Square-decagonal duoprism

Square-decagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Squadedip
Coxeter diagram	x4o x10o ()
Elements
Cells	10 cubes, 4 decagonal prisms
Faces	10+40 squares, 4 decagons
Edges	40+40
Vertices	40
Vertex figure	Digonal disphenoid, edge lengths √(5+√5)/2 (base 1) and √2 (base 2 and sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {4+{\sqrt {5}}}{2}}}\approx 1.76580$
Hypervolume	$5{\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\approx 7.69421$
Dichoral angles	Cube–4–cube: 144°
	Dip–10–dip: 90°
	Cube–4–dip: 90°
Height	1
Central density	1
Number of external pieces	14
Level of complexity	6
Related polytopes
Army	Squadedip
Regiment	Squadedip
Dual	Square-decagonal duotegum
Conjugate	Square-decagrammic duoprism
Abstract & topological properties
Flag count	960
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B2×I2(10), order 160
Flag orbits	6
Convex	Yes
Nature	Tame

The square-decagonal duoprism or squadedip, also known as the 4-10 duoprism, is a uniform duoprism that consists of 4 decagonal prisms and 10 cubes, with two of each joining at each vertex.

It is a prism based on the decagonal prism. As such it is also a CRF segmentochoron, designated K-4.97 on Richard Klitzing's list.

This polychoron can be alternated into a digonal-pentagonal duoantiprism, although it cannot be made uniform.

Gallery

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  Net

Vertex coordinates

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

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

Representations

A square-decagonal duoprism has the following Coxeter diagrams:

• x4o x10o () (full symmetry)
• x4o x5x () (B₂×H₂ symmetry, decagon as dipentagon)
• x x x10o () (I₂(10)×K₂ symmetry, square as rectangle)
• x x x5x () (H₂×K₂ symmetry, both above)
• xx xx10oo (I₂(10)×A₁ symmetry, decagon prismatic prism)
• xx xx5xx&#x (H₂×A₁ axial, as above)
• oqo xxx10ooo&#xt
• oqo xxx5xxx&#xt (H₂×A₁ axial, decagon-first)

External links

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

• Klitzing, Richard. "squadedip".