One finger turns it, two fingers pinch to zoom and drag to pan; a mouse wheel zooms, a double-tap recentres. Pick a pair on the pairs tab to keep just those two faces lit and see the edge — or the two edges — they hold in common.
Measured from the eight planes on screen, every frame. The published certificate is re-verified in exact integer arithmetic by poly.selftest.mjs — every quantity in that part stays below 2⁵³, so no rounding can occur and no tolerance is used.
The tetrahedron has four faces and every pair of them shares an edge. The Szilassi polyhedron has seven and does the same, on a torus. For decades those were the only two known, and whether any solid with more than seven faces could manage it was open.
This one has eight. Eight planar nonagons on a surface of genus three, 24 corners, 36 edges, three faces at every corner, and all 28 pairs of faces adjacent. It was published in September 2026 by Ruslan Mizhaev as an integer certificate — every corner an integer point, every plane an integer equation — so it can be checked by hand, which is what the check tab and this page's selftest do.
The classical count says a solid on a surface with h holes whose f faces all touch needs h = (f−4)(f−3)/12, which is a whole number only for f ≡ 0, 3, 4, 7 (mod 12). Eight is not in that list — so how?
Because that count assumes each pair of faces shares exactly one edge. Here twenty pairs do, and eight pairs share two. Grünbaum and Szilassi call such faces overarching, and set them aside by assumption; this solid is what you get when you don't. The pairs tab is the whole story in one table: every cell filled, eight of them holding a 2. Both edges of an overarching pair lie on the single line where the two faces' planes cross — they reach past each other and come back.
So it answers the question as usually written, and it does not settle the question the formula was about. Both of those are worth saying. The f = 12, h = 6 case — 44 corners, 66 edges, every pair meeting exactly once — is still nobody's.
Every corner has three faces, so every corner is where three face planes cross, and eight planes are the entire solid — the same trick the Szilassi page runs on seven. Eight planes × three numbers = 24; the similarities of space use 7; so this shape has 17 freedoms where its seven-faced cousin has 14. The shape tab hands them to you three at a time.
It is a far tighter object, though. Szilassi's solid has room to be pushed about; this one has 4.1% of its radius between the nearest edge and the nearest face it does not belong to, and nudging all eight planes by a twentieth of the radius breaks it more often than not. Try the sliders and watch how quickly the verdict turns red.
The paper gives the symmetry T(x, y, z) = (y, −x, −z) and notes ⟨T⟩ ≅ C₄, which is true as an abstract group. Worth being precise about what it is geometrically: its determinant is −1, so it is a four-fold rotary reflection — a quarter turn about the axis followed by a reflection across the plane through it — and it reverses the walk of all eight faces. Its square is the honest half-turn. In Schoenflies notation the point group is S₄, not C₄.
That has a consequence you can feel on the sliders: locking the symmetry is not a matter of copying one plane's knobs round its orbit. An improper map turns a tilt into the opposite tilt, so the sign flips at every step — and comes back after two, because T² is proper. With the lock on, two planes are free and four shape freedoms remain.
Every face touches every other, so no two may share a colour and eight are needed. Heawood's bound allows a map on a genus-3 surface up to nine — this one uses eight of them and touches everywhere.