# Pentagonal-pentagrammic duotegum

Pentagonal-pentagrammic duotegum
Rank4
TypeUniform dual
SpaceSpherical
Notation
Bowers style acronymStapedit
Coxeter diagramm5o2m5/2o
Elements
Cells25 tetrahedra
Faces25+25 triangles
Edges5+5+25
Vertices5+5
Vertex figure5 pentagonal tegums, 5 pentagrammic tegums
Measures (edge length 1)
Inradius${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Hypervolume${\displaystyle \frac{25}{48} ≈ 0.52083}$
Related polytopes
ArmyStapedit
RegimentStapedit
DualPentagonal-pentagrammic duoprism
ConjugatePentagonal-pentagrammic duotegum
Abstract & topological properties
OrientableYes
Properties
SymmetryH2×H2, order 100
ConvexNo
NatureTame

The pentagonal-pentagrammic duotegum, also known as the 5-5/2 duotegum, is a duotegum that consists of 25 regular tetrahedra and 10 vertices.

This is one of only two 4D duotegums that can be made to have regular tetrahedral cells and equilateral triangular faces. The other is the square duotegum, which is the regular hexadecachoron.

## Vertex coordinates

The vertices of a pentagonal-pentagrammic duotegum of edge length 1 are given by:

• ${\displaystyle \left(±\frac{1}{2},\, -\sqrt{\frac{5+2\sqrt{5}}{20}},\,0,\,0\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt{5}}{4},\, \sqrt{\frac{5-\sqrt{5}}{40}},\,0,\,0\right),}$
• ${\displaystyle \left(0,\, \sqrt{\frac{5+\sqrt{5}}{10}},\,0,\,0\right),}$
• ${\displaystyle \left(0,\,0,\,±\frac12,\,-\sqrt{\frac{5-2\sqrt5}{20}}\right),}$
• ${\displaystyle \left(0,\,0,\,±\frac{\sqrt5-1}{4},\,\sqrt{\frac{5+\sqrt5}{40}}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,-\sqrt{\frac{5-\sqrt5}{10}}\right).}$