# Gyroelongated triangular bicupola

Gyroelongated triangular bicupola
Rank3
TypeCRF
Notation
Bowers style acronymGyetibcu
Coxeter diagramoxos6soxo&#xt
Elements
Faces
Edges7×6
Vertices3×6
Vertex figures6 rectangles, edge lengths 1 and 2
12 irregular pentagons, edge lengths 1, 1, 1, 1, 2
Measures (edge length 1)
Volume${\displaystyle {\frac {5{\sqrt {2}}+3{\sqrt {2+2{\sqrt {3}}}}}{3}}\approx 4.69457}$
Dihedral angles3–3 join: ${\displaystyle \arccos \left({\frac {1}{3}}\right)+\arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 169.42821^{\circ }}$
3–4 join: ${\displaystyle \arccos \left({\frac {\sqrt {3}}{3}}\right)+\arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 153.63504^{\circ }}$
3–3 antiprismatic: ${\displaystyle \arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }}$
3–4 cupolaic: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Central density1
Number of external pieces26
Level of complexity28
Related polytopes
ArmyGyetibcu
RegimentGyetibcu
ConjugateGyroelongated triangular bicupola
Abstract & topological properties
Flag count168
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(A2×A1)+, order 6
Flag orbits28
ConvexYes
NatureTame

The gyroelongated triangular bicupola (OBSA: gyetibcu) is one of the 92 Johnson solids (J44). It consists of 2+6+6+6 triangles and 6 squares. It can be constructed by attaching triangular cupolas to the bases of the hexagonal antiprism.

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

## Vertex coordinates

Coordinates for the vertices of a gyroelongated triangular bicupola of edge length 1 are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,{\frac {2{\sqrt {6}}+3{\sqrt {{\sqrt {3}}-1}}}{6}}\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,{\frac {2{\sqrt {6}}+3{\sqrt {{\sqrt {3}}-1}}}{6}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,{\frac {\sqrt {{\sqrt {3}}-1}}{2}}\right)}$,
• ${\displaystyle \left(\pm 1,\,0,\,{\frac {\sqrt {{\sqrt {3}}-1}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,-{\frac {\sqrt {{\sqrt {3}}-1}}{2}}\right)}$,
• ${\displaystyle \left(0,\,\pm 1,\,-{\frac {\sqrt {{\sqrt {3}}-1}}{2}}\right)}$,
• ${\displaystyle \left(-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,-{\frac {2{\sqrt {6}}+3{\sqrt {{\sqrt {3}}-1}}}{6}}\right)}$,
• ${\displaystyle \left({\frac {\sqrt {3}}{3}},\,0,\,-{\frac {2{\sqrt {6}}+3{\sqrt {{\sqrt {3}}-1}}}{6}}\right)}$.