Seeing is believing

Pen-and-ink plates of parametric and implicit surfaces in the manner of Francis: silhouettes with cusps, hidden lines dashed, haloes at crossings, double curves of immersions, and hatching traced along principal curvature. Klein bottle, Boy’s surface, and a knotted torus are built in; type your own.

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.

A ball bounces inside a Platonic solid, sounding a steel-pan note at every face. Presets include orbits that strike every face exactly once — the icosahedron’s is the torus knot T(3,5).

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.

An endlessly zoomable Farey tessellation and its Ford circles in the upper half-plane, computed in exact arithmetic so that labels keep appearing all the way down to 10¹⁰⁰, with the cutting sequence of the centre line spelling out its continued fraction.

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).

Draw a graph — loops and parallel edges allowed — or pick one from a library of famous ones, and read off its adjacency, Laplacian, or Seidel matrix with eigenvalues, eigenvectors painted on the vertices, and an exact characteristic polynomial over ℤ that certifies every closed-form eigenvalue.

Zoomable Hasse diagrams built for Math 372: the quotient posets of unlabeled graphs, necklaces, simplicial complexes and finite spaces, the partition lattice and its quotient by the symmetric group, Young diagrams in a box as a quotient of a Boolean algebra, and the Tamari lattice drawn on binary trees, transfer systems on a chain, and noncrossing partitions.

The Klein, Poincaré, and half-plane models side by side with the hyperboloid they are projected from — drag triangles, polygons, and tessellations in one pane and watch the rays, the plaster model, and the Erlangen-style transformation group answer in the other.

Draw a graph — parallel edges allowed — or pick a classic, give every edge its own variable, and watch the determinant of the reduced weighted Laplacian expand into one monomial per spanning tree, each monomial and its tree painted in matching colours.

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.

Left, right, midpoint, trapezoid, and Simpson’s rules for any function you type, built for Math 111: see what each rule counts, compare all five at once, and watch the error fall like 1/n, 1/n², or 1/n⁴ on a log–log plot.

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

Every closed meander with a rainbow underneath, grown as an endless zoomable tree by cutting an outermost arch and wrapping it around everything, with a z² fold that turns each node into its semi-meander and picks out the genuine meanders.

Bott’s sphere of at-most-three-point subsets of the circle, drawn through Mostovoy’s lattices: the singletons form a trefoil, the pairs a Möbius band spanning it, and a three-stage homotopy contracts the trefoil through a dunce hat of triples — the null-homotopy of S¹ → Ran₃(S¹) that Ran₂(S¹) cannot provide.

Simple random walks on the hypercube Qₙ, one flipped coordinate at a time, animated on the cube while a tally counts how often ℓ steps bring the walker home. The exact probability — the spectral count of closed walks, 2⁻ⁿ n⁻ˡ Σ C(n,i)(n−2i)ˡ — is shown term by term beside the running frequency, so the convergence, and the odd-ℓ parity obstruction, are both visible.

The Boolean algebra Bₙ partitioned into chains by the linear-algebra method: for each level, a square submatrix of the up-operator with nonzero determinant is chosen, a nonzero term of its expansion is extracted, and that term is an order-matching. Following the matchings through the middle level gives C(n,⌊n/2⌋) chains, which is Sperner’s theorem. Reorder the elements and the determinants return a different partition; a bracketing rule gives the symmetric one for comparison.

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.

A loop around two rods of a taffy puller, pulled by an Anosov map of the torus and pushed to the sphere by the Weierstrass ℘-function — the rods are its branch points, and the Teichmüller flow moves them.

The standard proof of the Fundamental Theorem of Calculus, one step at a time, for any function you type: telescope the total change, find each piece’s Mean Value Theorem point, see the rises become a Riemann sum that never moves as n grows, and watch the argument break when F has a corner or a pole. Built for Math 111.

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

Spin S² once and n marked points drag out the full twist Δ²; spin it twice and the braid comes undone. The 4π rotation lifts to a closed loop in the simply connected S³, and contracting that loop is the isotopy that untangles the strands.

Ingredients from 1.8 million deduplicated recipes as a simplicial complex: a set of flavors spans a simplex when its ingredients appear together more often than chance predicts. Orbit it in 3D, slide the lift threshold to watch the Betti numbers change, trace its loops and hollow 2-spheres, and find the triples that keep it from being flag.

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 locker problem for a 4th-grade classroom: students take turns flipping every 2nd, 3rd, 4th… light, tally marks count the flips, the class taps in guesses before each turn, and tile rectangles pair up the students who cancel each other, leaving only the square numbers lit.

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.