geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Szilassi polyhedron is a nonconvex

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

, topologically a

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...

, with seven

hexagonal In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular hexagon'' has ...

faces.

Coloring and symmetry

The 14 vertices and 21 edges of the Szilassi polyhedron form an

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...

of the

Heawood graph Heawood is a surname. Notable people with the surname include: * Jonathan Heawood, British journalist * Percy John Heawood (1861–1955), British mathematician ** Heawood conjecture ** Heawood graph ** Heawood number See also *Heywood (surname) He ...

onto the surface of a torus. Each face of this polyhedron shares an edge with each other face. As a result, it requires seven colours to colour all adjacent faces. This example shows that, on surfaces topologically equivalent to a

, some subdivisions require seven colors, providing the lower bound for the seven colour theorem. The other half of the theorem states that all toroidal subdivisions can be colored with seven or fewer colors. The Szilassi polyhedron has an axis of 180-degree symmetry. This symmetry swaps three pairs of congruent faces, leaving one unpaired hexagon that has the same rotational symmetry as the polyhedron.

Complete face adjacency

The

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

and the Szilassi polyhedron are the only two known polyhedra in which each face shares an edge with each other face. If a polyhedron with ''f'' faces is embedded onto a surface with ''h'' holes, in such a way that each face shares an edge with each other face, it follows by some manipulation of the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...

that :

h = \frac.

This equation is satisfied for the tetrahedron with ''h'' = 0 and ''f'' = 4, and for the Szilassi polyhedron with ''h'' = 1 and ''f'' = 7. The next possible solution, ''h'' = 6 and ''f'' = 12, would correspond to a polyhedron with 44 vertices and 66 edges. However, it is not known whether such a polyhedron can be realized geometrically without self-crossings (rather than as an

abstract polytope In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be ...

). More generally this equation can be satisfied precisely when ''f'' is congruent to 0, 3, 4, or 7 modulo 12. File:Szilassi_polyhedron_3D_model.svg, link=, Interactive orthographic projection with each face a different colour. move the mouse left and right to rotate the model. File:Szilassi polyhedron.gif, Animation

History

The Szilassi polyhedron is named after Hungarian mathematician Lajos Szilassi, who discovered it in 1977. The dual to the Szilassi polyhedron, the

Császár polyhedron In geometry, the Császár polyhedron () is a nonconvex toroidal polyhedron with 14 triangular faces. This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron ...

, was discovered earlier by ; it has seven vertices, 21 edges connecting every pair of vertices, and 14 triangular faces. Like the Szilassi polyhedron, the Császár polyhedron has the topology of a torus.

References

External links

*. *. *{{MathWorld , urlname=SzilassiPolyhedron , title=Szilassi Polyhedron , mode=cs2
Szilassi Polyhedron
– Papercraft model a
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Nonconvex polyhedra Toroidal polyhedra Unsolved problems in mathematics