrically as the Clifford torus
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
airplug bottomless
Generative album artwork for Airplug's Bottomless
bottomless: animation
Real-time live visuals for Airplug's Bottomless performance
bottomless: animation II
Second set of real-time live visuals for Airplug's Bottomless performance
hyperbolic strip embedding
Strip of the hyperbolic plane isometrically embedded in 3-space, ruffling from negative curvature
hyperbolic cylinder embedding
Hyperbolic cylinder isometrically embedded in Euclidean space, with OBJ export for 3D printing
embedding test exponential curtain
Negatively curved curtain ruffling under gravity as excess area forces buckling
embedding test hyperbolic curtain
Strip of the hyperbolic plane draped as a curtain, buckling from its negative curvature
embedding test euclidean curtain
Flat curtain hanging under gravity, finding its natural shape in 3-space
embedding test hyperbolic cylinder
Hyperbolic cylinder embedded in Euclidean space, ruffling at its negatively curved edges
embedding test hyperbolic strip
Rectangular strip of the hyperbolic plane in 3-space, with geodesic spreading visible as buckling
embedding test hyperbolic disk
Disk of the hyperbolic plane forced into Euclidean space, showing characteristic saddle ruffling
euclidean cylinder embedding
Flat cylinder finding its round cross-section through isometric embedding in 3-space
spring charge square
Flat square mesh finding equilibrium under competing spring and charge forces
spring square
Flat square mesh relaxed by springs alone, without charges to prevent self-intersection
geodesic board: eggcarton
Parallel geodesics on an egg-carton surface, alternately focused and scattered by changing curvature
geodesic board: gaussian I
Geodesics focused by a single Gaussian bump, illustrating the effect of positive curvature
geodesic board: gaussian II
Geodesics strongly focused by a tall Gaussian bump, crossing sooner from higher curvature
geodesic board: gaussian III
Geodesics asymmetrically deflected by an off-center Gaussian bump
geodesic board: gaussian IV
Geodesics doubly lensed by two Gaussian bumps on a surface
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
geodesic board: waves
Parallel geodesics on a sinusoidal wave surface, bunching and spreading with the curvature
black holes octahedral geodesics
Light rays threading between six black holes at octahedral vertices, producing symmetric scattering patterns
black hole glass ball
Planar fan of light rays bending in a variable-index-of-refraction medium that is mathematically identical to a charged black hole, this time in a sphere.
black hole glass block
Planar fan of light rays bending in a variable-index-of-refraction medium that is mathematically identical to a charged black hole.
black hole glass block binary
Light rays scattered between two black holes with adjustable mass ratio, revealing chaotic capture boundaries, shown in equivalent classical optical system.
lightcone
Light cone surface warped by two charged black holes in a Majumdar-Papapetrou spacetime
lightcone bothsides
Light cones on opposite sides of a black hole pair, highlighting asymmetric causal distortion
lightcone double
Two light cones in a binary black hole spacetime, showing how causal structure varies with position
lightcone single
A single light cone tilting toward a charged black hole
lightcone quadruple
Four light cones mapping the tilt of causal structure across a two-black-hole spacetime
lightcone triple
A light cone between four black holes, revealing the varying causal geometry
black hole binary geodesics
Light rays scattered by three black holes in a triangular arrangement
black hole binary geodesics 2
Near-field scattering of light rays by two black holes, with a cutoff radius isolating the interaction
black hole geodesic project
Null geodesics near a black hole in full spacetime and spatial views.
black hole vision
Light rays near a charged black hole, colored by which hit the red object in the background.
black hole vision binary
Light rays threading between two black holes, colored by which reach the object in the background.
black hole vision binary
Light rays threading between two black holes, colored by which reach the object in the background.
black hole geodesic vision
Null geodesics near a black hole shown in spacetime
black hole binary horizon
Looking out from between two charged black holes, gravitational lensing distorting each horizon
mp black hole binary horizon 2
Interior view of a multi-black-hole spacetime with overlapping gravitational lensing from colored horizons
mp black hole octahedron horizon
View from inside a spacetime with six black holes at octahedral vertices, each horizon distinctly colored
knot dance with twistbar
Knot dance animation with a twist bar tracking writhe through each move
knot dance
Animated 3D knot diagrams for Danceability of Twisted Virtual Knots
knot dance: design
Design tool for choreographing the animated knot dance sequences