Runcinated 5-orthoplex

Runcinated 5-orthoplex
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Spat
Coxeter diagram	o4x3o3o3x ()
Elements
Tera	80 triangular-square duoprisms, 40 cuboctahedral prisms, 32 small prismatodecachora, 10 rectified tesseracts
Cells	80+160 tetrahedra, 320+320 triangular prisms, 240 cubes, 80 cuboctahedra
Faces	320+640 triangles, 240+960 squares
Edges	480+960
Vertices	320
Vertex figure	Triangular antifastegium, edge lengths 1 (base edges of triangular prism and top triangle) and √2 (side edges of base prism and lateral edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {14}}{2}}\approx 1.87083$
Hypervolume	${\frac {142{\sqrt {2}}}{3}}\approx 66.93944$
Diteral angles	Cope–cube–tisdip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Spid–trip–tisdip: $\arccos \left(-{\frac {\sqrt {15}}{5}}\right)\approx 140.76848^{\circ }$
	Rit–co–cope: 135°
	Spid–trip–cope: $\arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }$
	Spid–tet–spid: $\arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }$
	Rit–tet–spid: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
Central density	1
Number of external pieces	162
Level of complexity	25
Related polytopes
Army	Spat
Regiment	Spat
Dual	Small antitegmatonotched triacosiicosateron
Conjugate	None
Abstract & topological properties
Flag count	96000
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B5, order 3840
Convex	Yes
Nature	Tame

The runcinated 5-orthoplex, also called the small prismated triacontaditeron or spat, is a convex uniform 5-polytope. It consists of 80 triangular-square duoprisms, 40 cuboctahedral prisms, 10 rectified tesseracts, and 32 small prismatodecachora. One rectified tesseract, 2 small prismatodecachora, 3 cuboctahedral prisms, and 3 triangular-square duoprisms join at each vertex. As the name suggests, it is the runcination of the 5-orthoplex.

Vertex coordinates

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

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

Representations

A runcinated 5-orthoplex has the following Coxeter diagrams:

• o4x3o3o3x () (full symmetry)
• x3o3o3x *b3x () (D₅ symmetry)
• s4x3o3o3x () (D₅ symmetry, as alternated faceting)
• ooqoo4xxoxx3oooo3oxxxo&#xt (B₄ axial, rectified tesseract-first)
• xx(ox)(uo)xx3oo(ox)(xx)xo3ox(xx)(ox)oo3xx(uo)(ox)xx&#xt (A₄ axial, small prismatodecachoron-first)

Gallery

• 

  B₄ or D₅ orthographic projection

• 

  B₃, D₄, or A₂

• 

  B₂

• 

  A₃

Related polytopes

The regiment of the runcinated 5-orthoplex includes a total of 85 members plus 8 fissary members. Of these, 37 of the main members and one of the fissaries have full symmetry, while the rest have D₅ symmetry only.

External links

• Klitzing, Richard. "spat".
• Wikipedia contributors. "Runcinated 5-orthoplex".