Gyroelongated pentagonal pyramid

Gyroelongated pentagonal pyramid
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Gyepip
Coxeter diagram	oxo5oox&#xt
Elements
Faces	5+5+5 triangles, 1 pentagon
Edges	5+5+5+10
Vertices	1+5+5
Vertex figures	1+5 pentagons, edge length 1
	5 trapezoids, edge lengths 1, 1, 1, (1+√5)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{10+2\sqrt5}}4 ≈ 0.95106$
Volume	$\frac{25+9\sqrt5}{24} ≈ 1.88019$
Dihedral angles	3–3: $\arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°$
	3–5: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) 100.81232°$
Central density	1
Related polytopes
Army	Gyepip
Regiment	Gyepip
Dual	Order-5 monotruncated pentagonal trapezohedron
Conjugate	Retrogyroelongated pentagrammic pyramid
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The gyroelongated pentagonal pyramid is one of the 92 Johnson solids (J₁₁). It consists of 5+5+5 triangles and 1 pentagon. It can be constructed by attaching a pentagonal antiprism to the base of the pentagonal pyramid.

Alternatively, it can be constructed by diminishing one vertex from the regular icosahedron, which is why this polyhedron can also be called the diminished icosahedron. This means the icosahedron can also be thought of as a gyroelongated pentagonal bipyramid.

Vertex coordinates

A gyroelongated pentagonal pyramid of edge length 1 has the following vertices:

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

Alternative coordinates can be obtained by removing one vertex from the regular icosahedron:

• $\left(±\frac{1+\sqrt5}{4},\,+\frac12,\,0\right),$
• $\left(0,\,±\frac{1+\sqrt5}{4},\,±\frac12\right),$
• $\left(±\frac12,\,0,\,\frac{1+\sqrt5}{4}\right),$
• $\left(\frac12,\,0,\,-\frac{1+\sqrt5}{4}\right).$

External links

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#8 under ike).

• Klitzing, Richard. "gyepip".

• Quickfur. "The Gyroelongated Pentagonal Pyramid".

• Wikipedia Contributors. "Gyroelongated pentagonal pyramid".
• McCooey, David. "Gyroelongated Pentagonal Pyramid"