# Triangular cupofastegium

Triangular cupofastegium
Rank4
TypeSegmentotope
Notation
Bowers style acronymTricuf
Coxeter diagramox xx3xo&#x
Elements
Cells3 tetrahedra, 1+3 triangular prisms, 2 triangular cupolas
Faces2+6+6 triangles, 3+6 squares, 1 hexagon
Edges3+3+3+6+12
Vertices6+6
Vertex figures6 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)
Hypervolume${\displaystyle {\frac {5{\sqrt {5}}}{32}}\approx 0.34939}$
Dichoral anglesTrip–4–trip: ${\displaystyle \arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }}$
Tet–3–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }}$
Tet–3–tricu: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
Tricu–6–tricu: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
Tricu–4–trip: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{6}}\right)\approx 65.90516^{\circ }}$
Tricu–3–trip: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{4}}\right)\approx 52.23876^{\circ }}$
HeightsTrig atop tricu: ${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Trip atop hig: ${\displaystyle {\frac {\sqrt {15}}{6}}\approx 0.64550}$
Central density1
Related polytopes
ArmyTricuf
RegimentTricuf
DualTriangular cupolanotch
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA2×A1×I, order 12
ConvexYes
NatureTame

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:

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

## 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)