# Triangular cupolic prism

Triangular cupolic prism
Rank4
TypeSegmentotope
Notation
Bowers style acronymTricupe
Coxeter diagramxx ox3xx&#x
Elements
Cells1+3 triangular prisms, 3 cubes, 2 triangular cupolas, 1 hexagonal prism
Faces2+6 triangles, 3+3+3+6+6 squares, 2 hexagons
Edges3+6+6+6+6+12
Vertices6+12
Vertex figures6 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${\displaystyle {\frac {\sqrt {5}}{2}}\approx 1.11803}$
Hypervolume${\displaystyle {\frac {5{\sqrt {2}}}{6}}\approx 1.17851}$
Dichoral anglesTrip–4–cube: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)125.26439^{\circ }}$
Tricu–3–trip: 90°
Tricu–4–cube: 90°
Tricu–6–hip: 90°
Trip–4–hip: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52878^{\circ }}$
Cube–4–hip: ${\displaystyle \arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }}$
HeightsTricu atop tricu: 1
Trip atop hip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density1
Related polytopes
ArmyTricupe
RegimentTricupe
DualSemibisected hexagonal trapezohedral tegum
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA2×A1×I, order 12
ConvexYes
NatureTame

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.

A variant, known as the triangular cupolic wedge, has the same symmetry as the triangular cupolic prism, which occurs if the lacing edge lengths of the two prisms are unequal.

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:

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