Gyroelongated square pyramid

Gyroelongated square pyramid
(3D model) (OFF file)
Rank	3
Type	CRF
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	${\frac {{\sqrt {2}}+2{\sqrt {4+3{\sqrt {2}}}}}{6}}\approx 1.19270$
Dihedral angles	3–3 join: $\arccos \left({\frac {1-{\sqrt {2}}-2{\sqrt[{4}]{2}}}{3}}\right)\approx 158.57177^{\circ }$
	3–3 antiprismatic: $\arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }$
	3–3 pyramidal: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	3–4: $\arccos \left(-{\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }$
Central density	1
Number of external pieces	13
Level of complexity	10
Related polytopes
Army	Gyesp
Regiment	Gyesp
Dual	Order-4 monotruncated tetragonal trapezohedron
Conjugate	Gyroelongated square pyramid
Abstract & topological properties
Flag count	80
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B2×I, order 8
Flag orbits	10
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:

• $\left(0,\,0,\,{\frac {2{\sqrt {2}}+{\sqrt[{4}]{8}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}}\right)$,
• $\left(0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}}\right)$,
• $\left(\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}}\right)$.

External links

• Klitzing, Richard. "gyesp".
• Quickfur. "The Gyroelongated Square Pyramid".

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