Pentagonal-pentagrammic duoprism

Pentagonal-pentagrammic duoprism
	(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Starpedip
Coxeter diagram	x5o x5/2o ()
Elements
Cells	5 pentagonal prisms, 5 pentagrammic prisms
Faces	25 squares, 5 pentagons, 5 pentagrams
Edges	25+25
Vertices	25
Vertex figure	Digonal disphenoid, edge lengths (√5+1)/2 (base 1), (√5–1)/2 (base 2), √2 (sides)
Measures (edge length 1)
Circumradius	1
Hypervolume
Dichoral angles	Stip–5/2–stip: 108°
	Pip–4–stip: 90°
	Pip–5–pip: 36°
Central density	2
Number of external pieces	15
Level of complexity	12
Related polytopes
Army	Semi-uniform pedip
Regiment	Starpedip
Dual	Pentagonal-pentagrammic duotegum
Conjugate	Pentagonal-pentagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2×H2, order 100
Convex	No
Nature	Tame

The pentagonal-pentagrammic duoprism, also known as starpedip or the 5-5/2 duoprism, is a uniform duoprism that consists of 5 pentagonal prisms and 5 pentagrammic prisms, with 2 of each at each vertex.

This is the only duoprism aside from the tesseract to have a circumradius equal to its edge length.

The pentagonal-pentagrammic duoprism can be edge-inscribed into the small stellated hecatonicosachoron. The small stellated hecatonicosachoron's regiment contains a uniform compound of 24 pentagonal-pentagrammic duoprisms.

Vertex coordinates[edit | edit source]

The coordinates of a pentagonal-pentagrammic duoprism, centered at the origin and with unit edge length, are given by:

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