One finger turns it, two fingers pinch to zoom and drag to pan; a mouse wheel zooms. Double-tap recentres. The seven colours are not decoration: because every face touches every other, no fewer than seven will do — which is the whole reason this solid is famous.
Everything here is measured from the seven planes you are looking at, on every frame — nothing is asserted. The same code, unchanged, is held to these claims by poly.selftest.mjs.
Colour a map so that countries sharing a border get different colours. On a sphere four colours always suffice; on a torus you sometimes need seven. To need seven you want seven countries that all border one another — and the Szilassi polyhedron is that map, built as a solid: 7 hexagonal faces, 21 edges, 14 corners, every one of the C(7,2) = 21 face pairs sharing an edge.
Its dual, with seven corners and no diagonals at all, is on the next page: császár. The tetrahedron does the same trick with 4 triangles. Those two are the only polyhedra known where every face touches every other. Euler forces such a solid with f faces onto a surface with h = (f−4)(f−3)/12 holes, so the next candidate is 12 faces on a 6-holed surface — nobody has built one. Lajos Szilassi found this one in 1977; its dual, with 7 vertices and no diagonals at all, is Ákos Császár's 1949 polyhedron.
The 14 corners and 21 edges, taken as a graph, are the Heawood graph — the smallest cubic graph of girth 6. The check tab verifies that from the shape on screen: 3-regular, bipartite, girth 6, which pins it down uniquely.
Every corner has exactly three faces, so every corner is where three face planes cross — and there are only seven planes in the whole solid. Seven planes are the entire object. Nothing here is ever solved for: move a plane, re-intersect, done, and the faces are flat by construction rather than by approximation.
That makes the arithmetic easy. Seven planes × three numbers each = 21. Sliding, turning and scaling the whole solid in space uses up 7 of those without changing its shape. So the Szilassi polyhedron is not one shape but a 14-parameter family, and the published solid is one point in it. Hold the 180° half-turn symmetry and 7 are left. Both numbers are measured on the shape you are looking at — see check — as the rank of the map from knobs to corners, not counted on paper.
The shape tab hands you those knobs three at a time: pick a face, tilt its plane two ways, slide it along its own normal. What you cannot do is leave the family — flatness survives anything. What you can do is break the solid: the region of the 14 where the seven hexagons still bound a body rather than slicing through one another is open, and it is not very large. Push a slider far enough and the verdict under the object turns red and says which two faces crossed.
Two faces lie in two planes, so they can only meet on the line where those planes cross. Walk that line, mark every place either hexagon's boundary touches it, and test the middle of each stretch for being inside both. What both cover is the intersection. A solid needs it to be exactly their shared edge — and no more. Twenty-one pairs, and the answer is exact.
The naive shortcut fails here, which is worth knowing before you optimise it: faces 1 and 2 of the published solid have corners on both sides of each other's plane. They reach past one another without meeting. These hexagons are not convex.
Szilassi 1977 is Table 3 of Grünbaum & Szilassi's Geometric realizations of special toroidal complexes (2009), to the last decimal. Its faces are planar to the last bit of a double, which the selftest checks rather than trusts. The others were found by hill-climbing the knobs offline: roomy more than doubles the space between the tightest corner and the face it nearly touches, blunt opens the sharpest crease from 49° to 75°. Every shape you make is in the URL.