# Gyroelongated square pyramid

Gyroelongated square pyramid
Rank3
TypeCRF
Notation
Bowers style acronymGyesp
Coxeter diagramoxo4oox&#xt
Elements
Faces
Edges4+4+4+8
Vertices1+4+4
Vertex figures1 square, edge length 1
4 pentagons, edge length 1
4 trapezoids, edge lengths 1, 1, 1, 2
Measures (edge length 1)
Volume${\displaystyle {\frac {{\sqrt {2}}+2{\sqrt {4+3{\sqrt {2}}}}}{6}}\approx 1.19270}$
Dihedral angles3–3 join: ${\displaystyle \arccos \left({\frac {1-{\sqrt {2}}-2{\sqrt[{4}]{2}}}{3}}\right)\approx 158.57177^{\circ }}$
3–3 antiprismatic: ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }}$
3–3 pyramidal: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
3–4: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }}$
Central density1
Number of external pieces13
Level of complexity10
Related polytopes
ArmyGyesp
RegimentGyesp
DualOrder-4 monotruncated tetragonal trapezohedron
ConjugateGyroelongated square pyramid
Abstract & topological properties
Flag count80
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB2×I, order 8
Flag orbits10
ConvexYes
NatureTame

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:

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