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Szilassi polyhedron

 

 
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Szilassi polyhedron
 
 
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The Szilassi polyhedron is a nonconvex polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
, topologically a torus
Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
, with seven hexagonal
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
 faces.

Each face of this polyhedron shares an edge with each other face.






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Type 
Faces7 hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
s
Edges21
Vertices14
Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
0
Genus
Genus (mathematics)

In mathematics, genus has a few different, but closely related, meanings:...
1
Vertex configuration
Vertex configuration

In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
6.6.6
Symmetry group
List of spherical symmetry groups

List of symmetry groups on the sphere Spherical symmetry groups are also called point groups in three dimensions. This article is about Point_groups_in_three_dimensions#Finite_isometry_groups....
?
Dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
Császár polyhedron
Császár polyhedron

In geometry, the Cs?sz?r polyhedron is a nonconvex polyhedron, topologically a torus, with 14 triangular face .This polyhedron has no diagonals; every pair of vertex is connected by an edge....
PropertiesNonconvex
The Szilassi polyhedron is a nonconvex polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
, topologically a torus
Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
, with seven hexagonal
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
 faces.

Each face of this polyhedron shares an edge with each other face. It has an axis of 180-degree symmetry
Rotational symmetry

File:The armoured triskelion on the flag of the Isle of Man.svgGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation....
; three pairs of faces are congruent leaving one unpaired hexagon that has the same rotational symmetry as the polyhedron. The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the Heawood graph
Heawood graph

In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges. The graph is cubic graph, and all cycles in the graph have six or more edges....
 onto the surface of a torus.

The tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
 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
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
 that 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, but it is not known whether such a polyhedron exists. More generally this equation can be satisfied only when f  is congruent to 0, 3, 4, or 7 modulo 12.

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

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 to the Szilassi polyhedron, the Császár polyhedron
Császár polyhedron

In geometry, the Cs?sz?r polyhedron is a nonconvex polyhedron, topologically a torus, with 14 triangular face .This polyhedron has no diagonals; every pair of vertex is connected by an edge....
, was discovered earlier by Ákos Császár (1949); 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.

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