Triangular cupolic prism

Triangular cupolic prism
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Tricupe
Coxeter diagram	xx ox3xx&#x
Elements
Cells	1+3 triangular prisms, 3 cubes, 2 triangular cupolas, 1 hexagonal prism
Faces	2+6 triangles, 3+3+3+6+6 squares, 2 hexagons
Edges	3+6+6+6+6+12
Vertices	6+12
Vertex figures	6 rectangular pyramids, base edge lengths 1 and √2, side edge length √2
	12 irregular tetrahedra, edge lengths 1 (1), √2 (4), and √3 (1)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {5}}{2}}\approx 1.11803$
Hypervolume	${\frac {5{\sqrt {2}}}{6}}\approx 1.17851$
Dichoral angles	Trip–4–cube: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)125.26439^{\circ }$
	Tricu–3–trip: 90°
	Tricu–4–cube: 90°
	Tricu–6–hip: 90°
	Trip–4–hip: $\arccos \left({\frac {1}{3}}\right)\approx 70.52878^{\circ }$
	Cube–4–hip: $\arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }$
Heights	Tricu atop tricu: 1
	Trip atop hip: ${\frac {\sqrt {6}}{3}}\approx 0.81650$
Central density	1
Related polytopes
Army	Tricupe
Regiment	Tricupe
Dual	Semibisected hexagonal trapezohedral tegum
Conjugate	None
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A2×A1×I, order 12
Convex	Yes
Nature	Tame

The triangular cupolic prism, or tricupe, is a CRF segmentochoron (designated K-4.45 on Richard Klitzing's list). It consists of 2 triangular cupolas, 4 triangular prisms, 3 cubes, and 1 hexagonal prism.

As the name suggests, it is a prism based on the triangular cupola. As such, it is a segmentochoron between two triangular cupolas. It can also be viewed as a segmentochoron between a hexagonal prism and a triangular prism.

Two triangular cupolic prisms can be joined at their hexagonal prismatic cells in opposite orientations to form a cuboctahedral prism. By rotating one of the triangular cupolic prisms before joining, one can instead form a triangular orthobicupolic prism.

Vertex coordinates

Coordinates of the vertices of a triangular cupolic prism of edge length 1 centered at the origin are given by:

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

External links

• Klitzing, Richard. "tricupe".