Prismatointercepted pentachoron

Prismatointercepted pentachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Pinnip
Elements
Cells	10 triangular prisms, 5 octahedra
Faces	10+20 triangles, 15 squares
Edges	30
Vertices	10
Vertex figure	Triangular toroprism, edge lengths 1 (edges of squares) and √2 (long edges of triangles)
Edge figure	oct 3 oct 3 trip 4 trip 4 trip 3
Measures (edge length 1)
Circumradius	$\frac{\sqrt{15}}{5} ≈ 0.77460$
Hypervolume	$\frac{7\sqrt{5}}{48} ≈ 0.32609$
Dichoral angles	Oct–3–oct: $\arccos\left(\frac14\right) ≈ 75.52249°$
	Oct–3–trip: $\arccos\left(\frac{\sqrt6}{4}\right) ≈ 52.23876°$
	Trip–4–trip: $\arccos\left(\frac23\right) ≈ 48.18968°$
Number of pieces	25
Level of complexity	6
Related polytopes
Army	Rap
Regiment	Rap
Conjugate	None
Convex core	Joined pentachoron
Abstract properties
Euler characteristic	10
Topological properties
Orientable	Yes
Properties
Symmetry	A4, order 120
Convex	No
Nature	Wild

The prismatointercepted pentachoron, or pinnip, is a nonconvex uniform polychoron that consists of 5 regular octahedra and 10 triangular prisms. 3 octahedra and 6 triangular prisms join at each vertex.

It is the simplest wild uniform polytope. It is wild because it has octahedra with squares intercepting them.

Cross-sections

Vertex coordinates

Its vertices are the same as those of its regiment colonel, the rectified pentachoron.

External links

• Bowers, Jonathan. "Category 3: Triangular Rectates" (#41).

• Klitzing, Richard. "pinnip".