# Triangular prismatic pyramid

(Redirected from Wedge pyramid)
Triangular prismatic pyramid
Rank4
TypeSegmentotope
Notation
Bowers style acronymTrippy
Coxeter diagramox ox3oo&#x
Tapertopic notation[111]1
Elements
Cells2 tetrahedra, 3 square pyramids, 1 triangular prism
Faces2+3+6 triangles, 3 squares
Edges3+6+6
Vertices1+6
Vertex figures1 triangular prism, edge length 1
6 sphenoids, edge lengths 1 (4) and 2 (2)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {15}}{5}}\approx 0.77460}$
Hypervolume${\displaystyle {\frac {\sqrt {5}}{32}}\approx 0.069877}$
Dichoral anglesSquippy–3–tet: ${\displaystyle \arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }}$
Squippy–3–squippy: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
Trip–4–squippy: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{6}}\right)\approx 65.90516^{\circ }}$
Trip–3–tet: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{4}}\right)\approx 52.23876^{\circ }}$
HeightsTrig atop tet: ${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Trig atop inclined square: ${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Dyad atop squippy: ${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Point atop trip: ${\displaystyle {\frac {\sqrt {15}}{6}}\approx 0.64550}$
Central density1
Related polytopes
Armytrippy
Regimenttrippy
DualTriangular tegmatic pyramid
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA2×A1×I, order 12
ConvexYes
NatureTame

The triangular prismatic pyramid, or trippy, is a CRF segmentochoron (designated K-4.7 on Richard Klitzing's list). It has 2 regular tetrahedra, 3 square pyramids, and 1 triangular prism as cells. As the name suggests, it is a pyramid based on the triangular prism.

The triangular prismatic pyramid is the vertex pyramid of the rectified pentachoron, with the remainder of the original polychoron forming a triangular antifastegium.

## Vertex coordinates

The vertices of a triangular prismatic pyramid of edge length 1 are given by:

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

## Representations

A triangular prismatic pyramid has the following Coxeter diagrams:

• ox ox3oo&#x (full symmetry)
• oxx3ooo&#x (A2 symmetry, prism seen as frustum symmetry)
• oox oxx&#x (A1×A1 symmetry, wedge pyramid)
• oxxx&#x (bilateral symmetry only)