Pentagonal-decagonal duoprism

Pentagonal-decagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Padedip
Coxeter diagram	x5o x10o ()
Elements
Cells	10 pentagonal prisms, 5 decagonal prisms
Faces	50 squares, 10 pentagons, 5 decagons
Edges	50+50
Vertices	50
Vertex figure	Digonal disphenoid, edge lengths (1+√5)/2 (base 1), √(5+√5)/2 (base 2), and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {10+3{\sqrt {5}}}{5}}}\approx 1.82802$
Hypervolume	${\frac {25(2+{\sqrt {5}})}{8}}\approx 13.23771$
Dichoral angles	Pip–5–pip: 144°
	Dip–5–dip: 108°
	Dip–4–pip: 90°
Central density	1
Number of external pieces	15
Level of complexity	6
Related polytopes
Army	Padedip
Regiment	Padedip
Dual	Pentagonal-decagonal duotegum
Conjugate	Pentagrammic-decagrammic duoprism
Abstract & topological properties
Flag count	1200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2×I2(10), order 200
Flag orbits	6
Convex	Yes
Nature	Tame

The pentagonal-decagonal duoprism or padedip, also known as the 5-10 duoprism, is a uniform duoprism that consists of 5 decagonal prisms and 10 pentagonal prisms, with two of each joining at each vertex.

The convex hull of two orthogonal pentagonal-decagonal duoprisms is either the pentagonal duoexpandoprism or the pentagonal duotruncatoprism.

Gallery

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  Net

Vertex coordinates

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

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

Representations

A pentagonal-decagonal duoprism has the following Coxeter diagrams:

• x5o x10o () (full symmetry)
• x5x x5o () (H₂×H₂, decagons as dipentagons)
• ofx xxx10ooo&#xt I₂(10)×A₁ axial)
• ofx xxx5xxx&#xt (H₂×A₁ symmetry, dipentagonal axial)

External links

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

• Klitzing, Richard. "padedip".
• Quickfur. "The 5,10-Duoprism".