Pentagonal duotegum

Pentagonal duotegum
(OFF file)
Rank	4
Type	Noble
Space	Spherical
Notation
Bowers style acronym	Pedit
Coxeter diagram	m5o2m5o
Elements
Cells	25 tetragonal disphenoids
Faces	50 isosceles triangles
Edges	10+25
Vertices	10
Vertex figure	Pentagonal tegum
Measures (based on pentagons of edge length 1)
Edge lengths	Base (10): 1
	Lacing (25): $\sqrt{\frac{5+\sqrt5}{5}} ≈ 1.20300$
Circumradius	$\sqrt{\frac{5+\sqrt5}{10}} ≈ 0.85065$
Inradius	$\sqrt{\frac{5+2\sqrt5}{40}} ≈ 0.48662$
Central density	1
Related polytopes
Army	Pedit
Regiment	Pedit
Dual	Pentagonal duoprism
Conjugate	Pentagrammic duotegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2≀S2, order 200
Convex	Yes
Nature	Tame

The pentagonal duotegum or pedit, also known as the pentagonal-pentagonal duotegum, the 5 duotegum, or the 5-5 duotegum, is a noble duotegum that consists of 25 tetragonal disphenoids and 10 vertices, with 10 cells joining at each vertex. It is also the 10-4 step prism and the square funk tegum. It is the first in an infinite family of isogonal pentagonal hosohedral swirlchora and also the first in an infinite family of isochoric pentagonal dihedral swirlchora.

The ratio between the longest and shortest edges is 1:$\frac{\sqrt{25+5\sqrt5}}{5}$ ≈ 1:1.20300.

Vertex coordinates

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

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