Compound of two pentagonal prisms

Compound of two pentagonal prisms
Rank	3
Type	Uniform
Notation
Coxeter diagram	xo5ox xx
Elements
Components	2 pentagonal prisms
Faces	10 squares, 4 pentagons as 2 stellated decagons
Edges	10+20
Vertices	20
Vertex figure	Isosceles triangle, edge lengths (1+√5)/2, √2, √2
Measures (edge length 1)
Circumradius	${\sqrt {\frac {15+2{\sqrt {5}}}{20}}}\approx 0.98672$
Volume	${\frac {\sqrt {25+10{\sqrt {5}}}}{2}}\approx 3.44095$
Dihedral angles	4–4: 108°
	4–5: 90°
Height	1
Central density	2
Number of external pieces	22
Level of complexity	6
Related polytopes
Army	Semi-uniform Dip, edge lengths ${\sqrt {\frac {5-{\sqrt {5}}}{10}}}$ (base), 1 (sides)
Regiment	*
Dual	Compound of two pentagonal tegums
Conjugate	Compound of two pentagrammic prisms
Abstract & topological properties
Flag count	120
Orientable	Yes
Properties
Symmetry	I2(10)×A1, order 40
Convex	No
Nature	Tame

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:

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

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.