# Triangular pucofastegium

Triangular pucofastegium
Rank4
TypeSegmentotope
Notation
Bowers style acronymTripuf
Coxeter diagramox ox3xx&#x
Elements
Cells3 square pyramids, 3 triangular prisms, 2 triangular cupolas, 1 hexagonal prism
Faces1+6+6 triangles, 3+3+6 squares, 2 hexagons
Edges3+6+6+6+12
Vertices3+12
Vertex figures3 wedges, edge lengths 1 (base square and top edge) and 2 (remaining edges)
12 irregular tetrahedra, edge lengths 1 (2), 2 (3), and 3 (1)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {35}}{5}}\approx 1.18322}$
Hypervolume${\displaystyle {\frac {25{\sqrt {5}}}{96}}\approx 0.58231}$
Dichoral anglesSquippy–3–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }}$
Tricu–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }}$
Squippy–3–tricu: ${\displaystyle \arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }}$
Tricu–3–tricu: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
Hip–4–squippy: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{6}}\right)\approx 65.90516^{\circ }}$
Hip–6–tricu: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{4}}\right)\approx 52.23876^{\circ }}$
Trip–4–hip: ${\displaystyle \arccos \left({\frac {2}{3}}\right)\approx 48.18969^{\circ }}$
HeightsHig atop tricu: ${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Trig atop hip: ${\displaystyle {\frac {\sqrt {15}}{6}}\approx 0.64550}$
Central density1
Related polytopes
ArmyTripuf
RegimentTripuf
DualTriangular pucolanotch
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA2×A1×I, order 12
ConvexYes
NatureTame

The triangular pucofastegium, or tripuf, also sometimes called the triangular magnabicupolic ring, is a CRF segmentochoron (designated K-4.51 on Richard Klitzing's list). It consists of 3 triangular prisms, 3 square pyramids, 2 triangular cupolas, and 1 hexagonal prism.

The triangular pucofastegium occurs as the triangle-first cap of the small rhombated pentachoron.

## Vertex coordinates

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

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

## Representations

A triangular pucofastegium has the following Coxeter diagrams:

• ox ox3xx&#x (full symmetry)
• xxx3oxx&#x (A2 symmetry only, seen with hexagon atop triangular cupola)