Triangular orthobicupola

Triangular orthobicupola
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Tobcu
Coxeter diagram	xxx3oxo&#xt
Elements
Faces	2+6 triangles 6 squares
Edges	3+3+6+12
Vertices	6+6
Vertex figures	6 rectangles, edge lengths 1 and √2
	6 kites, edge lengths 1 and √2
Measures (edge length 1)
Circumradius	$1$
Volume	${\frac {5{\sqrt {2}}}{3}}\approx 2.35702$
Dihedral angles	3–3: $\arccos \left(-{\frac {7}{9}}\right)\approx 141.05756^{\circ }$
	3–4: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	4–4: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
Central density	1
Number of external pieces	14
Level of complexity	8
Related polytopes
Army	Tobcu
Regiment	Tobcu
Dual	Trapezo-rhombic dodecahedron
Conjugate	None
Abstract & topological properties
Flag count	96
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×A1, order 12
Flag orbits	8
Convex	Yes
Nature	Tame

The triangular orthobicupola (OBSA: tobcu) is one of the 92 Johnson solids (J₂₇). It consists of 2+6 triangles and 6 squares. It can be constructed by attaching two triangular cupolas at their hexagonal bases, such that the two triangular bases are in the same orientation.

If the cupolas are joined such that the bases are rotated 60°, the result is the triangular gyrobicupola, better known as the uniform cuboctahedron. Conversely, a triangular orthobicupola can be thought of as a gyrate cuboctahedron, since it is a cuboctahedron with a cupolaic segment rotated.

The triangular orthobicupola is the vertex figure of the gyrated tetrahedral-octahedral honeycomb.

Vertex coordinates

A triangular orthobicupola of edge length 1 has vertices given by the following coordinates:

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

Related polyhedra

A hexagonal prism can be inserted between the two halves of the triangular orthobicupola to produce the elongated triangular orthobicupola.

External links

• Klitzing, Richard. "tobcu".
• Quickfur. "The Triangular Orthobicupola".

• Wikipedia contributors. "Triangular orthobicupola".
• McCooey, David. "Triangular Orthobicupola"