“3 31 honeycomb”的意思、由来-开放百科全书

词条

3 31 honeycomb

释义

Construction
Kissing number
E7 lattice
Related honeycombs
See also
References

3₃₁ honeycomb
(no image)
Type	Uniform tessellation
Schläfli symbol	{3,3,3,3^3,1}
Coxeter symbol	3₃₁
Coxeter-Dynkin diagram	nodea_1\|3a\|nodea\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea}}
7-face types	3₂₁ {3⁶}
6-face types	2₂₁ {3⁵}
5-face types	2₁₁ {3⁴}
4-face type	{3³}
Cell type	{3²}
Face type	{3}
Face figure	0₃₁
Edge figure	1₃₁
Vertex figure	2₃₁
Coxeter group	, [3^3,3,1]
Properties	vertex-transitive

In 7-dimensional geometry, the 3₃₁ honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3^3,1} and is composed of 3₂₁ and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

Construction

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.

{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}

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

{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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

{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}

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

{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}

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

{{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea}}

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

{{CDD|branch_10|3a|nodea|3a|nodea|3a|nodea}}

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}.

{{CDD|node_1|2|node_1|3|node|3|node}}

Kissing number

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.

E7 lattice

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

contains as a subgroup of index 144.^[2] Both and can be seen as affine extension from from different nodes:

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

{{CDD|node|3|node|split1|nodes|3ab|nodes|3ab|nodes_10l}} = {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}} ∪ {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node_1}}

The E₇^* lattice (also called E₇²)^[3] has double the symmetry, represented by 3,3">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:

{{CDD|node|3|node|split1|nodes|3ab|nodes|3ab|nodes_10l}} ∪ {{CDD|node|3|node|split1|nodes|3ab|nodes|3ab|nodes_01l}} = {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}} ∪ {{CDD|node|split1|nodes|3ab|nodes_10lr|3ab|nodes|split2|node}} ∪ {{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node_1}} ∪ {{CDD|node|split1|nodes|3ab|nodes_01lr|3ab|nodes|split2|node}} = dual of {{CDD|node_1|3|node|split1|nodes|3ab|nodes|3ab|nodes}}.

Related honeycombs

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.

References

1. ^http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E7.html
2. ^N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p 177
3. ^http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html
4. ^The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin

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}} [https://books.google.com/books?id=fUm5Mwfx8rAC&lpg=PP1&dq=Coxeter&pg=PP1#v=onepage&q&f=false 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.
{{Cite book| first = John H. | last = Conway | authorlink = John Horton Conway |author2= Sloane, Neil J. A. | year = 1998 | title = Sphere Packings, Lattices and Groups | edition = (3rd ed.) | publisher = Springer-Verlag | location = New York | isbn = 0-387-98585-9}} p124-125, 8.2 The 7-dimensinoal lattices: E7 and E7
{{KlitzingPolytopes|flat.htm#7D|7D Heptacombs|x3o3o3o3o3o3o d3o - naquoh}}

1 : 8-polytopes

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Construction

Kissing number

E7 lattice

Related honeycombs

See also

References