# Gyroelongated square bipyramid

Gyroelongated square bipyramid
Rank3
TypeCRF
Notation
Bowers style acronymGyesqidpy
Coxeter diagramoxoo4ooxo&#xt
Elements
Faces8+8 triangles
Edges8+8+8
Vertices2+8
Vertex figures2 squares, edge length 1
8 pentagons, edge length 1
Measures (edge length 1)
Volume${\displaystyle {\frac {{\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}}{3}}\approx 1.42840}$
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: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Central density1
Number of external pieces16
Level of complexity6
Related polytopes
ArmyGyesqidpy
RegimentGyesqidpy
DualOrder-4 truncated tetragonal trapezohedron
ConjugateGyroelongated square bipyramid
Abstract & topological properties
Flag count96
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(8)×A1)/2, order 16
Flag orbits6
ConvexYes
NatureTame

The gyroelongated square bipyramid (OBSA: gyesqidpy) 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:

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