Gyroelongated square bipyramid

Gyroelongated square bipyramid
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Gyesqidpy
Coxeter diagram	oxoo4ooxo&#xt
Elements
Faces	8+8 triangles
Edges	8+8+8
Vertices	2+8
Vertex figures	2 squares, edge length 1
	8 pentagons, edge length 1
Measures (edge length 1)
Volume	$\frac{\sqrt2+\sqrt{4+3\sqrt2}}{3} ≈ 1.42840$
Dihedral angles	3–3 join: $\arccos\left(\frac{1-\sqrt2-2\sqrt[4]{2}}{3}\right) ≈ 158.57177°$
	3–3 antiprismatic: $\arccos\left(\frac{2\sqrt2-1}{3}\right) ≈ 127.55160°$
	3–3 pyramidal: $\arccos\left(-\frac13\right) ≈ 109.47122°$
Central density	1
Related polytopes
Army	Gyesqidpy
Regiment	Gyesqidpy
Dual	Order-4 truncated tetragonal trapezohedron
Conjugate	Gyroelongated square bipyramid
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(8)×A1)/2, order 16
Convex	Yes
Nature	Tame

The gyroelongated square bipyramid is one of the 92 Johnson solids (J₁₇). It consists of 8+8 triangles as faces. It can be constructed by inserting a square antiprism between the two pyramidal halves of the regular octahedron, seen as a square bipyramid.

Vertex coordinates

A gyroelongated square bipyramid of edge length 1 has the following vertices:

• $\left(0,\,0,\,±\frac{2\sqrt2+\sqrt[4]{8}}{4}\right),$
• $\left(±\frac12,\,±\frac12,\,\frac{\sqrt[4]{8}}{4}\right),$
• $\left(0,\,±\frac{\sqrt2}{2},\,-\frac{\sqrt[4]{8}}{4}\right),$
• $\left(±\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt[4]{8}}{4}\right).$

External links

• Klitzing, Richard. "gyesqidpy".

• Quickfur. "The Gyroelongated Square Bipyramid".

• Wikipedia Contributors. "Gyroelongated square bipyramid".
• McCooey, David. "Gyroelongated Square dipyramid"