5_{ 21} honeycomb
In geometry, the 5_{21} honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5_{21} is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.[1]
5_{21} honeycomb | |
---|---|
Type | Uniform honeycomb |
Family | k_{21} polytope |
Schläfli symbol | {3,3,3,3,3,3^{2,1}} |
Coxeter symbol | 5_{21} |
Coxeter-Dynkin diagram | |
8-faces | 5_{11} {3^{7}} |
7-faces | {3^{6}} |
6-faces | {3^{5}} |
5-faces | {3^{4}} |
4-faces | {3^{3}} |
Cells | {3^{2}} |
Faces | {3} |
Cell figure | 1_{21} |
Face figure | 2_{21} |
Edge figure | 3_{21} |
Vertex figure | 4_{21} |
Symmetry group | , [3^{5,2,1}] |
This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure[2] (Gosset regarded honeycombs in n dimensions as degenerate n+1 polytopes).
Each vertex of the 5_{21} honeycomb is surrounded by 2160 8-orthoplexes and 17280 8-simplices.
The vertex figure of Gosset's honeycomb is the semiregular 4_{21} polytope. It is the final figure in the k_{21} family.
This honeycomb is highly regular in the sense that its symmetry group (the affine Weyl group) acts transitively on the k-faces for k ≤ 6. All of the k-faces for k ≤ 7 are simplices.
Construction
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, 6_{11}.
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 4_{21} polytope.
The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 3_{21} polytope.
The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 2_{21} polytope.
The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 1_{21} polytope.
Kissing number
Each vertex of this tessellation is the center of a 7-sphere in the densest known packing in 8 dimensions; its kissing number is 240, represented by the vertices of its vertex figure 4_{21}.
E8 lattice
contains as a subgroup of index 5760.[3] Both and can be seen as affine extensions of from different nodes:
contains as a subgroup of index 270.[4] Both and can be seen as affine extensions of from different nodes:
The vertex arrangement of 5_{21} 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 D_{8}^{2} or D_{8}^{+} lattice), as well as the union of the vertices of three 8-simplex honeycombs (called an A_{8}^{3} lattice):[6]
= ∪ = ∪ ∪
Regular complex honeycomb
Using a complex number coordinate system, it can also be constructed as a regular complex polytope, given the symbol 3{3}3{3}3{3}3{3}3, and Coxeter diagram
Related polytopes and honeycombs
The 5_{21} 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.
k_{21} figures in n dimensional | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Finite | Euclidean | Hyperbolic | ||||||||
E_{n} | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||
Coxeter group |
E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||
Coxeter diagram |
|||||||||||
Symmetry | [3^{−1,2,1}] | [3^{0,2,1}] | [3^{1,2,1}] | [3^{2,2,1}] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||
Order | 12 | 120 | 192 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||
Graph | - | - | |||||||||
Name | −1_{21} | 0_{21} | 1_{21} | 2_{21} | 3_{21} | 4_{21} | 5_{21} | 6_{21} |
See also
Notes
- Coxeter, 1973, Chapter 5: The Kaleidoscope
- Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48.
- N.W. Johnson: Geometries and Transformations, (2018) 12.5: Euclidean Coxeter groups, p.294
- Johnson (2011) p.177
- http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html
- Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
- Coxeter Regular Convex Polytopes, 12.5 The Witting polytope
References
- 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
- (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)
Fundamental convex regular and uniform honeycombs in dimensions 2-9 | ||||||
---|---|---|---|---|---|---|
Space | Family | / / | ||||
E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |