Trirectified 8-cube

Trirectified 8-cube
Rank	8
Type	Uniform
Notation
Bowers style acronym	Tro
Coxeter diagram	o4o3o3x3o3o3o3o ()
Elements
Zetta	256 birectified octaexa, 16 hepteractihecatonicosoctaexa
Exa	1024 rectified heptapeta, 2048 birectified heptapeta, 112 birectified hexacontatetrapeta
Peta	1792 hexatera, 7168 rectified hexatera, 7168 dodecatera, 448 rectified triacontaditera
Tera	10752 pentachora, 14336+21504 rectified pentachora, 1120 hexadecachora
Cells	17920+26880 tetrahedra, 35840 octahedra
Faces	35840+35840 triangles
Edges	26880
Vertices	1792
Vertex figure	Octahedral-pentachoric duoprism, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {10}}{2}}\approx 1.58114$
Hypervolume	${\frac {35779}{2520}}\approx 14.19802$
Dizettal angles	Broc–ril–broc: $\arccos \left(-{\frac {3}{4}}\right)\approx 138.59038^{\circ }$
	Sez–bril–roc: $\arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }$
	Sez–brag–sez: 90°
Central density	1
Number of external pieces	272
Level of complexity	35
Related polytopes
Army	Tro
Regiment	Tro
Conjugate	None
Abstract & topological properties
Flag count	361267200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B8, order 10321920
Convex	Yes
Nature	Tame

The trirectified octeract, or tro, also called the trirectified 8-cube, is a convex uniform polyzetton. It consists of 16 hepteractihecatonicosoctaexa and 256 birectified octaexa. 8 rectified octaexa and 5 hepteractihecatonicosoctaexa join at each octahedral-pentachoric duoprismatic vertex. As the name suggests, it is the trirectification of the octeract.

Vertex coordinates

The vertices of a trirectified octeract of edge length 1 are given by all permutations of:

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

Representations

A trirectified octeract has the following Coxeter diagrams:

• o4o3o3x3o3o3o3o (full symmetry)
• o3o3o *b3x3o3o3o3o (D8 symmetry)
• ooo4ooo3oxo3xox3ooo3ooo3ooo&#xt (B7 axial, hepteractihecatonicosoctaexon-first)

External links

• Klitzing, Richard. "tro".
• Wikipedia contributors. "Trirectified 8-cube".