Pentagonal-hexagonal duoprism

Pentagonal-hexagonal duoprism
	(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Phiddip
Coxeter diagram	x5o2x6o ()
Elements
Cells	6 pentagonal prisms, 5 hexagonal prisms
Faces	30 squares, 6 pentagons, 5 hexagons
Edges	30+30
Vertices	30
Vertex figure	Digonal disphenoid, edge lengths (1+√5)/2 (base 1), √3 (base 2), and √2 (sides)
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral angles	Pip–5–pip: 120°
	Hip–6–hip: 108°
	Hip–4–pip: 90°
Central density	1
Number of external pieces	11
Level of complexity	6
Related polytopes
Army	Phiddip
Regiment	Phiddip
Dual	Pentagonal-hexagonal duotegum
Conjugate	Pentagrammic-hexagonal duoprism
Abstract & topological properties
Flag count	720
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2×G2, order 120
Convex	Yes
Nature	Tame

The pentagonal-hexagonal duoprism or phiddip, also known as the 5-6 duoprism, is a uniform duoprism that consists of 5 hexagonal prisms and 6 pentagonal prisms, with two of each joining at each vertex.

Vertex coordinates[edit | edit source]

Coordinates for the vertices of a pentagonal-hexagonal duoprism with edge length 1 are given by:

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