Decagonal duoprismatic prism

Decagonal duoprismatic prism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Daddip
Coxeter diagram	x x10o x10o ()
Elements
Tera	20 square-decagonal duoprisms, 2 decagonal duoprisms
Cells	100 cubes, 20+40 decagonal prisms
Faces	200+200 squares, 40 decagons
Edges	100+400
Vertices	200
Vertex figure	Tetragonal disphenoidal pyramid, edge lengths √(5+√5)/2 (disphenoid bases) and √2 (remaining edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {13+4{\sqrt {5}}}}{2}}\approx 2.34224$
Hypervolume	${\frac {25(5+2{\sqrt {5}})}{4}}\approx 59.20085$
Diteral angles	Squadedip–dip–squadedip: 144°
	Squadedip–cube–squadedip: 90°
	Dedip–dip–squadedip: 90°
Height	Dedip atop dedip: 1
Central density	1
Number of external pieces	22
Level of complexity	15
Related polytopes
Army	Daddip
Regiment	Daddip
Dual	Decagonal duotegmatic tegum
Conjugate	Decagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	I2(10)≀S2×A1, order 1600
Convex	Yes
Nature	Tame

The decagonal duoprismatic prism or daddip, also known as the decagonal-decagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 decagonal duoprisms and 20 square-decagonal duoprisms. Each vertex joins 4 square-decagonal duoprisms and 1 decagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

This polyteron can be alternated into a pentagonal duoantiprismatic antiprism, although it cannot be made uniform.

Vertex coordinates

The vertices of an decagonal duoprismatic prism of edge length 2sin(π/9) are given by:

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

Representations

A decagonal duoprismatic prism has the following Coxeter diagrams:

• x x10o x10o () (full symmetry)
• x x5x x5x () (decagons as dipentagons)
• xx10oo xx10oo&#x (decagonal duoprism atop decagonal duoprism)
• xx5xx xx5xx&#x