Triangular pucofastegium

{{Infobox polytope 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.
 * img=3g=6p.png
 * type=Segmentotope
 * dim = 4
 * obsa = Tripuf
 * cells = 3 square pyramids, 3 triangular prisms, 2 triangular cupolas, 1 hexagonal prism
 * faces = 1+6+6 triangles, 3+3+6 squares, 2 hexagons
 * edges = 3+6+6+6+12
 * vertices = 3+12
 * verf = 3 wedges, edge lengths 1 (base square and top edge) and $\sqrt{2}$ (remaining edges)
 * verf2 = 12 irregular tetrahedra, edge lengths 1 (2), $\sqrt{2}$ (3), and $\sqrt{3}$ (1)
 * coxeter = ox ox3xx&#x
 * army=Tripuf
 * reg=Tripuf
 * symmetry =A{{sub|2}}×A{{sub|1}}×I, order 12
 * circum = \frac{\sqrt{35}}{5} ≈ 1.18322
 * height = Hig atop tricu: $$\frac{\sqrt{10}}{4} ≈ 0.79057$$
 * height2 = Trig atop hip: $$\frac{\sqrt{15}}{6} ≈ 0.64550$$
 * hypervolume = $$\frac{25\sqrt5}{96} ≈ 0.58231$$
 * dich = Squippy–3–trip: $$\arccos\left(-\frac{\sqrt6}{4}\right) ≈ 127.76124°$$
 * dich2 = Tricu–4–trip: $$\arccos\left(-\frac{\sqrt6}{6}\right) ≈ 114.09484°$$
 * dich3 = Squippy–3–tricu: $$\arccos\left(-\frac14\right) ≈ 104.47751°$$
 * dich4 = Tricu–3–tricu: $$\arccos\left(\frac14\right) ≈ 75.52249°$$
 * dich5 = Hip–4–squippy: $$\arccos\left(\frac{\sqrt6]{6}\right) ≈ 65.90516°$$
 * dich6 = Hip–6–tricu: $$\arccos\left(\frac{\sqrt6}{4}\right) ≈ 52.23876°$$
 * dich7 = Trip–4–hip: $$\arccos\left(\frac23\right) ≈ 48.18969°$$
 * dual=Triangular pucolanotch
 * conjugate=None
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

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:
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,0,\,\frac{\sqrt{15}}{6}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,0,\,\frac{\sqrt{15}}{6}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac12,\,0\right),$$
 * $$\left(±1,\,0,\,±\frac12,\,0\right).$$

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)