It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.

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

Removing the node on the end of the 2-length branch leaves the 8-orthoplex, 6₁₁.

Removing the node on the end of the 1-length branch leaves the 8-simplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 4₂₁ polytope.

The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 3₂₁ polytope.

The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 2₂₁ polytope.

The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 1₂₁ polytope.

Each vertex of this tessellation is the center of a 7-sphere in the densest packing in 8 dimensions; its kissing number is 240, represented by the vertices of its vertex figure 4₂₁.

${\displaystyle {\tilde {E}}_{8}}$  contains ${\displaystyle {\tilde {A}}_{8}}$  as a subgroup of index 5760.^[3] Both ${\displaystyle {\tilde {E}}_{8}}$  and ${\displaystyle {\tilde {A}}_{8}}$  can be seen as affine extensions of ${\displaystyle A_{8}}$  from different nodes:  

${\displaystyle {\tilde {E}}_{8}}$  contains ${\displaystyle {\tilde {D}}_{8}}$  as a subgroup of index 270.^[4] Both ${\displaystyle {\tilde {E}}_{8}}$  and ${\displaystyle {\tilde {D}}_{8}}$  can be seen as affine extensions of ${\displaystyle D_{8}}$  from different nodes:  

The vertex arrangement of 5₂₁ is called the E8 lattice.^[5]

The E8 lattice can also be constructed as a union of the vertices of two 8-demicube honeycombs (called a D₈² or D₈⁺ lattice), as well as the union of the vertices of three 8-simplex honeycombs (called an A₈³ lattice):^[6]

                =               ∪               =           ∪           ∪          

Using a complex number coordinate system, it can also be constructed as a ${\displaystyle \mathbb {C} ^{4}}$  regular complex polytope, given the symbol 3{3}3{3}3{3}3{3}3, and Coxeter diagram          . Its elements are in relative proportion as 1 vertex, 80 3-edges, 270 ₃{3}₃ faces, 80 ₃{3}₃{3}₃ cells and 1 ₃{3}₃{3}₃{3}₃ Witting polytope cells.^[7]

The 5₂₁ is seventh in a dimensional series of semiregular polytopes, identified in 1900 by Thorold Gosset. Each member of the sequence has the previous member as its vertex figure. All facets of these polytopes are regular polytopes, namely simplexes and orthoplexes.

• E8 lattice
• 1₅₂ honeycomb
• 2₅₁ honeycomb

• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Coxeter, H. S. M. (1973). Regular Polytopes ((3rd ed.) ed.). New York: Dover Publications. ISBN 0-486-61480-8.
• 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]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: Geometries and Transformations, (2015)