# Pentagrammic disphenoid

Pentagrammic disphenoid
Rank5
TypeScaliform
Notation
Coxeter diagramxo5/2oo ox5/2oo&#x
Elements
Tera10 pentagrammic scalenes
Cells25 tetrahedra, 10 pentagrammic pyramids
Faces50 triangles, 2 pentagrams
Edges10+25
Vertices10
Vertex figurePentagrammic scalene
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {10-{\sqrt {5}}}{20}}}\approx 0.62305}$
Height${\displaystyle {\frac {1}{\sqrt[{4}]{5}}}\approx 0.66874}$
Related polytopes
ArmyPentagonal disphenoid
DualPentagrammic disphenoid
ConjugatePentagonal disphenoid
Abstract & topological properties
OrientableYes
Properties
SymmetryH2▲S2, order 200
ConvexNo
NatureTame

The pentagrammic disphenoid, or stadow, is a noble scaliform polyteron with 10 pentagrammic scalenes and 10 vertices. 7 facets join at each vertex. It can be constructed as the pyramid product of 2 regular pentagrams with a height chosen so that all edge lengths are equal.

## Vertex coordinates

The vertices of a pentagrammic disphenoid of edge length 1 are given by:

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