Pentagonal duoprism

The pentagonal duoprism or pedip, also known as the pentagonal-pentagonal duoprism, the 5 duoprism or the 5-5 duoprism, is a noble uniform duoprism that consists of 10 pentagonal prisms, with 4 meeting at each vertex. It is also the 10-4 gyrochoron and the square funk prism. It is the first in an infinite family of isogonal pentagonal dihedral swirlchora and also the first in an infinite family of isochoric pentagonal hosohedral swirlchora.

Pentagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Pedip
Coxeter diagram	x5o x5o ()
Elements
Cells	10 pentagonal prisms
Faces	25 squares, 10 pentagons
Edges	50
Vertices	25
Vertex figure	Tetragonal disphenoid, edge lengths (1+√5)/2 (bases) and √2 (sides)
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5+\sqrt5}{5}} ≈ 1.20300$
Inradius	$\sqrt{\frac{5+2\sqrt5}{20}} ≈ 0.68819$
Hypervolume	$\frac{25+10\sqrt5}{16} ≈ 2.96004$
Dichoral angles	Pip–5–pip: 108°
	Pip–4–pip: 90°
Central density	1
Number of external pieces	10
Level of complexity	3
Related polytopes
Army	Pedip
Regiment	Pedip
Dual	Pentagonal duotegum
Conjugate	Pentagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2≀S2, order 200
Convex	Yes
Nature	Tame

A pentagonal duoprism of edge length 1 contains the vertices of a regular pentachoron of edge length $\sqrt{\frac{5+\sqrt5}{2}}$, due to the fact the pentachoron is also the 5-2 step prism.

Gallery

• 

  Wireframe, cell, net

• 

  Wireframe, cell, net as 10-4 gyrochoron

Vertex coordinates

The vertices of a pentagonal duoprism of edge length 1, centered at the origin, are given by:

• $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$
• $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$
• $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$
• $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$
• $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$
• $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right).$

Representations

A pentagonal duoprism has the following Coxeter diagrams:

• x5o x5o (full symmetry)
• ofx xxx5ooo&#xt (pentagonal axial)

External links

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

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

• Klitzing, Richard. "pedip".

• Wikipedia Contributors. "5-5 duoprism".