5-orthoplex

5-orthoplex
Rank5
TypeRegular
Notation
Bowers style acronymTac
Coxeter diagramo4o3o3o3x ()
Schläfli symbol{3,3,3,4}
Bracket notation<IIIII>
Elements
Tera32 pentachora
Cells80 tetrahedra
Faces80 triangles
Edges40
Vertices10
Petrie polygons192 skew decagonal-decagrammic coils
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Edge radius${\displaystyle {\frac {1}{2}}=0.5}$
Face radius${\displaystyle {\frac {\sqrt {6}}{6}}\approx 0.40825}$
Cell radius${\displaystyle {\frac {\sqrt {2}}{4}}\approx 0.35355}$
Inradius${\displaystyle {\frac {\sqrt {10}}{10}}\approx 0.31623}$
Hypervolume${\displaystyle {\frac {\sqrt {2}}{30}}\approx 0.047140}$
Diteral angle${\displaystyle \arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }}$
Height${\displaystyle {\frac {\sqrt {10}}{5}}\approx 0.63246}$
Central density1
Number of external pieces32
Level of complexity1
Related polytopes
ArmyTac
RegimentTac
DualPenteract
ConjugateNone
Abstract & topological properties
Flag count3840
Euler characteristic2
OrientableYes
Properties
SymmetryB5, order 3840
Flag orbits1
ConvexYes
Net count9694
NatureTame

The 5-orthoplex, also called the pentacross, triacontaditeron, tac, or square-octahedral duotegum, is a regular 5-polytope. It has 32 regular pentachora as facets, joining 16 to a vertex in a 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 5-orthoplex of edge length 1, centered at the origin, are given by all permutations of:

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

Representations

A 5-orthoplex has the following Coxeter diagrams:

• o4o3o3o3x () (full symmetry)
• o3o3o3x *b3o () (D5 symmetry, has demitesseract verf)
• xo3oo3oo3ox&#x (A4 axial, pentachoric antiprism)
• ooo4ooo3ooo3oxo&#xt (B4 axial, as hexadecachoric bipyramid)
• qo oo4oo3oo3ox&#zx (B4×A1 symmetry, hexadecachoric bipyramid)
• oxo3ooo3oo *b3oo&#zx (D4 symmetry, demitesseractic bipyramid)
• qo ox3oo3oo *c3oo&#zx (D4×A1 symmetry, demitesseractic bipyramid)
• xox ooo4ooo3oxo&#xt (B3×A1 symmetry, edge-first)
• xox ooo3oxo3ooo&#xt (A3×A1 axial, edge-first)
• xoo3oox oxo4ooo&#xt (B2×A2 symmetry, face-first)
• oxo oxo xoo3oox&#xt (B2×K2 axial, triangle-first)
• xoo3ooo3oox oqo&#xt (A3×A1 symmetry, cell-first)
• oxoo3oooo3ooox&#xr (A3 symmetry)
• xoxo oxoo3ooox&#xr (A2×A1 symmetry)
• xo4oo oo4oo3ox&#zx (B3×B2 symmetry, square-octahedral duotegum)
• xo xo oo3ox3oo&#zx (A3×K2 symmetry, square-octahedral duotegum)
• o(xo)o o(xo)o o(ox)o o(ox)o&#xt (B2×B2 symmetry, square duotegmatic bipyramid)

Related polytopes

The regiment of the 5-orthoplex 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.