Triangular prismatic pyramid

Triangular prismatic pyramid
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Trippy
Coxeter diagram	ox ox3oo&#x
Tapertopic notation	[111]1
Elements
Cells	2 tetrahedra, 3 square pyramids, 1 triangular prism
Faces	2+3+6 triangles, 3 squares
Edges	3+6+6
Vertices	1+6
Vertex figures	1 triangular prism, edge length 1
	6 sphenoids, edge lengths 1 (4) and √2 (2)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {15}}{5}}\approx 0.77460$
Hypervolume	${\frac {\sqrt {5}}{32}}\approx 0.069877$
Dichoral angles	Squippy–3–tet: $\arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }$
	Squippy–3–squippy: $\arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }$
	Trip–4–squippy: $\arccos \left({\frac {\sqrt {6}}{6}}\right)\approx 65.90516^{\circ }$
	Trip–3–tet: $\arccos \left({\frac {\sqrt {6}}{4}}\right)\approx 52.23876^{\circ }$
Heights	Trig atop tet: ${\frac {\sqrt {10}}{4}}\approx 0.79057$
	Trig atop inclined square: ${\frac {\sqrt {10}}{4}}\approx 0.79057$
	Dyad atop squippy: ${\frac {\sqrt {10}}{4}}\approx 0.79057$
	Point atop trip: ${\frac {\sqrt {15}}{6}}\approx 0.64550$
Central density	1
Related polytopes
Army	trippy
Regiment	trippy
Dual	Triangular tegmatic pyramid
Conjugate	None
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A2×A1×I, order 12
Convex	Yes
Nature	Tame

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:

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

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)

External links

• Klitzing, Richard. "trippy".
• Hi.gher.Space Wiki Contributors. "Triangular prismic pyramid".