# W'

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${\displaystyle {\frac {15+7{\sqrt {5}}}{6}}\approx 5.10875}$
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

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,{\frac {3{\sqrt {3}}+{\sqrt {15}}}{6}}\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,{\frac {3{\sqrt {3}}+{\sqrt {15}}}{6}}\right)}$,
• ${\displaystyle \left(\pm 1,\,{\frac {\sqrt {3}}{6}},\,{\frac {3{\sqrt {3}}+{\sqrt {15}}}{6}}\right)}$,
• ${\displaystyle \left(0,\,-{\frac {2{\sqrt {3}}}{3}},\,{\frac {3{\sqrt {3}}+{\sqrt {15}}}{6}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,{\frac {2{\sqrt {3}}+{\sqrt {15}}}{6}},\,{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,-{\frac {{\sqrt {15}}-{\sqrt {3}}}{12}},\,{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\frac {5{\sqrt {3}}+{\sqrt {15}}}{12}},\,{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,0\right)}$,
• ${\displaystyle \left(\pm 1,\,0,\,0\right)}$.

## Related polytopes

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.