Cubic honeycomb

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Cubic honeycomb
Rank4
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymChon
Coxeter diagramx4o3o4o ()
Schläfli symbol{4,3,4}
Elements
CellsN cubes
Faces3N squares
Edges3N
VerticesN
Vertex figureOctahedron, edge length 2
Measures (edge length 1)
Vertex density
Dual cell volume
Related polytopes
ArmyChon
RegimentChon
DualCubic honeycomb
Petrie dualMucubic honeycomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes
NatureTame

The cubic honeycomb, or chon, is the only regular honeycomb or tessellation of 3D Euclidean space. 8 cubes join at each vertex of this honeycomb. It is also the 3D hypercubic honeycomb.

This honeycomb can be alternated into a tetrahedral-octahedral honeycomb, which is uniform.

Vertex coordinates[edit | edit source]

The vertices of a cubic honeycomb of edge length 1 are given by

  • in which .

Representations[edit | edit source]

A cubic honeycomb has the following Coxeter diagrams:

  • x4o3o4o () (regular)
  • x4o3o4x () (as expanded cubic honecyomb)
  • x4o3o2o3*b () (S4 symmetry)
  • xØo2x4o4o () (various square prismatic honeycombs)
  • xØo2o4x4o ()
  • xØo2x4o4x ()
  • xØx2x4o4o ()
  • xØx2o4x4o ()
  • xØx2x4o4x ()
  • xØo2xØo2xØo () (various apeirogonal triprismatic honeycombs)
  • xØx2xØo2xØo ()
  • xØx2xØx2xØo ()
  • xØx2xØx2xØx ()
  • qo3oo3oq3oo3*a&#zx (as hull of two alternate tetrahedral-octahedral honeycombs)

Gallery[edit | edit source]

External links[edit | edit source]