Tetrahedron atop triangular cupola

Tetrahedron atop triangular cupola
Rank4
TypeSegmentotope
SpaceSpherical
Notation
Bowers style acronymTetatricu
Coxeter diagramooxx3oxxo&#xr
Elements
Cells2 tetrahedra, 6 triangular prisms, 2 triangular cupolax
Faces2+6+6 triangles, 6+6 squares, 1 hexagon
Edges6+6+6+12
Vertices1+6+6
Vertex figures1 triangular antiprism, edge lengths 1 (base) and 2 (sides)
6 skewed rectangular pyramids, edge lengths, base edge lengths 1, 2, 1, 2, side edge lengths 1, 1, 2, 2
6 phyllic disphenoids, edge lengths 1 (2), 2 (3), and 3 (1)
Measures (edge length 1)
Hypervolume${\displaystyle \frac{5\sqrt5}{24} \approx 0.46586}$
Dichoral anglesTrip-4-trip: ${\displaystyle \arccos\left(-\frac23\right) \approx 131.81031^\circ}$
Tet-3-trip: ${\displaystyle \arccos\left(-\frac{\sqrt6}{4}\right) \approx 127.76124^\circ}$
Tricu-6-tricu: ${\displaystyle \arccos\left(-\frac14\right) \approx 104.477512^\circ}$
Tet-3-tricu: ${\displaystyle \arccos\left(\frac14\right) \approx 75.52249^\circ}$
Trip-4-tricu: ${\displaystyle \arccos\left(\frac{\sqrt6}{6}\right) \approx 65.905157^\circ}$
trip-3-tricu: ${\displaystyle \arccos\left(\frac{\sqrt6}{4}\right) \approx 52.23876^\circ}$
Height${\displaystyle \frac{\sqrt{10}}{4} \approx 0.79057}$
Central density1
Related polytopes
ArmyTetatricu
RegimentTetatricu
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(G2×A1)/2×I, order 12
ConvexYes
NatureTame

Tetrahedron atop triangular cupola, or tetatricu, is a CRF segmentochoron (designated K-4.24 on Richard Klitzing's list). As the name suggests, it consists of a tetrahedron and a triangular cupola as bases, connected by 1 additional tetrahedron, 1 additional triangular cupola, and 6 triangular prisms.

It can be formed by diminishing a tetrahedron atop cuboctahedron segmentochoron by a triangular cupofastegium, leaving a further triangular cupola behind while removing several tetrahedral and triangular prism cells. Therefore, its vertices are a subset of those of the uniform small prismatodecachoron.

Vertex coordinates

The vertices of a tetrahedron atop triangular cupola segmentochoron of edge length 1 are given by:

• ${\displaystyle \pm\left(0,\,0,\,0,\,\pm1\right)}$,
• ${\displaystyle \pm\left(0,\,0,\,\pm\frac{\sqrt3}{2},\,\pm\frac12\right)}$,
• ${\displaystyle \left(0,\,-\frac{\sqrt6}{3},\,\frac{\sqrt3}{3},\,0\right)}$,
• ${\displaystyle \left(0,\,-\frac{\sqrt6}{3},\,-\frac{\sqrt3}{6},\,\pm\frac12\right)}$,
• ${\displaystyle \left(\frac{\sqrt{10}}{4},\,-\frac{\sqrt6}{4},\,0,\,0\right)}$,
• ${\displaystyle \left(\frac{\sqrt{10}}{4},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right)}$,
• ${\displaystyle \left(\frac{\sqrt{10}}{4},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,\pm\frac12\right)}$.