Heawood map
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Heawood map  

Rank  3 
Type  Regular map, Regular toroid, Chiral polyhedron 
Elements  
Faces  7 hexagons 
Edges  21 
Vertices  14 
Vertex figure  Triangle 
Petrie polygons  3 tetradecagons 
Abstract & topological properties  
Flag count  84 
Euler characteristic  0 
Schläfli type  {6,3} 
Orientable  Yes 
Genus  1 
Skeleton  Heawood graph 
Properties  
Symmetry  C_{7}⋊C_{6}, order 42 
Flag orbits  2 
The Heawood map is a regular map. Its automorphism group darttransitive, thus it is a regular toroid and a regular map. However is not an abstract regular polytope since its automorphism group is not flagtransitive.
With seven hexagonal faces, it has the least number of faces out of any toroid. It also has the unusual property that each of its faces is adjacent to all of the other faces.
It can be realized in 3dimensional space with planar faces and without selfintersection as the Szilassi polyhedron.
Gallery[edit  edit source]

The Heawood map with a hexagonal fundamental polygon
External links[edit  edit source]
 Wedd, N. The Heawood map