One finger turns it, two fingers pinch to zoom and drag to pan; a mouse wheel zooms, and a double-tap recentres. Star of corner keeps only the six triangles that meet the corner picked on the shape tab — the ring of them is what has to stay unfolded for the surface to be a surface there.
The seven colours belong to the seven corners, and each triangle blends the three it hangs from. Because every pair of corners is joined by an edge, every pair of colours meets along one.
Measured from the seven points on screen, every frame. The same module, unchanged, is held to these claims by poly.selftest.mjs — including by a second, differently-built test that has to agree with the first.
In 1949 a Hungarian competition asked students to prove that the tetrahedron is the only polyhedron in which every pair of corners is joined by an edge — no diagonals anywhere. The proof is short: V − E + F = 2 forces V = 4. The question forgot to say simple polyhedron. On a torus the 2 becomes a 0, and out falls V = 7, E = 21, F = 14. Ákos Császár built it.
So here are seven points with every one of the 21 segments between them an edge of the solid, and fourteen triangles hung on them. Its skeleton is the complete graph K₇, drawn on a torus without crossings. Its dual is the Szilassi polyhedron: seven hexagons, every pair touching.
On the dual page, flatness is the hard part — seven hexagons have to stay planar, and the trick is that seven planes are the whole solid. Here it is the other way round. Faces are triangles, and three points are always coplanar, so you cannot make a face non-flat. Drag the corners anywhere you like.
The arithmetic comes out the same, as duals must: 7 points × 3 numbers = 21, minus the 7 similarities of space that move the solid without changing its shape, leaves 14 shape degrees of freedom. Measured on both pages, live.
What can go wrong is the surface passing through itself, and the way it goes wrong has structure. For each of the 35 quadruples of corners, record which side of the plane of the first three the fourth lies on. Those 35 signs are the shape's oriented matroid, and here is the point:
No solid can have four corners in one plane. Every pair of corners is an edge, so four coplanar corners would put two whole edges in one plane — and two edges in a plane, in a complete graph, have to cross. A crossing is not a solid. So along any continuous path of solids the 35 signs never pass through zero, which means they never change at all.
Different signs, different piece of the space. The four published realizations have four different sign patterns, so no amount of careful dragging takes you from one to another — the walk tab walks straight at a target and shows you exactly where the wall is. The four differ in one visible way: the order in which the three mirror pairs and the lone corner stack up the symmetry axis.
Bokowski and Eggert answered this in 1991, in All realizations of Möbius' torus with 7 vertices: up to equivalence there are 72, and all of them can be built. So the four here are four of seventy-two — Lajos Szilassi's models, the first being Császár's own 1949 coordinates. Anywhere you read that there are only four essentially different Császár polyhedra, that is four models being mistaken for the classification.
You can go and look. Throw seven points into a cube at random and hang the Möbius triangulation on them: about one roll in 5,800 is already an embedded torus. Twenty-four million rolls gave 4,139 solids in 62 of the 72 types, all four of the published ones among them — the missing ten sit in slices of the space too thin for that many darts to hit. csaszar/census.mjs is that experiment, and it prints those numbers.
Two triangles lie in two planes, so they can only meet on the line where those planes cross. Walk that line, find the stretch each triangle covers, and where the stretches overlap is exactly where the triangles touch. A solid needs that overlap to be no more than what the two already share — an edge, a corner, or nothing. Parallel planes are the case to be careful with: distinct ones never meet and are perfectly fine, and the two far triangles of an octahedron are the everyday example.