Demos
O(2,2) on the Isotropic Grassmannian
Limit sets of O(2,2) on the two ruling circles, in circle coordinates
O(4,2) on the Isotropic Grassmannian
Limit sets of 154 monodromy groups acting on 2-planes in R⁴ʼ²
Borsuk–Ulam
Antipodal points sharing an image, found as a ribbon twists from a band into a Möbius strip
Brouwer Fixed Point
The fixed point of a self-map of the disk, caught the moment its graph links with the diagonal
Disk Map Playground
Fold and stretch a map of the disk to itself; the fixed-point indices always sum to 1
The Hairy Ball Theorem
Combing a hairy ball: the zero that survives any combing, since the indices always sum to 2
Hypergeometric Atlas, Degree ≤ 6
Every hypergeometric monodromy group of degree ≤ 6, from O(1) up to Sp(6,ℝ)
Hypergeometric Atlas, Degree 7
509 hypergeometric monodromy groups of degree 7, with limit sets in RP⁶
Canonical Hyperbolic Coxeter Polygons
Hyperbolic Coxeter polygons built from their vertex orders alone, reflected to any depth
Starscapes: Integer Quadratics
Roots of integer quadratic polynomials scattered across the upper half-plane
Stable Norm: the Exclusion Picture
Where the lower bound clears the modular torus's upper bound, excluding the minimum
Stable Norm up the Cusp
Stable-norm volume climbing monotonically out the cusp of moduli space
Stable Norm Volume over Moduli Space
Stable-norm ball volume across moduli space: wells at the hexagonal tori, passes at the square
Stable Norm: Inner Bound
The convex hull of boundary samples, giving a lower bound on the stable-norm ball
Stable Norm: Outer Bound
Supporting lines at each rational corner, circumscribing the stable-norm ball
Stable Norm: Boundary Samples
Boundary samples ±(p,q)/ℓ of the stable-norm ball, filling in as the slope cutoff grows
The Markoff Cubic, Flat
The Markoff cubic flattened, beside the stable-norm ball of each hyperbolic structure
Cayley Graphs in the Hyperbolic Plane
Cayley graphs of 2D Coxeter groups drawn in H², with edges colored by generator
Cayley Graphs in Hyperbolic Space
Cayley graphs of 3D Coxeter groups inside the ball model, with edges colored by generator
L-Surface Geodesics
The L-shaped translation surface relaxed into a smooth embedded genus-2 surface
Deformations that Keep the Trefoil
Deformations of z² + w³ that leave the singularity link a trefoil
O(5) Hypergeometric Atlas
All 77 degree-5 orthogonal monodromy groups, sorted thin, arithmetic, open, or finite
Figure-Eight Character Variety in SL(4,ℝ)
The SL(4,ℝ) character variety of the figure-eight knot group, as a projected point cloud
Curated Moduli
Eight vertex flat embedded tori whose moduli lie on the boundary of the fundamental domain.
Parametric Cubic Surface
A parametric representation of a cubic surface derived from blowing up the plane at six points.
Parametric Cubic Surface (Stereo)
A parametric representation of a cubic surface derived from blowing up the plane at six points, rendered in stereographic projection from the double cover.
Happy Birthday Rich
Live version of birthday present for Rich Schwartz, on the occasion of his 60th brithday conference.
Seifert Surface Point Cloud
Preimages of the real line under complex polynomials with a singularity at 0.
Hyperbolic Voronoi: Expanding
The Voronoi diagram of a collection of points as they move towards the ideal boundary of hyperbolic space.
Hyperbolic Voronoi: Poisson Process
Voronoi tessellation of the hyperbolic plane for a Poisson process of points
SO(3,4) Representations of Δ(2,3,r)
Searching numerically for representations of this triangle group.
M4 Geodesic Linkage
Physical motion as geodesic flow on the configuration sphere of the 4-rod linkage; release the ball with momentum to launch a trajectory.
M3 Geodesic Linkage
Unforced motion of the 3-rod linkage, pacing around its configuration circle as a geodesic of the kinetic-energy metric.
M3 Parametric Linkage
The configuration space of the 3-rod linkage is a circle: drag a point around it and watch the chain flex in step.
M4 Parametric Linkage
The configuration space of the 4-rod linkage is a sphere: drag a point across it and watch the chain flex in step.
Featured Cubic Lines Live
Blowing up the plane at six points to give a cubic surface