Császár polyhedron

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Császár polyhedron
Rank3
TypeRegular toroid, Regular map
Notation
Schläfli symbol
Elements
Faces2+2+2+2+2+2+2 scalene triangles
Edges1+1+1+2+2+2+2+2+2+2+2+2
Vertices1+2+2+2
Vertex figure2+2+2 irregular hexagons, 1 bilaterally-symmetric hexagon
Related polytopes
DualSzilassi polyhedron
Abstract & topological properties
Flag count84
Euler characteristic0
OrientableYes
Genus1
SkeletonK7
Properties
SymmetryK2+×I, order 2
Flag orbits42
ConvexNo
History
Discovered byÁkos Császár
First discovered1949

The Császár polyhedron (Hungarian: [ˈt͡ʃaːsaːr], approximate English pronunciation CHA-sar) is a toroidal polyhedron without any diagonals; that is, every pair of vertices is connected by an edge. It is the dual of the Szilassi polyhedron. It is a regular toroid and regular map, but it is not abstractly regular.

It has 14 scalene triangular faces, 7 vertices, and 21 edges. 6 faces meet at each vertex.

Its vertices and edges are an embedding of the complete graph K7 onto a torus of genus 1.

It has the same edge skeleton as the 7-2 step prism.

Vertex coordinates[edit | edit source]

External links[edit | edit source]

Bibliography[edit | edit source]

  • Brehm, Ulrich; Kühnel, Wolfgang (2008), "Equivelar maps on the torus", European journal of combinatorics, doi:10.1016/j.ejc.2008.01.010