Square-pentagonal duoprism

Square-pentagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Squipdip
Coxeter diagram	x4o x5o ()
Elements
Cells	5 cubes, 4 pentagonal prisms
Faces	5+20 squares, 4 pentagons
Edges	20+20
Vertices	20
Vertex figure	Digonal disphenoid, edge lengths (1+√5)/2 (base 1) and √2 (base 2 and sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {10+{\sqrt {5}}}{10}}}\approx 1.10617$
Hypervolume	${\frac {\sqrt {25+10{\sqrt {5}}}}{4}}\approx 1.72047$
Dichoral angles	Cube–4–cube: 108°
	Pip–5–pip: 90°
	Cube–4–pip: 90°
Height	1
Central density	1
Number of external pieces	9
Level of complexity	6
Related polytopes
Army	Squipdip
Regiment	Squipdip
Dual	Square-pentagonal duotegum
Conjugate	Square-pentagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B2×H2, order 80
Convex	Yes
Nature	Tame

The square-pentagonal duoprism or squipdip, also known as the 4-5 duoprism, is a uniform duoprism that consists of 4 pentagonal prisms and 5 cubes, with two of each joining at each vertex.

It is also a CRF segmentochoron, being a pentagonal prism atop pentagonal prism. It is designated K-4.42 on Richard Klitzing's list. It can thus be thought of as a prism based on the pentagonal prism.

Vertex coordinates

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

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

Representations

A suare-pentagonal duoprism has the following Coxeter diagrams:

• x4o x5o (full symmetry)
• x x x5o (H2×A1×A1 symmetry, square as rectangle)
• xx xx5oo#x (H2×A1 axial, pentagonal prism prism)
• ofx xxx4ooo&#xt (BC2×A1 axial, square-first)
• ofx xxx xxx&#xt (A1×A1×A1, as above with rectangle symmetry)
• oqo xxx5ooo&#xt (H2×A1 axial, pentagon-first)

External links

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

• Klitzing, Richard. "squipdip".