It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram.

Removing the node on the short branch leaves the 6-simplex facet:

Removing the node on the end of the 3-length branch leaves the 3₂₁ facet:

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 2₃₁ polytope.

The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (1₃₁).

The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (0₃₁).

The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes tetrahedral prism {}×{3,3}.

Each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 dimensions; its kissing number is 126, represented by the vertices of its vertex figure 2₃₁.

The 3₃₁ honeycomb's vertex arrangement is called the E₇ lattice.^[1]

${\displaystyle {\tilde {E}}_{7}}$  contains ${\displaystyle {\tilde {A}}_{7}}$  as a subgroup of index 144.^[2] Both ${\displaystyle {\tilde {E}}_{7}}$  and ${\displaystyle {\tilde {A}}_{7}}$  can be seen as affine extension from ${\displaystyle A_{7}}$  from different nodes:  

The E₇ lattice can also be expressed as a union of the vertices of two A₇ lattices, also called A₇²:

          =           ∪          

The E₇^* lattice (also called E₇²)^[3] has double the symmetry, represented by [[3,3^3,3]]. The Voronoi cell of the E₇^* lattice is the 1₃₂ polytope, and voronoi tessellation the 1₃₃ honeycomb.^[4] The E₇^* lattice is constructed by 2 copies of the E₇ lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A₇^* lattices, also called A₇⁴:

          ∪           =           ∪           ∪           ∪           = dual of          .

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.

• 8-polytope
• 1₃₃ honeycomb

• H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1] GoogleBook
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• R. T. Worley, The Voronoi Region of E7*. SIAM J. Discrete Math., 1.1 (1988), 134-141.
• Conway, John H.; Sloane, Neil J. A. (1998). Sphere Packings, Lattices and Groups ((3rd ed.) ed.). New York: Springer-Verlag. ISBN 0-387-98585-9. p124-125, 8.2 The 7-dimensinoal lattices: E7 and E7*
• Klitzing, Richard. "7D Heptacombs x3o3o3o3o3o3o *d3o - naquoh".