Truncated 5-orthoplex

Truncated 5-orthoplex
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Tot
Coxeter diagram	o4o3o3x3x ()
Elements
Tera	10 hexadecachora, 32 truncated pentachora
Cells	160 tetrahedra, 80 truncated tetrahedra
Faces	320 triangles, 80 hexagons
Edges	40+240
Vertices	80
Vertex figure	Octahedral pyramid, edge lengths 1 (base) and √3 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {10}}{2}}\approx 1.58114$
Hypervolume	${\frac {119{\sqrt {2}}}{15}}\approx 11.21943$
Diteral angles	Tip–tut–tip: $\arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }$
	Tip–tet–hex: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
Central density	1
Number of external pieces	42
Level of complexity	5
Related polytopes
Army	Tot
Regiment	Tot
Dual	Octakis penteract
Conjugate	None
Abstract & topological properties
Flag count	19200
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B5, order 3840
Convex	Yes
Nature	Tame

The truncated 5-orthoplex, also called the truncated triacontaditeron or tot, is a convex uniform 5-polytope. It consists of 10 regular hexadecachora and 32 truncated pentachora. One hexadecachoron and 8 truncated pentachora join at each vertex. As the name suggests, it is the truncation of the 5-orthoplex.

Vertex coordinates

The vertices of a truncated 5-orthoplex of edge length 1 are given by all permutations of:

• $\left(\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,0\right)$.

Representations

A truncated 5-orthoplex has the following Coxeter diagrams:

• o4o3o3x3x () (full symmetry)
• o3o3x3x *b3o () (D₅ symmetry)
• ooooo4ooooo3ooxoo3xuxux&#xt (B₄ axial, hexadecachoron-first)
• xuxux3ooxoo3ooooo *b3ooooo&#xt (D₄ axial, hexadecachoron-first)

Gallery

• 

  B₄ or D₅ orthographic projection

• 

  B₃, D₄, or A₂

• 

  B₂

• 

  A₃

Related polytopes

The truncated 5-orthoplex is the colonel of a two-member regiment that also includes the truncated hexadecahemidecateron.

External links

• Bowers, Jonathan. "Category 2: Truncates" (#15).

• Klitzing, Richard. "tot".
• Wikipedia contributors. "Truncated 5-orthoplex".