Compound of two pentagonal prisms

Compound of two pentagonal prisms
Rank3
TypeUniform
Notation
Coxeter diagramxo5ox xx
Elements
Components2 pentagonal prisms
Faces10 squares, 4 pentagons as 2 stellated decagons
Edges10+20
Vertices20
Vertex figureIsosceles triangle, edge lengths (1+5)/2, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {15+2{\sqrt {5}}}{20}}}\approx 0.98672}$
Volume${\displaystyle {\frac {\sqrt {25+10{\sqrt {5}}}}{2}}\approx 3.44095}$
Dihedral angles4–4: 108°
4–5: 90°
Height1
Central density2
Number of external pieces22
Level of complexity6
Related polytopes
ArmySemi-uniform Dip, edge lengths ${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{10}}}}$ (base), 1 (sides)
Regiment*
DualCompound of two pentagonal tegums
ConjugateCompound of two pentagrammic prisms
Abstract & topological properties
Flag count120
OrientableYes
Properties
SymmetryI2(10)×A1, order 40
ConvexNo
NatureTame

The stellated decagonal prism or compound of two pentagonal prisms is a prismatic uniform polyhedron compound. It consists of 2 stellated decagons and 10 squares. Each vertex joins one stellated decagon and two squares. As the name suggests, it is a prism based on a stellated decagon.

Its quotient prismatic equivalent is the pentagonal antiprismatic prism, which is four-dimensional.

Vertex coordinates

Coordinates for the vertices of a stellated decagonal prism of edge length 1 centered at the origin are given by:

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

Variations

This compound has variants where the bases are non-regular compounds of two pentagons. In these cases the compound has only pentagonal prismatic symmetry and the convex hull is a dipentagonal prism.