In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the 2
22 honeycomb is a
uniform tessellation of the six-dimensional Euclidean space. It can be represented by the
Schläfli symbol . It is constructed from
221 facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
and has a
122 vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
, with 54 2
21 polytopes around every vertex.
Its
vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
is the ''
E6 lattice'', and the
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
of the
E6 Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
so it can also be called the E
6 honeycomb.
Construction
It is created by a
Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.
Construction process
...
upon a set of 7
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
mirrors in 6-dimensional space.
The facet information can be extracted from its
Coxeter–Dynkin diagram, .
Removing a node on the end of one of the 2-node branches leaves the
221, its only
facet
Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...
type,
The
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is determined by removing the ringed node and ringing the neighboring node. This makes
122, .
The
edge figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
is the vertex figure of the vertex figure, here being a
birectified 5-simplex, ''t''
2, .
The
face figure is the vertex figure of the edge figure, here being a triangular
duoprism
In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...
, ×, .
Kissing number
Each vertex of this tessellation is the center of a 5-sphere in the densest known
packing in 6 dimensions, with
kissing number
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...
72, represented by the vertices of its
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
122.
E6 lattice
The 2
22 honeycomb's
vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
is called the E
6 lattice.
The E
62 lattice, with
3,3,32,2 symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
, can be constructed by the union of two E
6 lattices:
: ∪
The E
6* lattice (or E
63) with
[32,2,2_symmetry._The_
[32,2,2_symmetry._The_Voronoi_cell">
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22 polytope, and the Voronoi tessellation is a #Bitruncated 2 22 honeycomb, bitruncated 2
22 honeycomb.
The Voronoi Cells of the E6* and E7* Lattices
, Edward Pervin It is constructed by 3 copies of the E6 lattice vertices, one from each of the three branches of the Coxeter diagram.
: ∪ ∪ = dual to .
Geometric folding
The group is related to the by a geometric folding
Fold, folding or foldable may refer to:
Arts, entertainment, and media
* ''Fold'' (album), the debut release by Australian rock band Epicure
* Fold (poker), in the game of poker, to discard one's hand and forfeit interest in the current pot
*Abov ...
, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.
Related honeycombs
The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with symmetry. 24 of them have doubled symmetry 3,3,32,2 with 2 equally ringed branches, and 7 have sextupled (3 !) symmetry [32,2,2_with_identical_rings_on_all_3_branches._There_are_no_regular_honeycombs_in_the_family_since_its_Coxeter_diagram_a_nonlinear_graph,_but_the_222_and_#Bitruncated_2_22_honeycomb.html" ;"title="2,2,2.html" ;"title=" [32,2,2_with_identical_rings_on_all_3_branches._There_are_no_regular_honeycombs_in_the_family_since_its_Coxeter_diagram_a_nonlinear_graph,_but_the_222_and_#Bitruncated_2_22_honeycomb">birectified_222_are_Face-transitive#Related_terms.html" ;"title="2,2,2">[32,2,2 with identical rings on all 3 branches. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but the 222 and #Bitruncated 2 22 honeycomb">birectified 222 are Face-transitive#Related terms">isotopic, with only one type of facet: 221, and Rectified 1 22 polytope, rectified 122 polytopes respectively.
Birectified 222 honeycomb
The birectified 222 honeycomb , has rectified 1 22 polytope facets, , and a proprism
In geometry of 4 dimensions or higher, a proprism is a polytope resulting from the Cartesian product of two or more polytopes, each of two dimensions or higher. The term was coined by John Horton Conway for ''product prism''. The dimension of the ...
×× vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
.
Its facets are centered on the vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
of E6* lattice, as:
: ∪ ∪
Construction
The facet information can be extracted from its Coxeter–Dynkin diagram, .
The vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is determined by removing the ringed node and ringing the neighboring node. This makes a proprism ××, .
Removing a node on the end of one of the 3-node branches leaves the 122, its only facet
Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...
type, .
Removing a second end node defines 2 types of 5-faces: birectified 5-simplex, 022 and birectified 5-orthoplex
In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.
There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the 5-cube, and the 4th and last being the ...
, 0211.
Removing a third end node defines 2 types of 4-faces: rectified 5-cell
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In t ...
, 021, and 24-cell, 0111.
Removing a fourth end node defines 2 types of cells: octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
, 011, and tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
, 020.
k22 polytopes
The 222 honeycomb, is fourth in a dimensional series of uniform polytopes, expressed by Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
as k22 series. The final is a paracompact hyperbolic honeycomb, 322. Each progressive uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
is constructed from the previous as its vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
.
The 222 honeycomb is third in another dimensional series 22k.
Notes
References
* Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes)
* Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
''Regular Polytopes'' (1963), Macmillan Company
** ''Regular Polytopes'', Third edition, (1973), Dover edition, (Chapter 5: The Kaleidoscope)
* 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,
GoogleBook
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3–45* R. T. Worley, ''The Voronoi Region of E6*''. J. Austral. Math. Soc. Ser. A, 43 (1987), 268-278.
* p125-126, 8.3 The 6-dimensional lattices: E6 and E6*
*
*
{{Honeycombs
7-polytopes