Decagonal duoprism

Decagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Dedip
Coxeter diagram	x10o x10o ()
Elements
Cells	20 decagonal prisms
Faces	100 squares, 20 decagons
Edges	200
Vertices	100
Vertex figure	Tetragonal disphenoid, edge lengths √(5+√5)/2 (bases) and √2 (sides)
Measures (edge length 1)
Circumradius	$\frac{\sqrt2+\sqrt{10}}{2} ≈ 2.28825$
Inradius	$\frac{\sqrt{5+2\sqrt5}}{2} ≈ 1.53884$
Hypervolume	$\frac{25(5+2\sqrt5)}{4}≈ 59.20085$
Dichoral angles	Dip–10–dip: 144º
	Dip–4–dip: 90°
Central density	1
Number of external pieces	20
Level of complexity	3
Related polytopes
Army	Dedip
Regiment	Dedip
Dual	Decagonal duotegum
Conjugate	Decagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(10)≀S2, order 800
Convex	Yes
Nature	Tame

The decagonal duoprism or dedip, also known as the decagonal-decagonal duoprism, the 10 duoprism or the 10-10 duoprism, is a noble uniform duoprism that consists of 20 decagonal prisms, with four at each vertex. It is also the 20-9 gyrochoron. It is the first in an infinite family of isogonal decagonal dihedral swirlchora and also the first in an infinite family of isochoric decagonal hosohedral swirlchora.

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

A unit decagonal duoprism can be edge-inscribed into the small ditetrahedronary hexacosihecatonicosachoron.

Gallery

• 

  Wireframe, cell, net

Vertex coordinates

Coordinates for the vertices of a decagonal duoprism of edge length 1, centered at the origin, are given by:

• $\left(0,\,±\frac{1+\sqrt5}{2},\,0,\,±\frac{1+\sqrt5}{2}\right),$
• $\left(0,\,±\frac{1+\sqrt5}{2},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$
• $\left(0,\,±\frac{1+\sqrt5}{2},\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$
• $\left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,0,\,±\frac{1+\sqrt5}{2}\right),$
• $\left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$
• $\left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$
• $\left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,0,\,±\frac{1+\sqrt5}{2}\right),$
• $\left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$
• $\left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right).$

Representations

A decagonal duoprism has the following Coxeter diagrams:

• x10o x10o (full symmetry)
• x5x x10o (one decagon as dipentagon)
• x5x x5x (both decagons have pentagonal symmetry)

External links

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

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

• Klitzing, Richard. "dadip".

• Wikipedia Contributors. "10-10 duoprism".