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Coxeter-Dynkin diagram
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a Coxeter-Dynkin diagram is a graph
Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
 with labelled edges. It represents the spatial relations between a collection of mirrors (or reflecting hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
s), and describes a kaleidoscopic
Kaleidoscope

A kaleidoscope is a tube of mirrors containing loose colored beads, pebbles or other small colored objects. The viewer looks in one end and light enters the other end, Reflection off the mirrors....
 construction.

The diagram represents a Coxeter group
Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
. Each graph node represents a mirror (domain facet
Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
) and the label attached to a graph edge encodes the dihedral angle
Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
 order between two mirrors (on a domain ridge
Ridge (geometry)

In geometry, a ridge is an -dimensional element of an n-dimensional polytope. It is also sometimes called a subfacet for having one lower dimension than a Facet ....
).

In addition, when used to represent a specific uniform polytope
Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope Facet . A uniform polytope must also have only regular polygon faces....
, the diagram has rings (circles) around nodes for active mirrors and hollow nodes (holes) to represent alternation.

Dynkin diagrams are used to classify root systems and therefore Lie algebras.

Description
The diagram can also represent polytopes by adding rings (circles)
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 around nodes.






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Encyclopedia


In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a Coxeter-Dynkin diagram is a graph
Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
 with labelled edges. It represents the spatial relations between a collection of mirrors (or reflecting hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
s), and describes a kaleidoscopic
Kaleidoscope

A kaleidoscope is a tube of mirrors containing loose colored beads, pebbles or other small colored objects. The viewer looks in one end and light enters the other end, Reflection off the mirrors....
 construction.

The diagram represents a Coxeter group
Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
. Each graph node represents a mirror (domain facet
Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
) and the label attached to a graph edge encodes the dihedral angle
Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
 order between two mirrors (on a domain ridge
Ridge (geometry)

In geometry, a ridge is an -dimensional element of an n-dimensional polytope. It is also sometimes called a subfacet for having one lower dimension than a Facet ....
).

In addition, when used to represent a specific uniform polytope
Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope Facet . A uniform polytope must also have only regular polygon faces....
, the diagram has rings (circles) around nodes for active mirrors and hollow nodes (holes) to represent alternation.

Dynkin diagrams are used to classify root systems and therefore Lie algebras.

Description


The diagram can also represent polytopes by adding rings (circles)
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 around nodes. Every diagram needs at least one active node to represent a polytope.

The rings express information on whether a generating point is on or off the mirror. Specifically a mirror is active (creates reflections) only when points are off the mirror, so adding a ring means a point is off the mirror and creates a reflection.

Hollow rings (holes) are also used. A polytope with an alternation operator applied has all the ringed nodes replaced by holes. If all the nodes are holes, the figure is considered a snub.

Edges are labeled with an integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 n (or sometimes more generally a rational number
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
 p/q) representing a dihedral angle
Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
 of 180/n. If an edge is unlabeled, it is assumed to be 3. If n=2 the angle is 90 degrees and the mirrors have no interaction, and the edge can be omitted. Two parallel mirrors can be marked with "∞".

In principle, n mirrors can be represented by a complete graph in which all n*(n-1)/2 edges are drawn. In practice interesting configurations of mirrors will include a number of right angles, and the corresponding edges can be omitted.

Polytopes and tessellation
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
s can be generating using these mirrors and a single generator point. Mirror images create new points as reflections. Edges can be created between points and a mirror image. Faces
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
 can be constructed by cycles of edges created, etc.

Examples

  • A single node represents a single mirror. This is called group A1. If ringed this creates a line segment
    Line segment

    In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
     perpendicular to the mirror, represented as .
  • Two unattached nodes represent two perpendicular
    Perpendicular

    In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
     mirrors. If both nodes are ringed, a rectangle
    Rectangle

    In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
     can be created, or a square
    Square (geometry)

    In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
     if the point is equal distance from both mirrors.
  • Two nodes attached by an order-n edge can creates an n-gon
    Polygon

    In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
     if the point is on one mirror, and a 2n-gon if the point is off both mirrors. This forms the I1(n) group.
  • Two parallel mirrors can represent an infinite polygon I1(∞) group, also called I~1.
  • Three mirrors in a triangle form images seen in a traditional kaleidoscope
    Kaleidoscope

    A kaleidoscope is a tube of mirrors containing loose colored beads, pebbles or other small colored objects. The viewer looks in one end and light enters the other end, Reflection off the mirrors....
     and be represented by 3 nodes connected in a triangle. Repeating examples will have edges labeled as (3 3 3), (2 4 4), (2 3 6), although the last two can be drawn in a line with the 2 edge ignored. These will generate uniform tilings
    Tiling by regular polygons

    Plane Tessellation by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler in Harmonices Mundi....
    .
  • Three mirrors can generate uniform polyhedron
    Uniform polyhedron

    A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
    s, including rational numbers is the set of Schwarz triangle
    Schwarz triangle

    In mathematics, a Schwarz triangle is a spherical triangle that can be used to tessellation a sphere. Each Schwarz triangle defines a finite group — its triangle group....
    s.
  • Three mirrors with one perpendicular to the other two can form the uniform prisms
    Prism (geometry)

    In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
    .


In general all regular n-polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
s, represented by Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 symbol can have their fundamental domain
Fundamental domain

In geometry, the fundamental domain of a symmetry group of an object or pattern is a part of the pattern, as small as possible, which, based on the symmetry, determines the whole object or pattern....
s represented by a set of n mirrors and a related in a Coxeter-Dynkin diagram in a line of nodes and edges labeled by p,q,r...

Finite Coxeter groups


Families of convex uniform polytopes are defined by Coxeter group
Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
s.

Notes:
  • Three different symbols are given for the same groups - as a letter/number, as a bracketed set of numbers, and as the Coxeter diagram.
  • The bifurcated Dn groups are also given an h[] notation representing the fact it is half or alternated version of the regular Cn groups.
  • The bifurcated Dn and En groups are also labeled by a superscript form [3a,b,c] where a,b,c are the number of segments in each of the 3 branches.


nA1+C2+D3+E4-8F4H2-4I2(p)
1A1=[]
      
2A2=[3]
C2=[4]
   H2=[5]
I2(p)=[p]

3A3=[32]

C3=[4,3]
D3=A3=[30,1,1]

  H3=[5,3]
 
4A4=[33]

C4=[4,32]

D4=h[4,3,3]=[31,1,1]

E4=A4=[30,2,1]

F4=[3,4,3]
H4=[5,3,3]
 
5A5=[34]

C5=[4,33]

D5=h[4,33]=[32,1,1]

E5=B5=[31,2,1]

   
6A6=[35]

C6=[4,34]

D6=h[4,34]=[33,1,1]

E6=[32,2,1]

   
7A7=[36]

C7=[4,35]

D7=h[4,35]=[34,1,1]

E7=[33,2,1]

   
8A8=[37]

C8=[4,36]

D8=h[4,36]=[35,1,1]

E8=[34,2,1]

   
9A9=[38]

C9=[4,37]

D9=h[4,37]=[36,1,1]

    
10+......


  1. An forms the simplex
    Simplex

    In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
     polytope family.
  2. Dn is the family of demihypercubes, beginning at n=4 with the 16-cell
    16-cell

    In Fourth dimension geometry, a 16-cell, is a regular convex polychora, or polytope existing in four dimensions. It is also known as the hexadecachoron....
    , and n=5 with the demipenteract
    Demipenteract

    In Fifth dimension geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted....
    . (Also named Bn)
  3. Bn or Cn forms the hypercube
    Hypercube

    In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
    /orthoplex polytope family.
  4. I2n forms the regular polygon
    Regular polygon

    A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
    s. (Also named D2n)
  5. E6,E7,E8 are the generators of the Gosset Semiregular polytopes
  6. F4 is the 24-cell
    24-cell

    In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....
     polychoron family.
  7. H3 is the dodecahedron
    Dodecahedron

    A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
    /icosahedron
    Icosahedron

    In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
     polyhedron family. (Also named G3)
  8. H4 is the 120-cell
    120-cell

    In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....
    /600-cell
    600-cell

    In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
     polychoron family. (Also named G4)


Infinite Coxeter groups


Families of convex uniform tessellations are defined by Coxeter group
Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
s.

Note:
  1. A~n-1 is a cyclic group. (Also named Pn)
  2. B~n-1 forms the alternated hypercubic tessellation
    Alternated hypercubic honeycomb

    In geometry, the alternated hypercube honeycomb is a dimensional infinite series of Honeycomb s, based on the hypercube honeycomb with an Alternation operation....
     family. (Also named Sn) Also labeled by a h[] notation as a half of the regular one.
  3. C~n-1 forms the hypercube regular tessellation
    Hypercubic honeycomb

    In geometry, a hypercubic honeycomb is a family of List_of_regular_polytopes#Tessellations in n-dimensions with the Schl?fli symbols and containing the symmetry of Coxeter_diagram#Infinite_Coxeter_groups Rn for n>=3....
     family family. (Also named Rn)
  4. D~n-1 (Also named Qn) Also labeled by a q[] notation as a quarter of the regular one.
  5. E~6,E~7,E~8,E~9 are Gosset tessellations. (Also named T7,T8,T9,T10) T10 exists in hyperbolic space. Also labeled by a superscript form [3a,b,c] where a,b,c are the number of segments in each of the 3 branches.
  6. F~4 is the 24-cell
    24-cell

    In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....
      regular tessellation. (Also named U5)
  7. H~2 is the hexagonal tiling
    Hexagonal tiling

    In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....
    . (Also named V3)
  8. I~1 is two parallel mirrors. (Also named W2)


nA~2+B~3+C~2+D~4+E~6-9F~4H~2I~1
1       I~1=[∞]

2A~2=h[6,3]

 C~2=[4,4]

   H~2=[6,3]

 
3A~3=q[4,3,4]

B~3=h[4,3,4]

C~3=[4,3,4]

    
4A~4

B~4=h[4,32,4]

C~4=[4,32,4]

D~4=q[4,32,4]

 F~4=[3,4,3,3]

  
5A~5

B~5=h[4,33,4]

C~5=[4,33,4]

D~5=q[4,33,4]

    
6A~6

B~6=h[4,34,4]

C~6=[4,34,4]

D~6=q[4,34,4]

E~6=[32,2,2]

   
7A~7

B~7=h[4,35,4]

C~7=[4,35,4]

D~7=q[4,35,4]

E~7=[33,3,1]

   
8A~8

B~8=h[4,36,4]

C~8=[4,36,4]

D~8=q[4,36,4]

E~8=[35,2,1]

   
9A~9

B~9=h[4,37,4]

C~9=[4,37,4]

D~9=q[4,37,4]

E~9=[36,2,1]

   
10............    

Hyperbolic infinite Coxeter groups


There are many infinite Coxeter group
Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
s whose symmetry can tessellate hyperbolic space. There are infinitely many linear groups for order 2, in the form [p,q], and no compact groups beyond order 5.

The two bifurcating groups have doubled fundamental domains as linear ones. [5,3,4] --> [5,31,1], and [5,3,3,4] --> [5,3,31,1].

nLinearBifurcatingCyclic
3∞:
List of regular polytopes

This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....
 , 2(p+q)


...


...


...
0:∞: , p+q+r=10


...


...

...

...
43:
List of regular polytopes

This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....



1:
5:

53:
List of regular polytopes

This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....



1:
1:


See also

  • Coxeter group
    Coxeter group

    In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
  • Root system
    Root system

    In mathematics, a root system is a configuration of vector spaces in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras....
  • Satake diagram
    Satake diagram

    In the mathematics study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram whose configurations classify semisimple Lie algebra Lie algebras over the field of real numbers....
  • Uniform polytope
    Uniform polytope

    A uniform polytope is a vertex-transitive polytope made from uniform polytope Facet . A uniform polytope must also have only regular polygon faces....
    • Wythoff symbol
      Wythoff symbol

      In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a Wythoff construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane....
    • Uniform polyhedron
      Uniform polyhedron

      A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
    • List of uniform polyhedra
      List of uniform polyhedra

      Uniform polyhedra and tilings form a well studied group. They are listed here for quick comparison of their properties and varied naming schemes and symbols....
    • List of uniform planar tilings
      List of uniform planar tilings

      This table shows the 11 convex Uniform tessellations of the Euclidean geometry, and their dual tilings.There are three regular, and eight semiregular, Tiling by regular polygons in the plane....
    • Uniform polychoron
      Uniform polychoron

      In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
    • Convex uniform honeycomb
      Convex uniform honeycomb

      In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
  • Wythoff construction
    Wythoff construction

    File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
     and Wythoff symbol
    Wythoff symbol

    In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a Wythoff construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane....


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