# W'

W'
Rank3
TypeAcrohedron
SpaceSpherical
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:

• ${\displaystyle \left(\pm\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,\frac{3\sqrt{3}+\sqrt{15}}{6}\right)}$,
• ${\displaystyle \left(0,\,\frac{\sqrt{3}}{3},\,\frac{3\sqrt{3}+\sqrt{15}}{6}\right)}$,
• ${\displaystyle \left(\pm1,\,\frac{\sqrt{3}}{6},\,\frac{3\sqrt{3}+\sqrt{15}}{6}\right)}$,
• ${\displaystyle \left(0,\,-\frac{2\sqrt{3}}{3},\,\frac{3\sqrt{3}+\sqrt{15}}{6}\right)}$,
• ${\displaystyle \left(\pm\frac12,\,\frac{2\sqrt{3}+\sqrt{15}}{6},\,\frac{\sqrt{3}+\sqrt{15}}{6}\right)}$,
• ${\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,-\frac{\sqrt{15}-\sqrt{3}}{12},\,\frac{\sqrt{3}+\sqrt{15}}{6}\right)}$,
• ${\displaystyle \left(\pm\frac{1+\sqrt5}{4},\,-\frac{5\sqrt{3}+\sqrt{15}}{12},\,\frac{\sqrt{3}+\sqrt{15}}{6}\right)}$,
• ${\displaystyle \left(\pm\frac{1}{2},\,\pm\frac{\sqrt{3}}{2},\,0\right)}$,
• ${\displaystyle \left(\pm1,\,0,\,0\right)}$.

## Related polytopes

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.