Gyroelongated triangular bicupola

Gyroelongated triangular bicupola
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Gyetibcu
Coxeter diagram	oxos6soxo&#xt
Elements
Faces	2+3×6 triangles 6 squares
Edges	7×6
Vertices	3×6
Vertex figures	6 rectangles, edge lengths 1 and √2
	12 irregular pentagons, edge lengths 1, 1, 1, 1, √2
Measures (edge length 1)
Volume	${\frac {5{\sqrt {2}}+3{\sqrt {2+2{\sqrt {3}}}}}{3}}\approx 4.69457$
Dihedral angles	3–3 join: $\arccos \left({\frac {1}{3}}\right)+\arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 169.42821^{\circ }$
	3–4 join: $\arccos \left({\frac {\sqrt {3}}{3}}\right)+\arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 153.63504^{\circ }$
	3–3 antiprismatic: $\arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }$
	3–4 cupolaic: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
Central density	1
Number of external pieces	26
Level of complexity	28
Related polytopes
Army	Gyetibcu
Regiment	Gyetibcu
Dual	Trideltohexapentagonal octadecahedron
Conjugate	Gyroelongated triangular bicupola
Abstract & topological properties
Flag count	168
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(A2×A1)+, order 6
Flag orbits	28
Convex	Yes
Nature	Tame

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:

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

External links

• Klitzing, Richard. "gyetibcu".
• Quickfur. "The Gyroelongated Triangular Bicupola".

• Wikipedia contributors. "Gyroelongated triangular bicupola".
• McCooey, David. "Gyroelongated Triangular Bicupola"