Tetrahedral-octahedral honeycomb

The tetrahedral-octahedral honeycomb (OBSA: octet) also known as the alternated cubic honeycomb, is a convex uniform honeycomb. 6 octahedra and 8 tetrahedra join at each vertex of this honeycomb, with a cuboctahedron as the vertex figure. As one of its names suggests, it can be formed by alternation of the cubic honeycomb. It is also the 3D simplicial honeycomb.

Tetrahedral-octahedral honeycomb
Rank4
TypeUniform
SpaceEuclidean
Notation
Bowers style acronymOctet
Coxeter diagramx3o4o2o3*b ()
Elements
Cells
Faces8N triangles
Edges6N
VerticesN
Vertex figureCuboctahedron, edge length 1
Measures (edge length 1)
Vertex density
Dual cell volume
Related polytopes
ArmyOctet
RegimentOctet
DualRhombic dodecahedral honeycomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryS4
ConvexYes
NatureTame

The vertex locations of this honeycomb are known as the face-centered cubic or FCC lattice, which has the important property that placing spheres at each of the points that touch each other results in a maximally dense packing of equal spheres. (There are infinitely many cubic close packings, but the FCC lattice is notable for its symmetry, along with the HCP lattice.) This geometric property makes the FCC lattice ubiquitous in chemistry, such as in the structure of sodium chloride crystals (as found in table salt). Taking the convex hull of a set of vertices enclosed by a sphere of any location or size results in a Waterman polyhedron.

Vertex coordinates edit

The vertices of a tetrahedral-octahedral honeycomb of edge length 1 are given by

  •  ,

where i , j , and k  are integers, and i+j+k  is even.

Integral vertex coordinates for the tetrahedral-octahedral honeycomb can be given in 4D as:

  •  ,

where i , j  and k  range over the integers.

Representations edit

A tetrahedral-octahedral honeycomb has the following Coxeter diagrams:

  • x3o4o2o3*b (     ) (full symmetry)
  • x3o3o3o3*a (   ) (P4 symmetry, cyclotetrahedral honeycomb)
  • s4o3o4o (       ) (as alternated cubic honeycomb)
  • s4o3o4s (       )
  • s4o3o2o3*b (     )
  • sØs2s4o4o (         ) (as alternated square prismatic honeycomb)
  • sØs2o4s4o (         )
  • sØs2sØs2sØs (           ) (as alternated product of three diapeirogons)

Gallery edit

External links edit