Tetrahedraloctahedral honeycomb
The tetrahedraloctahedral 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.
Tetrahedraloctahedral honeycomb  

Rank  4 
Type  Uniform 
Space  Euclidean 
Notation  
Bowers style acronym  Octet 
Coxeter diagram  x3o4o2o3*b () 
Elements  
Cells 

Faces  8N triangles 
Edges  6N 
Vertices  N 
Vertex figure  Cuboctahedron, edge length 1 
Measures (edge length 1)  
Vertex density  
Dual cell volume  
Related polytopes  
Army  Octet 
Regiment  Octet 
Dual  Rhombic dodecahedral honeycomb 
Conjugate  None 
Abstract & topological properties  
Orientable  Yes 
Properties  
Symmetry  S_{4} 
Convex  Yes 
Nature  Tame 
The vertex locations of this honeycomb are known as the facecentered 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 tetrahedraloctahedral 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 tetrahedraloctahedral honeycomb can be given in 4D as:
 ,
where i , j and k range over the integers.
Representations edit
A tetrahedraloctahedral 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

Wireframe
External links edit
 Klitzing, Richard. "octet".
 Wikipedia contributors. "Tetrahedraloctahedral honeycomb".
 Binnendyk, Eric. "Category 1: Primaries" (#3).