Pentagonal duoantiprism

Pentagonal duoantiprism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Pedap
Coxeter diagram	s10o2s10o ()
Elements
Cells	50 tetragonal disphenoids, 20 pentagonal antiprisms
Faces	200 isosceles triangles, 20 pentagons
Edges	100+100
Vertices	50
Vertex figure	Gyrobifastigium
Measures (based on pentagons of edge length 1)
Edge lengths	Lacing (100): $\sqrt{\frac{5-\sqrt5}{5}} ≈ 0.74350$
	Edges of pentagons (100): 1
Circumradius	$\sqrt{\frac{5+\sqrt5}{5}} ≈ 1.20300$
Central density	1
Related polytopes
Army	Pedap
Regiment	Pedap
Dual	Pentagonal duoantitegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(10)≀S2)/2, order 400
Convex	Yes
Nature	Tame

The pentagonal duoantiprism or pedap, also known as the pentagonal-pentagonal duoantiprism, the 5 duoantiprism or the 5-5 duoantiprism, is a convex isogonal polychoron that consists of 20 pentagonal antiprisms and 50 tetragonal disphenoids. 4 pentagonal antiprisms and 4 tetragonal disphenoids join at each vertex. It can be obtained through the process of alternating the decagonal duoprism. However, it cannot be made uniform, and has two edge lengths. It is the second in an infinite family of isogonal pentagonal dihedral swirlchora.

The ratio between the longest and shortest edges is 1:$\frac{\sqrt{5+\sqrt5}}{2}$ ≈ 1:1.34500.

Vertex coordinates

The vertices of a pentagonal duoantiprism based on pentagons of edge length 1, centered at the origin, are given by:

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