Elongated pentagonal rotunda

Elongated pentagonal rotunda
	(3D model); (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Epro
Coxeter diagram	ofxx5xoxx&#xt
Elements
Faces	5+5 triangles, 5+5 squares, 1+5 pentagons, 1 decagon
Edges	5+5+5+5+5+10+10+10
Vertices	5+5+10+10
Vertex figures	5 rectangles, edge lengths 1 and (1+√5)/2
	10 irregular tetragons, edge lengths 1, (1+√5)/2, √2, √2
	10 isosceles triangles, edge lengths √(5+√5)/2, √2, √2
Measures (edge length 1)
Volume
Dihedral angles	3–4:
	5–4:
	4–4: 144°
	3–5:
	4–10: 90°
Central density	1
Related polytopes
Army	Epro
Regiment	Epro
Dual	Decakis order-10 truncated semibisected pentagonal rhombitrapezohedron
Conjugate	Elongated pentagrammic rotunda
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The elongated pentagonal rotunda is one of the 92 Johnson solids (J₂₁). It consists of 5+5 triangles, 5+5 squares, 1+5 pentagons, and 1 decagon. It can be constructed by attaching a decagonal prism to the decagonal base of the pentagonal rotunda.

If a second rotunda is attached to the other decagonal base of the prism in the same orientation, the result is the elongated pentagonal orthobirotunda. If the second rotunda is rotated 36º instead, the result is the elongated pentagonal gyrobirotunda.

Vertex coordinates[edit | edit source]

An elongated pentagonal rotunda of edge length 1 has the following vertices:

$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$
$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12\right),$
$\left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12\right),$
$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),$
$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),$
$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),$
$\left(0,\,-\sqrt{\frac{5+2\sqrt5}{5}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right),$
$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{25+11\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right),$
$\left(±\frac{3+\sqrt5}{4},\,-\sqrt{\frac{5+\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right).$

The three tunnellings of the elongated pentagonal rotunda. The convex hull is shown as a transparency.

Three quasi-convex Stewart toroids are made by tunnelling the elongated pentagonal rotunda. Two of these are distinct elongations of the tunnelled pentagonal rotunda.