5-orthoplex

From Polytope Wiki
Jump to navigation Jump to search
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
Vertex figureHexadecachoron, edge length 1
Petrie polygons192 skew decagonal-decagrammic coils
Measures (edge length 1)
Circumradius
Edge radius
Face radius
Cell radius
Inradius
Hypervolume
Diteral angle
Height
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[edit | edit source]

The vertices of a regular 5-orthoplex of edge length 1, centered at the origin, are given by all permutations of:

  • .

Representations[edit | edit source]

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[edit | edit source]

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.

External links[edit | edit source]