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 ...
, an E
9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space.
, also (E
10) 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 cut ...
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 line ...
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 refle ...
s with a bifurcated
Coxeter-Dynkin diagram of lengths 6,2,1. There are 1023 unique E
10 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: 6
21, 2
61, 1
62.
621 honeycomb
The 6
21 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-fa ...
and
9-orthoplex facets within the symmetry of the E
10 Coxeter group.
This honeycomb is highly regular in the sense that its symmetry group (the affine E
9 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, and ...
in 1900, listing polytopes and honeycombs constructed entirely of regular facets, although his list ended with the 8-dimensional the Euclidean honeycomb, 5
21.
[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 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, 7
11.
:
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-fa ...
.
:
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 line ...
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 lines ...
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 6
21 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 Polyt ...
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, and ...
. 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 line ...
. 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
simplexes 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 octahed ...
es.
261 honeycomb
The 2
61 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-fa ...
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 ...
. 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 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-fa ...
.
:
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 line ...
is determined by removing the ringed node and ringing the neighboring node. This makes the
9-demicube, 1
61.
:
The ''edge figure'' is the vertex figure of the edge figure. This makes the
rectified 8-simplex, 0
51.
:
The ''face figure'' is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the
5-simplex prism.
:
Related polytopes and honeycombs
The 2
61 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 vert ...
s and honeycombs.
162 honeycomb
The 1
62 honeycomb contains
152 (9-honeycomb) and 1
61 9-demicube 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 ...
. 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 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, 1
61.
:
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 line ...
is determined by removing the ringed node and ringing the neighboring node. This makes the
birectified 9-simplex, 0
62.
:
Related polytopes and honeycombs
The 1
62 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 vert ...
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