# 6-cube

(Redirected from Hexeract)
6-cube
Rank6
TypeRegular
Notation
Bowers style acronymAx
Coxeter diagramx4o3o3o3o3o ()
Schläfli symbol{4,3,3,3,3}
Tapertopic notation111111
Toratopic notationIIIIII
Bracket notation[IIIIII]
Elements
Peta12 5-cubes
Tera60 tesseracts
Cells160 cubes
Faces240 squares
Edges192
Vertices64
Vertex figure5-simplex, edge length 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {6}}{2}}\approx 1.22475}$
Edge radius${\displaystyle {\frac {\sqrt {5}}{2}}\approx 1.11803}$
Cell radius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Teron radius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Inradius${\displaystyle {\frac {1}{2}}=0.5}$
Hypervolume1
Dipetal angle90°
Height1
Central density1
Number of external pieces12
Level of complexity1
Related polytopes
ArmyAx
RegimentAx
Dual6-orthoplex
κ ?Alternative 6-cube
ConjugateNone
Abstract & topological properties
Flag count46080
Euler characteristic0
OrientableYes
Properties
SymmetryB6, order 46080
Flag orbits1
ConvexYes
Net count502110[1]
NatureTame

The 6-cube (OBSA: ax), also called the hexeract or dodecapeton, is a regular 6-polytope. It has twelve 5-cubes as facets, joining 6 to a vertex. It is the 6-dimensional hypercube.

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

It can be alternated into a 6-demicube, which is uniform.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A 6-cube has the following Coxeter diagrams:

• x4o3o3o3o3o () (full symmetry)
• x4o3o2x4o3o () (B3×B3 symmetry, cubic duoprism)
• x4o2x4o3o3o () (B4×B2 symmetry, square-tesseractic duoprism)
• x2x4o3o3o3o () (B5×B1 symmetry, 5-cube prism)
• x4o2x4o2x4o () (B2×B2×B2 symmetry, square trioprism)
• x2x4o2x4o3o () (B3×B3×A1 symmetry, square-cubic duoprismatic prism)
• x2x2x4o3o3o () (B4×K2 symmetry, tesseract prismatic prism)
• x2x2x4o2x4o () (B2×B2×K2 symmetry, square duoprismatic prismatic prism)
• x2x2x2x4o3o () (B3×K3 symmetry, cubic prism prism prism)
• x2x2x2x2x4o () (B2×K4 symmetry, five dimensions different)
• x2x2x2x2x2x () (K6 symmetry, all six dimensions different)
• xx4oo3oo3oo3oo&#x (B5 axial, 5-cube prism)
• qo3oo3oq *b3oo3oo3oo&#zx (D4 symmetry, hull of two demihexeracts)
• oqooooo3ooqoooo3oooqooo3ooooqoo3oooooqo&#xt (A5 axial, vertex-first)