W'

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W'
Rank3
TypeAcrohedron
Notation
Stewart notationW'
Elements
Faces1 hexagon, 3 squares, 3 pentagons, 6+3+3+1 triangles
Edges3+3+3+3+6+6+6+6
Vertices3+3+6+6
Measures (edge length 1)
Volume
Central density1
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
ConvexYes
NatureWild

W' is a convex[note 1] polyhedron with all regular faces. However it is not a Johnson solid, since several of its faces are coplanar. It is notable for being the first discovered example of a 6-5-4 acrohedron. It was discovered by John Horton Conway and given its name by Bonnie Stewart.

Vertex coordinates[edit | edit source]

A W' of edge length 1 has vertices given by the following coordinates:

  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • .

Related polytopes[edit | edit source]

W' and J92 can both be built from the same 4 parts.

W' is closely related to the triangular hebesphenorotunda, a Johnson solid. The two have the same face counts, symmetry and volume.[1]

Its coplanar faces are resolvable by excavating them with four tetrahedra, resulting in a 28-faced 6-5-4 acrohedron. Stewart names this polyhedron W''. W'' is weakly quasi-convex.[1]

Alex Doskey found that W' can be augmented with a square pyramid to produce a 6-5-3-3 acrohedron.

W' is non-self-intersecting. Richard Klitzing found a self-intersecting 6-5-4 acrohedron with the same symmetry, that has only 17 faces.

External links[edit | edit source]

Notes[edit | edit source]

  1. If the definition that a convex polytope is the convex hull of its vertices is used, W' is not convex. However the interior of W' is a convex set, so it satisfies the interior definition. Stewart considered W' to be convex.

References[edit | edit source]

  1. 1.0 1.1 Stewart (1964:168)

Bibliography[edit | edit source]

  • Stewart, Bonnie (1964). Adventures Amoung the Toroids (2 ed.). ISBN 0686-119 36-3.