Gyroelongated square pyramid

Gyroelongated square pyramid
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Gyesp
Coxeter diagram	oxo4oox&#xt
Elements
Faces	4+4+4 triangles, 1 square
Edges	4+4+4+8
Vertices	1+4+4
Vertex figures	1 square, edge length 1
	4 pentagons, edge length 1
	4 trapezoids, edge lengths 1, 1, 1, √2
Measures (edge length 1)
Volume	$\frac{\sqrt2+2\sqrt{4+3\sqrt2}}{6} ≈ 1.19270$
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: $\arccos\left(-\frac13\right) ≈ 109.47122°$
	3–4: $\arccos\left(-\frac{\sqrt6-\sqrt3}{3}\right) ≈ 103.83616°$
Central density	1
Related polytopes
Army	Gyesp
Regiment	Gyesp
Dual	Order-4 monotruncated tetragonal trapezohedron
Conjugate	Gyroelongated square pyramid
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B2×I, order 8
Convex	Yes
Nature	Tame

The gyroelongated square pyramid is one of the 92 Johnson solids (J₁₀). It consists of 4+4+4 triangles and 1 square. It can be constructed by attaching a square antiprism to the base of the square pyramid.

If a second pyramid is attached to the other base of the square antiprism, the result is the gyroelongated square bipyramid.

Vertex coordinates

A gyroelongated square pyramid 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. "gyesp".

• Quickfur. "The Gyroelongated Square Pyramid".

• Wikipedia Contributors. "Gyroelongated square pyramid".
• McCooey, David. "Gyroelongated Square Pyramid"