# Gyroelongated square bicupola

Gyroelongated square bicupola
Rank3
TypeCRF
Notation
Bowers style acronymGyesquibcu
Coxeter diagramsoxo8oxos&#xt
Elements
Faces
Edges7×8
Vertices3×8
Vertex figures8 isosceles trapezoids, edge lengths 1, 2, 2, 2
16 irregular pentagons, edge lengths 1, 1, 1, 1, 2
Measures (edge length 1)
Volume${\displaystyle 2{\frac {3+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}+{\sqrt {146+103{\sqrt {2}}}}}}}{3}}\approx 8.15357}$
Dihedral angles3–3 antiprismatic: ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2+{\sqrt {2}}}}}{3}}\right)\approx 153.96238^{\circ }}$
3–3 join: ${\displaystyle \arccos \left({\frac {\sqrt {3}}{3}}\right)+\arccos \left(-{\sqrt {\frac {7+4{\sqrt {2}}-2{\sqrt {20+14{\sqrt {2}}}}}{3}}}\right)\approx 151.33013^{\circ }}$
3–4 cupolaic: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
3–4 join: ${\displaystyle \arccos \left({\frac {\sqrt {2}}{2}}\right)+\arccos \left(-{\sqrt {\frac {7+4{\sqrt {2}}-2{\sqrt {20+14{\sqrt {2}}}}}{3}}}\right)\approx 141.59451^{\circ }}$
4–4: 135°
Central density1
Number of external pieces34
Level of complexity28
Related polytopes
ArmyGyesquibcu
RegimentGyesquibcu
Abstract & topological properties
Flag count224
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(B2×A1)+, order 8
Flag orbits28
ConvexYes
NatureTame

The gyroelongated square bicupola (OBSA: gyesquibcu) is one of the 92 Johnson solids (J45). It consists of 8+8+8 triangles and 2+8 squares. It can be constructed by attaching square cupolas to the bases of the octagonal antiprism.

It is one of the five Johnson solids to be chiral.

## Vertex coordinates

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

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

where ${\displaystyle H={\sqrt {\frac {-2-2{\sqrt {2}}+{\sqrt {20+14{\sqrt {2}}}}}{8}}}}$ is the distance between the octagonal antiprism's center and the center of one of its bases.