Szilassi Polyhedron: The Toroidal Big Brother of the Tetrahedron
What Do a Tetrahedron and a 7-Faced Polyhedron Have in Common?
What does a tetrahedron share with a bizarre polyhedron made of 7 hexagons? Both are true unicorns of the geometric world: in each of them, every two faces share an edge. Only two such solids exist — and the second one is called the Szilassi polyhedron.
It was discovered in 1977 by Hungarian mathematician Lajos Szilassi and has held a special place in topology ever since.
The Topology of the Szilassi Polyhedron: Structure and Properties
Here is what makes it remarkable:
7 hexagonal faces
21 edges
14 vertices
The topology of a torus, not a sphere
Striking symmetry: three pairs of faces are congruent, and one face is symmetric with respect to itself
An embedded Heawood graph — a celebrated construction from graph theory
Duality with another legendary solid: the Császár polyhedron
Unlike ordinary polyhedra, the Szilassi polyhedron cannot be unfolded onto a flat plane without distortion — it is fundamentally a toroidal object, shaped like a donut rather than a ball.
The Seven Color Theorem and Toroidal Maps
Thanks to the Szilassi polyhedron, mathematicians gained a clearer path to the “seven color theorem”: any map drawn on a torus can be colored using at most seven colors so that no two adjacent regions share the same color.
On a flat plane, four colors suffice. But a torus is topologically richer — it demands more. The Szilassi polyhedron provided the key to understanding exactly why.
The Szilassi Polyhedron Beyond Mathematics
The solid has even been immortalized in bronze — a sculpture of the Szilassi polyhedron stands in Maykop, the capital of the Republic of Adygea. And in a Moscow school, a mathematics and physics class took on the symbolic name “Silaedr” in its honor.
So if you thought the tetrahedron was the ultimate minimalist marvel of geometry — know that it has a big brother. One that lives not on a flat plane, but on a torus, and makes topology genuinely beautiful.
