# Hexeract

(Redirected from 6-hypercube)
Hexeract
Rank6
TypeRegular
SpaceSpherical
Bowers style acronymAx
Info
Coxeter diagramx4o3o3o3o3o
Schläfli symbol{4,3,3,3,3}
SymmetryBC6, order 46080
ArmyAx
RegimentAx
Elements
Vertex figureHexateron, edge length 2
Peta12 penteracts
Tera60 tesseracts
Cells160 cubes
Faces240 squares
Edges192
Vertices64
Measures (edge length 1)
Hypervolume1
Dipetal angle90º
Height1
Central density1
Euler characteristic0
Related polytopes
DualHexacontatetrapeton
ConjugateHexeract
Properties
ConvexYes
OrientableYes
NatureTame

The hexeract, or ax, also called the 6-cube or dodecapeton, is one of the 3 regular polypeta. It has 12 penteracts as facets, joining 6 to a vertex. It is the 6-dimensional hypercube.

It is also a penteractic prism, cubic duoprism, square trioprism, dodecahedron-small stellated dodecahedral gyropeton and 12-3-5 gyropeton.

It can be alternated into a demihexeract, which is uniform.

## Vertex coordinates

The vertices of a hexeract of edge length 1, centered at the origin, are given by:

• (±1/2, ±1/2, ±1/2, ±1/2, ±1/2, ±1/2).

## Representations

A hexeract has the following Coxeter diagrams:

• x4o3o3o3o3o (full symmetry)
• x4o3o x4o3o (BC3×BC3 symmetry, cubic duoprism)
• x4o x4o3o3o (BC4×BC2 symmetry, square-tesseractic duoprism_)
• x x4o3o3o3o (BC5×A1 symmetry, penteractic prism)
• x4o x4o x4o (BC2×BC2×BC2 symmetry, square trioprism)
• x x4o x4o3o (BC3×BC3×A1 symmetry, square-cubic duoprismatic prism)
• x x x4o3o3o (BC4×A1×A1 symmetry, tesseract prismatic prism)
• x x x4o x4o (BC2×BC2×A1×A1 symmetry, square duoprismatic prismatic prism)
• x x x x4o3o (BC3×A1×A1×A1 symmetry, cubic prism prism prism)
• x x x x x4o (five dimensions different)
• x x x x x x (all six dimensions different)
• xx4oo3oo3oo3oo&#x (BC5 axial, penteract prism)
• qo3oo3oq *b3oo3oo3oo&#zx *D4 symmetry, hull of two demihexeracts)
• oqooooo3ooqoooo3oooqooo3ooooqoo3oooooqo&#xt (A5 axial, vertex-first)

## Related polytopes

o4o3o3o3o3o truncations
Name OBSA CD diagram Picture
Hexeract ax
Rectified hexeract rax
Birectified hexeract brox
Birectified hexacontatetrapeton brag
Rectified hexacontatetrapeton rag
Hexacontatetrapeton gee
Truncated hexeract tox
Bitruncated hexeract botox
Hexeractihexacontatetrapeton xog
Bitruncated hexacontatetrapeton botag
Truncated hexacontatetrapeton tag
Small rhombated hexeract srox
Small birhombated hexeract saborx
Small birhombated hexacontatetrapeton siborg
Small rhombated hexacontatetrapeton srog
Great rhombated hexeract grox
Great birhombated hexeract gaborx
Great birhombated hexacontatetrapeton gaborg
Great rhombated hexacontatetrapeton grog
Small prismated hexeract spox
Small biprismated hexeract sobpoxog
Small prismated hexacontatetrapeton spog
Prismatotruncated hexeract potax
Biprismatorhombated hexacontatetrapeton boprag
Prismatorhombated hexacontatetrapeton prog
Prismatorhombated hexeract prox
Biprismatorhombated hexeract boprax
Prismatotruncated hexacontatetrapeton potag
Great prismated hexeract gippox
Great biprismated hexeract gobpoxog
Great prismated hexacontatetrapeton gopog
Small cellated hexeract scox
Small cellated hexacontatetrapeton scag
Cellitruncated hexeract catax
Celliprismated hexacontatetrapeton copog
Cellirhombated hexeract crax
Cellirhombated hexacontatetrapeton crag
Celligreatorhombated hexeract cagorx
Celliprismatorhombated hexacontatetrapeton coprag
Celliprismated hexeract copox
Cellitruncated hexacontatetrapeton catog
Celliprismatotruncated hexeract captix
Celliprismatotruncated hexacontatetrapeton captog
Celliprismatorhombated hexeract coprix
Celligreatorhombated hexacontatetrapeton cagorg
Great cellated hexeract gocax
Great cellated hexacontatetrapeton gocog
Small terated hexeract stoxog
Tericellated hexacontatetrapeton tacog
Teriprismated hexacontatetrapeton topag
Terigreatorhombated hexeract togrix
Teriprismated hexeract tapox
Tericellirhombated hexacontatetrapeton tocrag
Teriprismatorhombated hexeract tiprixog
Terigreatoprismated hexeract tagpox
Tericellated hexeract tacox
Tericellitruncated hexeract tactaxog
Tericellirhombated hexeract tocrax
Tericelligreatorhombated hexeract tocagrax
Terigreatorhombated hexacontatetrapeton togrig
Tericelligreatorhombated hexacontatetrapeton tecagorg
Terigreatoprismated hexacontatetrapeton tagpog
Great terated hexeract gotaxog