Triacontaditeron

Triacontaditeron
(OFF file)
Rank	5
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Tac
Coxeter diagram	o4o3o3o3x ()
Schläfli symbol	{3,3,3,4}
Bracket notation	<IIIII>
Elements
Tera	32 pentachora
Cells	80 tetrahedra
Faces	80 triangles
Edges	40
Vertices	10
Vertex figure	Hexadecachoron, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt2}{2} \approx 0.70711$
Edge radius	$\frac12 = 0.5$
Face radius	$\frac{\sqrt6}{6} \approx 0.40825$
Cell radius	$\frac{\sqrt2}{4} \approx 0.35355$
Inradius	$\frac{\sqrt{10}}{10} \approx 0.31623$
Hypervolume	$\frac{\sqrt2}{30} \approx 0.047140$
Diteral angle	$\arccos\left(-\frac35\right) \approx 126.86990^\circ$
Height	$\frac{\sqrt{10}}{5} \approx 0.63246$
Central density	1
Number of pieces	32
Level of complexity	1
Related polytopes
Army	Tac
Regiment	Tac
Dual	Penteract
Conjugate	None
Abstract properties
Flag count	3840
Net count	9694
Euler characteristic	2
Topological properties
Orientable	Yes
Properties
Symmetry	B5, order 3840
Convex	Yes
Nature	Tame

The triacontaditeron, or tac, also called the pentacross, 5-orthoplex, or square-octahedral duotegum, is one of the 3 regular polytera. It has 32 regular pentachora as facets, joining 16 to a vertex in an hexadecachoral arrangement. It is the 5-dimensional orthoplex.

It can also be seen as a convex segmentoteron, as a pentachoric antiprism. It is also the hexadecachoric tegum.

Vertex coordinates

The vertices of a regular triacontaditeron of edge length 1, centered at the origin, are given by all permutations of:

• $\left(\pm\frac{\sqrt2}{2},\,0,\,0,\,0,\,0\right).$

Representations

A triacontaditeron has the following Coxeter diagrams:

• o4o3o3o3x (full symmetry)
• o3o3o *b3o3x (D5 symmetry, h has demitesseract verf)
• xo3oo3oo3ox&#x (A4 axial, pentachoric antiprism)
• ooo4ooo3ooo3oxo&#xt (BC4 axial, as hexadecachoric bipyramid)
• qo oo4oo3oo3ox&#zx (BC4×A1 symmetry, hexadecachoric bipyramid)
• oxo3ooo3oo *b3oo&#zx (D4 symmetry, demitesseractic bipyramid)
• qo ox3oo3oo *c3oo&#zx (D4×1 symmetry, demitesseractic bipyramid)
• xox ooo4ooo3oxo&#xt (BC3×A1 symmetry, edge-first)
• xox ooo3oxo3ooo&#xt (A3×A1 axial, edge-first)
• xoo3oox oxo4ooo&#xt (BC2×A2 symmetry, face-first)
• oxo oxo xoo3oox&#xt (BC2×A1×A1 axial, triangle-first)
• xoo3ooo3oox oqo&#xt (A3×A1 symmetry, cell-first)
• oxoo3oooo3ooox&#xr (A3 symmetry)
• xoxo oxoo3ooox&#xr (A2×A1 symmetry)
• xo4oo oo4oo3ox&#zx (BC3×BC2 symmetry, square-octahedral duotegum)
• xo xo oo3ox3oo&#zx (A3×A1×A1 symmetry, square-octahedral duotegum)
• o(xo)o o(xo)o o(ox)o o(ox)o&#xt (BC2×BC2 symmetry, square duotegmatic bipyramid)

Related polytopes

The regiment of the triacontaditeron contains 4 uniform members, including itself, one with D5 symmetry (the hexadecahemidecateron), and 2 with pentachoric antiprism symmetry (the pentachoric hemiantiprism and spinopentachoric hemiantiprism). THere are also 2 scaliform members known with 5-2 step prism alterprismatic symmetry.

External links

• Bowers, Jonathan. "Category 1: Primary Polytera" (#3).

• Klitzing, Richard. "tac".

• Wikipedia Contributors. "5-orthoplex".
• Hi.gher.Space Wiki Contributors. "Aeroteron".