Seeing is believing

A curve and its dual, side by side. Drag the coefficients and watch inflection points and bitangents migrate between the two pictures.

Every way to parenthesize an expression, enumerated and evaluated — Catalan numbers doing their day job.

Iterated barycentric subdivision of a simplex, and the alarming thinness of the simplices it produces.

Closed geodesics in the space of lattices attached to totally real cubic fields, drawn against ℝP².

Real plane curves drawn on a disc model of ℝP², where the line at infinity is just another line.

Two generic rotations generate a free group. Here are its words, plotted in ℝP³, drifting toward the Banach–Tarski paradox.

The cyclic, dihedral, tetrahedral, octahedral, and icosahedral groups, each shown acting on the sphere it was born to rotate.

The directed graph of b ↦ b + a modulo m. Change a and m and watch the cycle structure reorganize itself around gcd(a, m).

Closed curves crossing a line, counted and drawn, with the associahedron and multiplication table that organize them.

Unfoldings of the regular 4-polytopes, including all 261 nets of the tesseract.

Orbs tracing a periodic flow on a lattice, because sometimes the right way to understand a group action is to let it run.

ℤ/12 ≅ ℤ/4 × ℤ/3 as a pair of gears — the Chinese remainder theorem as a mechanism rather than a proof.

Cubic and quartic surfaces rendered in projective 3-space, including the ones whose singularities only make sense once you leave affine coordinates.

Temperley–Lieb diagrams and the Markov trace that turns them into knot invariants.

Watching a polynomial map with constant nonzero Jacobian fold the plane — the picture that makes the conjecture feel plausible and hard at the same time.

The multiplicative norm of the swap action, made interactive — an equivariant construction that usually only lives in a diagram.

Take the short diagonals of a convex polygon and keep the polygon they cut out. A pentagon returns projectively equivalent to itself, so the picture falls inward forever.

Drag a loop around two punctures through a braid word and watch it stretch. With σ₁σ₂⁻¹ on three rods the intersection counts grow like Fibonacci, and their growth rate is the dilatation.

Polyhedra whose graphs are bipartite, turned in space so you can check the two-coloring by eye.

The space of pentagons with fixed side lengths, up to rotation — a surface you can walk around by bending a linkage.