Pentagrammic-decagonal duoprism

Pentagrammic-decagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Stardedip
Coxeter diagram	x5/2o x10o ()
Elements
Cells	10 pentagrammic prisms, 5 decagonal prisms
Faces	50 squares, 10 pentagrams, 5 decagons
Edges	50+50
Vertices	50
Vertex figure	Digonal disphenoid, edge lengths (√5–1)/2 (base 1), √(5+√5)/2 (base 2), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {10+2{\sqrt {5}}}{5}}}\approx 1.70130$
Hypervolume	${\frac {25}{8}}=3.125$
Dichoral angles	Stip–5/2–stip: 144°
	Stip–4–dip: 90°
	Dip–10–dip: 36°
Central density	2
Number of external pieces	20
Level of complexity	12
Related polytopes
Army	Semi-uniform padedip
Regiment	Stardedip
Dual	Pentagrammic-decagonal duotegum
Conjugate	Pentagonal-decagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2×I2(10), order 200
Convex	No
Nature	Tame

The pentagrammic-decagonal duoprism, also known as stardedip or the 5/2-10 duoprism, is a uniform duoprism that consists of 10 pentagrammic prisms and 5 decagonal prisms, with 2 of each at each vertex.

Vertex coordinates

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

• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\right)$,
• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right)$,
• $\left(0,\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\right)$,
• $\left(0,\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\right)$,
• $\left(0,\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right)$.

Representations

A pentagrammic-decagonal duoprism has the following Coxeter diagrams:

• x5/2o x10o () (full symmetry)
• x5x x5/2o () (H₂×H₂ symmetry, decagons as dipentagons)

External links

• Bowers, Jonathan. "Category A: Duoprisms".

• Klitzing, Richard. "stardedip".