the fusion — K₇ on a torus (soft-GL)
7 vertices wind around a torus; every pair is joined by an edge (that\u2019s all 21 — the complete graph K₇, exactly a simplex\u2019s defining property); 14 triangles tile the surface. it\u2019s a closed torus (the hole is real), built from nothing but triangles with a complete skeleton. drag to rotate; toggle the layers.
the two-of-a-kind — tetrahedron ↔ Császár
a polyhedron with no diagonals — every vertex pair already joined by an edge — is the polyhedral form of a simplex-skeleton. Exactly two are known: the tetrahedron (K₄, on a sphere, no hole) and the Császár polyhedron (K₇, on a torus, one hole). Same idea, one genus apart.
tetrahedron — K₄ · 4 vertices · genus 0 · the minimal container
Császár — K₇ · 7 vertices · genus 1 · minimal container + the hole
Kₙ needs genus ⌈(n−3)(n−4)/12⌉ ; n=7 → 1, exactly a torus.
toddler corner
ELI5: we had two "simplest cages." One was the tetrahedron — 4 corners, every corner roped to every other corner, the least you need to trap a blob (but solid, no hole). The other was the donut — the shape that\u2019s famous for its hole. Fuse them and you get a magic donut made entirely of triangles, with SEVEN corners — and the wild part: every single corner still has a rope straight to every other corner, all 21 ropes, no corner left out. That "everybody connected to everybody" is the tetrahedron\u2019s superpower, and here it is living on a donut. It turns out you almost can\u2019t do this: in the whole universe of shapes, only TWO are known where every corner touches every other corner directly — the little tetrahedron, and this seven-cornered donut. They\u2019re a matched pair, one flat-and-solid, one holey, separated by exactly one hole. So "fuse the skeleton and the hole" isn\u2019t a poem — it\u2019s a specific, rare, real object, and there are only two of its kind.