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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 ...
, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. _9, also (E10) is a paracompact hyperbolic group, so either
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 cu ...
s or
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 of a polyhedron. Mark a point somewhere along each connected edge. Draw l ...
s will not be bounded. E10 is last of the series of
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...
s with a bifurcated Coxeter-Dynkin diagram of lengths 6,2,1. There are 1023 unique E10 honeycombs by all combinations of its Coxeter-Dynkin diagram. There are no regular honeycombs in the family since its Coxeter diagram is a nonlinear graph, but there are three simplest ones, with a single ring at the end of its 3 branches: 621, 261, 162.


621 honeycomb

The 621 honeycomb is constructed from alternating
9-simplex In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces ...
and
9-orthoplex In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells ''4-faces'', 5376 5-simplex ''5-faces'', 4608 6-simplex ''6-faces'', 2304 7-simplex ...
facets within the symmetry of the E10 Coxeter group. This honeycomb is highly regular in the sense that its symmetry group (the affine E9 Weyl group) acts transitively on the ''k''-faces for ''k'' ≤ 7. All of the ''k''-faces for ''k'' ≤ 8 are simplices. This honeycomb is last in the series of k21 polytopes, enumerated by
Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, an ...
in 1900, listing polytopes and honeycombs constructed entirely of regular facets, although his list ended with the 8-dimensional the Euclidean honeycomb, 521.Conway, 2008, The Gosset series, p 413


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 10
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 hyp ...
mirrors in 9-dimensional hyperbolic 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
9-orthoplex In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells ''4-faces'', 5376 5-simplex ''5-faces'', 4608 6-simplex ''6-faces'', 2304 7-simplex ...
, 711. : Removing the node on the end of the 1-length branch leaves the
9-simplex In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces ...
. : 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw l ...
is determined by removing the ringed node and ringing the neighboring node. This makes the 521 honeycomb. : 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 line ...
is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 421 polytope. : The ''face figure'' is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 321 polytope. : The ''cell figure'' is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 221 polytope. :


Related polytopes and honeycombs

The 621 is last in a dimensional series of
semiregular polytope In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Poly ...
s and honeycombs, identified in 1900 by
Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, an ...
. Each member of the sequence has the previous member 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw l ...
. All facets of these polytopes are
regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
s, namely
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
es and
orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahe ...
es.


261 honeycomb

The 261 honeycomb is composed of 251 9-honeycomb and
9-simplex In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces ...
facets. It is the final figure in the 2k1 family.


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 10
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 hyp ...
mirrors in 9-dimensional hyperbolic space. The facet information can be extracted from its Coxeter-Dynkin diagram. : Removing the node on the short branch leaves the
9-simplex In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces ...
. : Removing the node on the end of the 6-length branch leaves the 251 honeycomb. This is an infinite facet because E10 is a paracompact hyperbolic group. : 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw l ...
is determined by removing the ringed node and ringing the neighboring node. This makes the
9-demicube In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it ...
, 161. : The ''edge figure'' is the vertex figure of the edge figure. This makes the rectified 8-simplex, 051. : The ''face figure'' is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the
5-simplex In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The ...
prism. :


Related polytopes and honeycombs

The 261 is last in a dimensional series of
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 v ...
s and honeycombs.


162 honeycomb

The 162 honeycomb contains 152 (9-honeycomb) and 161
9-demicube In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it ...
facets. It is the final figure in the 1k2 polytope family.


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 10
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 hyp ...
mirrors in 9-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
9-demicube In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it ...
, 161. : Removing the node on the end of the 6-length branch leaves the 152 honeycomb. : 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw l ...
is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 9-simplex, 062. :


Related polytopes and honeycombs

The 162 is last in a dimensional series of
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 v ...
s and honeycombs.


Notes


References

* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,

* Harold Scott MacDonald Coxeter, Coxeter ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes) * Harold Scott MacDonald Coxeter, Coxeter ''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,

** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45 {{Polytopes 10-polytopes