Triangular cupofastegium

Triangular cupofastegium
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Tricuf
Coxeter diagram	ox xx3xo&#x
Elements
Cells	3 tetrahedra, 1+3 triangular prisms, 2 triangular cupolas
Faces	2+6+6 triangles, 3+6 squares, 1 hexagon
Edges	3+3+3+6+12
Vertices	6+6
Vertex figures	6 skewed rectangular pyramids, base edge lengths 1, √2, 1, √2, side edge lengths 1, 1, √2. √2
	6 sphenoids, edge lengths 1 (3), √2 (2), and √3 (1)
Measures (edge length 1)
Circumradius	1
Hypervolume	${\frac {5{\sqrt {5}}}{32}}\approx 0.34939$
Dichoral angles	Trip–4–trip: $\arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }$
	Tet–3–trip: $\arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }$
	Tet–3–tricu: $\arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }$
	Tricu–6–tricu: $\arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }$
	Tricu–4–trip: $\arccos \left({\frac {\sqrt {6}}{6}}\right)\approx 65.90516^{\circ }$
	Tricu–3–trip: $\arccos \left({\frac {\sqrt {6}}{4}}\right)\approx 52.23876^{\circ }$
Heights	Trig atop tricu: ${\frac {\sqrt {10}}{4}}\approx 0.79057$
	Trip atop hig: ${\frac {\sqrt {15}}{6}}\approx 0.64550$
Central density	1
Related polytopes
Army	Tricuf
Regiment	Tricuf
Dual	Triangular cupolanotch
Conjugate	None
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A2×A1×I, order 12
Convex	Yes
Nature	Tame

The triangular cupofastegium, or tricuf, also called the triangular orthobicupolic ring, is a CRF segmentochoron (designated K-4.25 on Richard Klitzing's list). It consists of 1+3 triangular prisms, 3 tetrahedra, and 2 triangular cupolas.

The triangular cupofastegium can be thought of as a wedge of the small prismatodecachoron, or as a part of the larger segmentochoron tetrahedron atop cuboctahedron, with the remainder forming the segmentochoron tetrahedron atop triangular cupola.

Vertex coordinates

The vertices of a triangular cupofastegium with edge length 1 are given by:

• $\left(\pm {\frac {1}{2}},\,--{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt {15}}{6}}\right),$
• $\left(0,\,{\frac {\sqrt {3}}{3}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt {15}}{6}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,0,\,0\right),$
• $\left(\pm 1,\,0,\,0,\,0\right).$

Representations

A triangular cupofastegium has the following Coxeter diagrams:

• ox xx3xo&#x (full symmetry)
• xxx3oxo&#x (A2 symmetry only, seen with triangle atop triangular cupola)

External links

• Klitzing, Richard. "tricuf".