Coxeter–Dynkin diagram: Difference between revisions

Revision as of 07:11, 16 September 2006

Latest revision as of 21:46, 31 August 2025

Line 1:

{{Short description|Pictorial representation of symmetry}}

In [[geometry]], a '''Coxeter-Dynkin diagram''' is a [[Graph (mathematics)|graph]] representing a relational set of [[mirror]] (or [[reflection]]al [[hyperplane]]s) in space for a [[Kaleidoscope|kaleidoscopic]] construction. Each node in the graph represents one mirror, and each edge represents the [[dihedral angle]] between two mirrors.

{{Multiple issues|

{{Technical|date=December 2022}}

{{More citations needed|date=December 2023}}

{{MOS|article|date=July 2025| [[MOS:FORMULA]] - avoid mixing {{tag|math}} and {{tl|math}} in the same expression}}

}}

[[Image:Finite coxeter.svg|400px|right|thumb|Coxeter–Dynkin diagrams for the fundamental finite Coxeter groups]]

[[File:Affine coxeter.svg|400px|right|thumb|Coxeter–Dynkin diagrams for the fundamental affine Coxeter groups]]

In [[geometry]], a '''[[Harold Scott MacDonald Coxeter|Coxeter]]–[[Eugene Dynkin|Dynkin]] diagram''' (or '''Coxeter diagram''', '''Coxeter graph''') is a [[Graph (discrete mathematics)|graph]] with numerically labeled edges (called '''branches''') representing a [[Coxeter group]] or sometimes a [[uniform polytope]] or [[uniform tiling]] constructed from the group.

A class of closely related objects is the [[Dynkin diagram]]s, which differ from Coxeter diagrams in two respects: firstly, branches labeled "{{math|4}}" or greater are [[Directed graph|directed]], while Coxeter diagrams are [[Undirected graph|undirected]]; secondly, Dynkin diagrams must satisfy an additional ([[Crystallographic restriction theorem|crystallographic]]) restriction, namely that the only allowed branch labels are {{math|2, 3, 4,}} and {{math|6.}} Dynkin diagrams correspond to and are used to classify [[root system]]s and therefore [[semisimple Lie algebra]]s.<ref>{{citation |first=Brian C. |last=Hall |title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction |publisher=Springer |year=2003 |isbn=978-0-387-40122-5}}</ref>

As a graph itself, the diagram represents [[Coxeter group]]s.

== Description ==

The diagram can also represent [[polytopes]] by adding [[ring]]s (circles) around nodes. 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.

A Coxeter group is a group that admits a presentation:

<math display="block">\langle r_0,r_1,\dots,r_n \mid (r_i r_j)^{m_{i,j}} = 1 \rangle</math>

where the {{mvar|m_i,j}} are integers that are elements of some [[symmetric matrix]] {{mvar|M}} which has {{math|1}}s on its [[diagonal]]. (Thus each generator <math>r_i</math> has order 2.){{efn|<math>(R_i \circ R_i)^1 = {\rm Id}.</math>}} This matrix {{mvar|M}}, the '''Coxeter matrix''', completely determines the Coxeter group.

Since the Coxeter matrix is symmetric, it can be viewed as the [[adjacency matrix]] of an [[Graph labeling|edge-labeled graph]] that has vertices corresponding to the generators {{mvar|r{{sub|i}}}}, and edges labeled with {{mvar|m{{sub|i,j}}}} between the vertices corresponding to {{mvar|r{{sub|i}}}} and {{mvar|r{{sub|j}}}}. In order to simplify these diagrams, two changes can be made:

Edges are labeled with an [[integer]] ''n'' (or sometimes more generally a [[rational number]] ''p/q'') representing a [[dihedral angle]] of ''180/n''. If an edge is unlabeled, it is assumed to be ''3''. If ''n'' is two the angle is 90 degrees and the mirrors have no interaction, and the edge can be left unmarked. Two parallel mirrors can be marked with an infinity (&infin;) symbol.

* Edges that are labeled with {{math|2}} can be omitted, with the missing edges being implied to be {{math|2}}s. A label {{math|2}} indicates that the corresponding two generators commute; {{math|2}} is the smallest number that can be used to label an edge.

* Edges labeled {{math|3}} can be left unlabeled, with the implication that an unlabeled edge acts as a {{math|3}}.

The resulting graph is a Coxeter-Dynkin diagram that describes the considered Coxeter group.

== Schläfli matrix ==

In general ''n'' mirrors can be represented in an n-[[simplex]] graph where all ''n*(n-1)/2'' edges are drawn. In practice interesting sets of mirrors will have a number of right angles, and such edges can be left undrawn.

Every Coxeter diagram has a corresponding '''Schläfli matrix''' (so named after [[Ludwig Schläfli]]), {{math|''A'',}} with matrix elements {{math|''a_i,j'' {{=}} ''a_j,i'' {{=}} −2 cos({{pi}}/''p_i,j'')}} where {{mvar|p_i,j}} is the branch order between mirrors {{mvar|i}} and {{math|''j'';}} that is, {{math|{{pi}}/''p_i,j''}} is the [[dihedral angle]] between mirrors {{mvar|i}} and {{mvar|j.}} As a ''matrix of cosines'', {{mvar|A}} is also called a [[Gramian matrix]]. All [[Coxeter group]] Schläfli matrices are symmetric because their root vectors are normalized. {{mvar|A}} is closely related to the [[Cartan matrix]], used in the similar but directed graph: the [[Dynkin diagram]], in the limited cases of {{math|''p'' {{=}} 2,3,4,}} and {{math|6,}} which are generally ''not'' symmetric.

The determinant of the Schläfli matrix is called the '''Schläflian''';{{cn|date=January 2024}} the Schläflian and its sign determine whether the group is finite (positive), affine (zero), or indefinite (negative).<ref>{{cite book|last1=Borovik|first1=Alexandre|last2=Borovik|first2=Anna|title=Mirrors and reflections|year=2010|publisher=Springer|pages=118–119}}</ref> This rule is called '''Schläfli's Criterion'''.<ref>Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}}, sec. 7.7, p. 133, ''Schläfli's Criterion''</ref>{{Failed verification|date=November 2021}} {{^|Schläfli's Criterion as described in section 7.7 applies only to linear Coxeter diagrams.}}

[[Image:Example fundamental ___domain coxeter dynkin diagram.png|320px|thumb]]

[[Image:Example2 fundamental ___domain coxeter dynkin diagram.png|320px|thumb]]

[[Polytopes]] and [[tessellation]]s can be generating using these mirrors and a single generator point. Mirror images create new points as reflections. [[Edge]]s can be created between points and a mirror image. [[Face (geometry)|Faces]] can be constructed by cycles of edges created, etc.

The [[eigenvalues]] of the Schläfli matrix determine whether a Coxeter group is of ''finite type'' (all positive), ''affine type'' (all non-negative, at least one is zero), or ''indefinite type'' (otherwise). The indefinite type is sometimes further subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups. We use the following definitions:

Examples:

* A Coxeter group with connected diagram is ''hyperbolic'' if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type.

* A single node represents a single mirror. All points off the mirror are drawn with a ring. Connecting such an off mirror point to its reflection creates a [[digon]] or [[edge]] perpendicular to the mirror.

* A hyperbolic Coxeter group is '''compact''' if all its subgroups are finite (i.e. have positive determinants), and '''paracompact''' if all its subgroups are finite or affine (i.e. have nonnegative determinants).

* Two unattached nodes represent two [[perpendicular]] mirrors. If both nodes are ringed, a [[rectangle]] can be created, or a [[square]] if the point is equal distance from both mirrors.

* Two nodes attached by an ''n'' edge can create an [[polygon|n-gon]] if the point is on one mirror, and a ''2n-gon'' if the point is off both mirrors.

Finite and affine groups are also called ''elliptical'' and ''parabolic'' respectively. Hyperbolic groups are also called Lannér, after F. Lannér who enumerated the compact hyperbolic groups in 1950,<ref>Lannér F., ''On complexes with transitive groups of automorphisms'', Medd. Lunds Univ. Mat. Sem. [Comm. Sem. Math. Univ. Lund], 11 (1950), 1–71</ref> and Koszul (or quasi-Lannér) for the paracompact groups.

* Three mirrors in a triangle form images seen in a traditional [[kaleidoscope]] 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 [[Tiling by regular polygons|uniform tilings]].

* Three mirrors with a common point can generate [[uniform polyhedron]]s, including rational numbers is the set of [[Schwarz triangle]]s.

=== Rank 2 Coxeter groups ===

The type of a rank {{math|2}} Coxeter group, i.e. generated by two different mirrors, is fully determined by the determinant of the Schläfli matrix, as this determinant is simply the product of the eigenvalues: finite (positive determinant), affine (zero determinant), or hyperbolic (negative determinant) type. Coxeter uses an equivalent [[Coxeter notation|bracket notation]] which lists sequences of branch orders as a substitute for the node-branch graphic diagrams. Rational solutions {{math|[''p''/''q''],}} {{CDD|node|p|rat|q|node}}, also exist, with [[Greatest common divisor|{{math|gcd}}]]{{math|(''p'',''q'') {{=}} 1;}} these define overlapping fundamental domains. For example, {{math|3/2, 4/3, 5/2, 5/3, 5/4,}} and {{math|6/5.}}

{| class=wikitable

!Type

!colspan=5|Finite

!Affine

!colspan=2|Hyperbolic

|- align=center

!Geometry

|[[File:Dihedral_symmetry_domains_1.svg|80px]]

|[[File:Dihedral_symmetry_domains_2.svg|80px]]

|[[File:Dihedral_symmetry_domains_3.png|80px]]

|[[File:Dihedral_symmetry_domains_4.svg|80px]]

|...

|[[File:Dihedral_symmetry_domains_infinity.png|80px]]

|[[File:Horocycle_mirrors.png|80px]]

|[[File:Dihedral_symmetry_ultra.png|80px]]

|- align=center

!Coxeter diagram<BR>Bracket notation

|{{CDD|node_c1}}<BR>[ ]

|{{CDD|node_c1|2|node_c3}}<BR>[2]

|{{CDD|node_c1|3|node_c3}}<BR>[3]

|{{CDD|node_c1|4|node_c3}}<BR>[4]

|{{CDD|node_c1|p|node_c3}}<BR>[''p'']

|{{CDD|node_c1|infin|node_c3}}<BR>[&infin;]

|{{CDD|node_c2|infin|node_c3}}<BR>[&infin;]

|{{CDD|node_c2|ultra|node_c3}}<BR>[i&pi;/''&lambda;'']

|-

![[Group order|Order]]

!2

!4

!6

!8

!2''p''

!colspan=3|&infin;

|- align=center

|colspan=9|Mirror lines are colored to correspond to Coxeter diagram nodes.<BR>Fundamental domains are alternately colored.

|}

{| class="wikitable collapsible collapsed" style="width:720px;"

!colspan=8| Rank 2 Coxeter group diagrams

|-

!rowspan=2|[[Group order|Order]]<BR>''p''

!rowspan=2|Group

!rowspan=2 colspan=2|Coxeter diagram

!colspan=2|Schläfli matrix

|-

!<math>\left[\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}\right]</math>

!Determinant<BR><math>(4 - a_{21}a_{12})</math>

|- align=center

!colspan=7|[[Dynkin diagram#Finite Dynkin diagrams|Finite]] (Determinant > 0)

|- align=center

!2

!I₂(2) = A₁×A₁

|{{CDD|node|2|node}}||[2]

|<math>\left[\begin{smallmatrix}2&0\\0&2\end{smallmatrix}\right]</math>

|4

|- align=center

!3

!I₂(3) = A₂

|{{CDD|node|3|node}}||[3]

|<math>\left[\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}\right]</math>

|rowspan=2|3

|- align=center

!3/2

!

|{{CDD|node|3x|rat|2x|node}}||[3/2]

|<math>\left[\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right]</math>

|- align=center

!4

!I₂(4) = B₂

|{{CDD|node|4|node}}||[4]

|<math>\left[\begin{smallmatrix}2&-\sqrt{2}\\-\sqrt{2}&2\end{smallmatrix}\right]</math>

|rowspan=2|2

|- align=center

!4/3

!

|{{CDD|node|4|rat|3x|node}}||[4/3]

|<math>\left[\begin{smallmatrix}2&\sqrt{2}\\\sqrt{2}&2\end{smallmatrix}\right]</math>

|- align=center

!5

!I₂(5) = H₂

|{{CDD|node|5|node}}||[5]

|<math>\left[\begin{smallmatrix}2&-\phi\\-\phi&2\end{smallmatrix}\right]</math>

|rowspan=2|<math>(5-\sqrt{5})/2</math><BR>≈ 1.38196601125

|- align=center

!5/4

!

|{{CDD|node|5|rat|4|node}}||[5/4]

|<math>\left[\begin{smallmatrix}2&\phi\\\phi&2\end{smallmatrix}\right]</math>

|- align=center

!5/2

!

|{{CDD|node|5-2|node}}||[5/2]

|<math>\left[\begin{smallmatrix}2&1-\phi\\1-\phi&2\end{smallmatrix}\right]</math>

|rowspan=2|<math>(5+\sqrt{5})/2</math><BR>≈ 3.61803398875

|- align=center

!5/3

!

|{{CDD|node|5-3|node}}||[5/3]

|<math>\left[\begin{smallmatrix}2&\phi-1\\\phi-1&2\end{smallmatrix}\right]</math>

|- align=center

!6

!I₂(6) = G₂

|{{CDD|node|6|node}}||[6]

|<math>\left[\begin{smallmatrix}2&-\sqrt{3}\\-\sqrt{3}&2\end{smallmatrix}\right]</math>

|rowspan=2|1

|- align=center

!6/5

!

|{{CDD|node|6|rat|5|node}}||[6/5]

|<math>\left[\begin{smallmatrix}2&\sqrt{3}\\\sqrt{3}&2\end{smallmatrix}\right]</math>

|- align=center

!8

!I₂(8)

|{{CDD|node|8|node}}||[8]

|<math>\left[\begin{smallmatrix}2&-\sqrt{2+\sqrt2}\\-\sqrt{2+\sqrt2}&2\end{smallmatrix}\right]</math>

|<math>2-\sqrt{2}</math><BR>≈ 0.58578643763

|- align=center

!10

!I₂(10)

|{{CDD|node|10|node}}||[10]

|<math>\left[\begin{smallmatrix}2&-\sqrt{(5+\sqrt5)/2}\\-\sqrt{(5+\sqrt5)/2}&2\end{smallmatrix}\right]</math>

|<math>(3-\sqrt{5})/2</math><BR>≈ 0.38196601125

|- align=center

!12

!I₂(12)

|{{CDD|node|12|node}}||[12]

|<math>\left[\begin{smallmatrix}2&-\sqrt{2+\sqrt3}\\-\sqrt{2+\sqrt3}&2\end{smallmatrix}\right]</math>

|<math>2-\sqrt{3}</math><BR>≈ 0.26794919243

|- align=center

!''p''

!I₂(''p'')

|{{CDD|node|p|node}}||[''p'']

|<math>\left[\begin{smallmatrix}2&-2\cos(\pi/p)\\-2\cos(\pi/p)&2\end{smallmatrix}\right]</math>

|<math>4\sin^2(\pi/p)</math>

|- align=center

!colspan=6|[[Affine Dynkin diagram|Affine]] (Determinant = 0)

|- align=center

!∞

!I₂(∞) = <math>{\tilde{I}}_1</math> = <math>{\tilde{A}}_1</math>

|{{CDD|node|infin|node}}||[&infin;]

|<math>\left[\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}\right]</math>

|0

|- align=center

!colspan=6|Hyperbolic (Determinant ≤ 0)

|- align=center

!∞

!

|{{CDD|node|infin|node}}||[&infin;]

|<math>\left[\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}\right]</math>

|0

|- align=center

!∞

!

|{{CDD|node|ultra|node}}||[i&pi;/''&lambda;'']

|<math>\left[\begin{smallmatrix}2&-2\cosh(2\lambda)\\-2\cosh(2\lambda)&2\end{smallmatrix}\right]</math>

|<math>-4\sinh^2(2\lambda) \le 0</math>

|}

=== Geometric visualizations ===

The Coxeter–Dynkin diagram can be seen as a graphic description of the [[fundamental ___domain]] of mirrors. A mirror represents a [[hyperplane]] within a spherical, Euclidean, or hyperbolic space of given dimension. (In 2D spaces, a mirror is a line; in 3D, a mirror is a plane.)

These visualizations show the fundamental domains for 2D and 3D Euclidean groups, and for 2D spherical groups. For each, the Coxeter diagram can be deduced by identifying the hyperplane mirrors and labelling their connectivity, ignoring {{math|90}}-degree dihedral angles (order {{math|2;}} see footnote [a] below).

{| class="wikitable" style="width:720px;"

|colspan=2|[[Image:Coxeter-dynkin plane groups.png|720px]]<BR>Coxeter groups in the Euclidean plane with equivalent diagrams.

Here, ___domain vertices are labeled as graph branches {{math|1, 2,}} etc., and are colored by their reflection order (connectivity). Reflections are labeled as graph nodes {{math|R1, R2,}} etc. Reflections at {{math|90}} degrees are inactive in the sense that, together, they generate no new reflections;{{efn|If <math>R \perp R',</math> then <math>R' \circ R = {\rm sym}_{R \cap R'} = R \circ R',</math> so <math>R \circ R' \circ R = R \circ R \circ R' = {\rm Id} \circ R' = R'</math> and <math>R' \circ R \circ R' = R' \circ R' \circ R = {\rm Id} \circ R = R.</math>}} they are therefore not connected to each other by a branch on the diagram. Parallel mirrors are connected to each other by an {{math|∞}} labeled branch.

The square of the prismatic group <math>{\tilde{I}}_1</math>×<math>{\tilde{I}}_1</math> is shown as a doubling of the <math>{\tilde{C}}_2</math> triangle around its {{math|R2}} side*, but can also be created as a rectangular ___domain from doubling the <math>{\tilde{G}}_2</math> triangle around its {{math|R2}} side*. The <math>{\tilde{A}}_2</math> triangle is a doubling of the <math>{\tilde{G}}_2</math> triangle around its {{math|R3}} side*.<BR>*(this side disappears by doubling around itself)

|- valign=top

|colspan=2|[[File:Hyperbolic kaleidoscopes.png|720px]]<BR>Many Coxeter groups in the [[hyperbolic plane]] can be extended from the Euclidean cases as a series of hyperbolic solutions.

|- valign=top

|[[Image:Coxeter-Dynkin 3-space groups.png|360px]]<BR>Coxeter groups in 3-space with diagrams. Mirrors (triangle faces) are labeled by opposite vertex: {{math|0, ..., 3.}} Branches are colored by their reflection order.<BR><math>{\tilde{C}}_3</math> fills {{math|1/48}} of the cube. <math>{\tilde{B}}_3</math> fills {{math|1/24}} of the cube. <math>{\tilde{A}}_3</math> fills {{math|1/12}} of the cube.

|[[Image:Coxeter-Dynkin sphere groups.png|360px]]<BR>Coxeter groups in the sphere with equivalent diagrams. One fundamental ___domain is outlined in yellow. Domain vertices (and graph branches) are colored by their reflection order.

|}

== Application to uniform polytopes ==

{| class=wikitable align=right width=480

|- valign=top

|[[File:Coxeter_diagram_elements.png|240px]]<BR>In constructing uniform polytopes, nodes are marked as ''active'' by a ring if a generator point is off the mirror, creating a new edge between a generator point and its mirror image. An unringed node represents an ''inactive'' mirror that generates no new points. A ring with no node is called a ''hole''.

|[[File:Kaleidoscopic_construction_of_square.png|240px]]<BR>Two orthogonal mirrors can be used to generate a square, {{CDD|node_1|2|node_1}}, seen here with a red generator point and 3 virtual copies across the mirrors. The generator has to be off both mirrors in this orthogonal case to generate an interior. The ring markup presumes active rings have generators equal distance from all mirrors, while a [[rectangle]] can also represent a nonuniform solution.

|}

Coxeter–Dynkin diagrams can explicitly enumerate nearly all classes of [[uniform polytope]] and [[Uniform tiling|uniform tessellations]]. Every uniform polytope with pure reflective symmetry (all but a few special cases have pure reflectional symmetry) can be represented by a Coxeter–Dynkin diagram with permutations of ''markups''. Each uniform polytope can be generated using such mirrors and a single generator point: mirror images create new points as reflections, then polytope [[edge (geometry)|edges]] can be defined between points and a mirror image point. [[Face (geometry)|Faces]] are generated by the repeated reflection of an edge eventually wrapping around to the original generator; the final shape, as well as any higher-dimensional facets, are likewise created by the face being reflected to enclose an area.

To specify the generating vertex, one or more nodes are marked with rings, meaning that the vertex is ''not'' on the mirror(s) represented by the ringed node(s). (If two or more mirrors are marked, the vertex is equidistant from them.) A mirror is ''active'' (creates reflections) only with respect to points not on it. A diagram needs at least one active node to represent a polytope. An unconnected diagram (subgroups separated by order-2 branches, or orthogonal mirrors) requires at least one active node in each subgraph.

All [[regular polytope]]s, represented by [[Schläfli symbol]] {{math|{{mset|''p'', ''q'', ''r'', ...}}}}, can have their [[fundamental ___domain]]s represented by a set of ''n'' mirrors with a related Coxeter–Dynkin diagram of a line of nodes and branches labeled by {{math|''p'', ''q'', ''r'', ...,}} with the first node ringed.

Uniform polytopes with one ring correspond to generator points at the corners of the fundamental ___domain simplex. Two rings correspond to the edges of simplex and have a degree of freedom, with only the midpoint as the uniform solution for equal edge lengths. In general ''k''-ring generator points are on ''(k-1)''-faces of the simplex, and if all the nodes are ringed, the generator point is in the interior of the simplex.

The special case of uniform polytopes with non-reflectional symmetry is represented by a secondary markup where the central dot of a ringed node is removed (called a ''hole''). These shapes are [[Alternation (geometry)|alternations]] of polytopes with reflective symmetry, implying that every other vertex is deleted. The resulting polytope will have a subsymmetry of the original [[Coxeter group]]. A truncated alternation is called a [[Snub (geometry)|''snub'']].

* A single node represents a single mirror. This is called group A₁. If ringed this creates a [[line segment]] perpendicular to the mirror, represented as {}.

* Two unattached nodes represent two [[perpendicular]] mirrors. If both nodes are ringed, a [[rectangle]] can be created, or a [[square (geometry)|square]] if the point is at equal distance from both mirrors.

* Two nodes attached by an order-''n'' branch can create an [[polygon|''n''-gon]] if the point is on one mirror, and a 2''n''-gon if the point is off both mirrors. This forms the {{math|I₁(n)}} group.

* Two parallel mirrors can represent an infinite polygon {{math|I₁(∞)}} group, also called {{math|Ĩ₁}}.

* Three mirrors in a triangle form images seen in a traditional [[kaleidoscope]] and can be represented by three nodes connected in a triangle. Repeating examples will have branches labeled as (3 3 3), (2 4 4), (2 3 6), although the last two can be drawn as a line (with the ''2'' branches ignored). These will generate [[Tiling by regular polygons|uniform tilings]].

* Three mirrors can generate [[uniform polyhedron#(4 3 2) Oh Octahedral symmetry|uniform polyhedra]]; including rational numbers gives the set of [[Schwarz triangle]]s.

* Three mirrors with one perpendicular to the other two can form the [[Prism (geometry)|uniform prisms]].

{| class=wikitable width=800

In general all regular n-[[polytope]]s, represented by [[Schläfli symbol]] symbol {p,q,r,...} can have their [[fundamental ___domain]]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...

|- valign=top

|[[File:Wythoffian construction diagram.svg|400px]]<BR>There are 7 reflective uniform constructions within a general triangle, based on 7 topological generator positions within the fundamental ___domain. Every active mirror generates an edge, with two active mirrors have generators on the ___domain sides and three active mirrors has the generator in the interior. One or two degrees of freedom can be solved for a unique position for equal edge lengths of the resulting polyhedron or tiling.

|[[File:Polyhedron truncation example3.png|400px]]<BR>Example 7 generators on [[octahedral symmetry]], fundamental ___domain triangle (4 3 2), with 8th snub generation as an [[alternation (geometry)|alternation]]

|}

The duals of the uniform polytopes are sometimes marked up with a perpendicular slash replacing ringed nodes, and a slash-hole for hole nodes of the snubs. For example, {{CDD|node_1|2|node_1}} represents a [[rectangle]] (as two active orthogonal mirrors), and {{CDD|node_f1|2|node_f1}} represents its [[dual polygon]], the [[rhombus]].

Examples

* [[Image:Dynkins-100.png]] - {p,q} - a regular polyhedron or tiling

=== Examples with polyhedra and tilings ===

* [[Image:Dynkins-1000.png]] - {p,q,r} - a regular [[polychoron]] or honeycomb

For example, the B₃ [[Coxeter group]] has a diagram: {{CDD|node|4|node|3|node}}. This is also called [[octahedral symmetry]].

There are 7 convex [[uniform polyhedra]] that can be constructed from this symmetry group and 3 from its [[Alternation (geometry)|alternation]] subsymmetries, each with a uniquely marked up Coxeter–Dynkin diagram. The [[Wythoff symbol]] represents a special case of the Coxeter diagram for rank 3 graphs, with all 3 branch orders named, rather than suppressing the order 2 branches. The Wythoff symbol is able to handle the ''snub'' form, but not general alternations without all nodes ringed.

{{Octahedral truncations}}

The same constructions can be made on disjointed (orthogonal) Coxeter groups like the uniform [[Prism (geometry)|prisms]], and can be seen more clearly as tilings of [[dihedron]]s and [[hosohedra]] on the sphere, like this [6]×[] or [6,2] family:

{{Hexagonal dihedral truncations}}

In comparison, the [6,3], {{CDD|node|6|node|3|node}} family produces a parallel set of 7 uniform tilings of the Euclidean plane, and their dual tilings. There are again 3 alternations and some half symmetry version.

{{Hexagonal_tiling_table}}

In the hyperbolic plane [7,3], {{CDD|node|7|node|3|node}} family produces a parallel set of uniform tilings, and their dual tilings. There is only 1 alternation ([[Snub (geometry)|snub]]) since all branch orders are odd. Many other hyperbolic families of uniform tilings can be seen at [[uniform tilings in hyperbolic plane]].

{{Heptagonal tiling table}}

== Very-extended Coxeter diagrams ==

One usage includes a [[Dynkin diagram#Noncompact (Over-extended forms)|'''very-extended''']] definition from the direct [[Dynkin diagram]] usage which considers affine groups as '''extended''', hyperbolic groups '''over-extended''', and a third node as '''very-extended''' simple groups. These extensions are usually marked by an exponent of 1,2, or 3 + symbols for the number of extended nodes. This extending series can be extended backwards, by sequentially removing the nodes from the same position in the graph, although the process stops after removing branching node. The [[En (Lie algebra)|E₈]] extended family is the most commonly shown example extending backwards from E₃ and forwards to E₁₁.

The extending process can define a limited series of Coxeter graphs that progress from finite to affine to hyperbolic to Lorentzian. The determinant of the Cartan matrices determine where the series changes from finite (positive) to affine (zero) to hyperbolic (negative), and ending as a Lorentzian group, containing at least one hyperbolic subgroup.<ref>{{cite arXiv | eprint=hep-th/0608161 | title=Kac-Moody Algebras in M-theory | last1=De Buyl | first1=Sophie | date=2006 }}</ref> The noncrystallographic H_n groups forms an extended series where H₄ is extended as a compact hyperbolic and over-extended into a lorentzian group.

The determinant of the Schläfli matrix by rank are:<ref>[http://deepblue.lib.umich.edu/handle/2027.42/70011 Cartan–Gram determinants for the simple Lie groups], Wu, Alfred C. T, The American Institute of Physics, Nov 1982</ref>

* {{math|size=100%|1=det(''A''₁^''n'' = [2^''n''−1]) = 2^''n'' (Finite for all ''n'')}}

* {{math|size=100%|1=det(''A''_''n'' = [3^''n''−1]) = ''n'' + 1 (finite for all ''n'')}}

* {{math|size=100%|1=det(''B''_''n'' = [4,3^''n''−2]) = 2 (finite for all ''n'')}}

* {{math|size=100%|1=det(''D''_''n'' = [3^{''n''−3,1,1}]) = 4 (finite for all ''n'')}}

Determinants of the Schläfli matrix in exceptional series are:

* {{math|size=100%|1=det([[En (Lie algebra)|''E''_''n'']] = [3^{''n''−3,2,1}]) = 9 − ''n'' (finite for ''E''₃ (= ''A''₂''A''₁), ''E''₄ (= ''A''₄), ''E''₅ (= ''D''₅), [[E6 (mathematics)|''E''₆]], [[E7 (mathematics)|''E''₇]] and [[E8 (mathematics)|''E''₈]], affine at [[E8 lattice|''E''₉]] (<math>\tilde{E}_8</math>), hyperbolic at ''E''₁₀)}}

* {{math|size=100%|1=det([3^{''n''−4,3,1}]) = 2(8 − ''n'') (finite for ''n'' = 4 to 7, affine (<math>\tilde{E}_7</math>), and hyperbolic at ''n'' = 8.)}}

* {{math|size=100%|1=det([3^{''n''−4,2,2}]) = 3(7 − ''n'') (finite for ''n'' = 4 to 6, affine (<math>\tilde{E}_6</math>), and hyperbolic at ''n'' = 7.)}}

* {{math|size=100%|1=det(''F''_''n'' = [3,4,3^''n''−3]) = 5 − ''n'' (finite for F₃ (= ''B''₃) to [[F4 (mathematics)|''F''₄]], affine at [[F4 lattice|''F''₅]] (<math>\tilde{F}_4</math>), hyperbolic at ''F''₆)}}

* {{math|size=100%|1=det(''G''_''n'' = [6,3^''n''−2]) = 3 − ''n'' (finite for [[G2 (mathematics)|''G''₂]], affine at ''G''₃ (<math>\tilde{G}_2</math>), hyperbolic at ''G''₄)}}

{| class=wikitable

|+ Smaller extended series

|-

!Finite

!<math>A_2</math>

!<math>C_2</math>

![[G2 (mathematics)|<math>G_2</math>]]

!<math>A_3</math>

!<math>B_3</math>

!<math>C_3</math>

!<math>H_4</math>

|-

!Rank {{mvar|n}}

![3^[3],3^{''n''&minus;3}]

![4,4,3^{''n''&minus;3}]

!G_n=[6,3^{''n''&minus;2}]

![3^[4],3^{''n''&minus;4}]

![4,3^{1,''n''&minus;3}]

![4,3,4,3^{''n''&minus;4}]

!''H_n''=[5,3^{''n''&minus;2}]

|-

!2

| style="background:#ffffe0;"|[3]<br>A₂<br>{{CDD|branch}}

| style="background:#ffffe0;"|[4]<br>C₂<br>{{CDD|node|4|node}}

| style="background:#ffffe0;"|[6]<br>G₂<br>{{CDD|node|6|node}}

|

|[2]<br>A₁²<br>{{CDD|nodes}}

|[4]<br>C₂<br>{{CDD|node|4|node}}

|[5]<br>H₂<br>{{CDD|node|5|node}}

|-

!3

| style="background:#ffe0e0;"|[3^[3]]<br><math>A_{2}^{+}={\tilde{A}}_{2}</math><br>{{CDD|branch|split2|node_c1}}

| style="background:#ffe0e0;"|[4,4]<br><math>C_{2}^{+}={\tilde{C}}_{2}</math><br>{{CDD|node|4|node|4|node_c1}}

| style="background:#ffe0e0;"|[6,3]<br><math>G_{2}^{+}={\tilde{G}}_{2}</math><br>{{CDD|node|6|node|3|node_c1}}

| style="background:#ffffe0;"|[3,3]=A₃<br>{{CDD|node|split1|nodes}}

| style="background:#ffffe0;"|[4,3]<br>B₃<br>{{CDD|nodes|split2-43|node}}

| style="background:#ffffe0;"|[4,3]<br>C₃<br>{{CDD|node|4|node|3|node}}

|[5,3]<br>H₃<br>{{CDD|node|5|node|3|node}}

|-

!4

| style="background:#e0ffe0;"|[3^[3],3]<br><math>A_{2}^{++}={\overline{P}}_3</math><br>{{CDD|branch|split2|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[4,4,3]<br><math>C_{2}^{++}={\overline{R}}_3</math><br>{{CDD|node|4|node|4|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[6,3,3]<br><math>G_{2}^{++}={\overline{V}}_3</math><br>{{CDD|node|6|node|3|node_c1|3|node_c2}}

| style="background:#ffe0e0;"|[3^[4]]<br><math>A_{3}^{+}={\tilde{A}}_3</math><br>{{CDD|node|split1|nodes|split2|node_c1}}

| style="background:#ffe0e0;"|[4,3^1,1]<br><math>B_{3}^{+}={\tilde{B}}_{3}</math><br>{{CDD|nodes|split2-43|node|3|node_c1}}

| style="background:#ffe0e0;"|[4,3,4]<br><math>C_{3}^{+}={\tilde{C}}_{3}</math><br>{{CDD|node|4|node|3|node|4|node_c1}}

| style="background:#ffffe0;"|[5,3,3]<br>H₄<br>{{CDD|node|5|node|3|node|3|node}}

|-

!5

| style="background:#e0e0ff;"|[3^[3],3,3]<br>A₂⁺⁺⁺<br>{{CDD|branch|split2|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[4,4,3,3]<br>C₂⁺⁺⁺<br>{{CDD|node|4|node|4|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[6,3,3,3]<br>G₂⁺⁺⁺<br>{{CDD|node|6|node|3|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0ffe0;"|[3^[4],3]<br><math>A_{3}^{++}={\overline{P}}_4</math><br>{{CDD|node|split1|nodes|split2|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[4,3^2,1]<br><math>B_{3}^{++}={\overline{S}}_4</math><br>{{CDD|nodes|split2-43|node|3|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[4,3,4,3]<br><math>C_{3}^{++}={\overline{R}}_4</math><br>{{CDD|node|4|node|3|node|4|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[5,3³]<br>H₅=<math>{\overline{H}}_4</math><br>{{CDD|node|5|node|3|node|3|node|3|node}}

|-

!6

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|[3^[4],3,3]<br>A₃⁺⁺⁺<br>{{CDD|node|split1|nodes|split2|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[4,3^3,1]<br>B₃⁺⁺⁺<br>{{CDD|nodes|split2-43|node|3|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[4,3,4,3,3]<br>C₃⁺⁺⁺<br>{{CDD|node|4|node|3|node|4|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[5,3⁴]<br>H₆<br>{{CDD|node|5|node|3|node|3|node|3|node|3|node}}

|- align=center

!Det({{mvar|M_n}})

|3(3&minus;''n'')

|2(3&minus;''n'')

|3&minus;''n''

|4(4&minus;''n'')

|colspan=2|2(4&minus;''n'')

|

|}

{| class=wikitable

|+ Middle extended series

|-

!Finite

!<math>A_4</math>

!<math>B_4</math>

!<math>C_4</math>

!<math>D_4</math>

![[F4 (mathematics)|<math>F_4</math>]]

!<math>A_5</math>

!<math>B_5</math>

!<math>D_5</math>

|-

!Rank {{mvar|n}}

![3^[5],3^{''n''&minus;5}]

![4,3,3^{''n''&minus;4,1}]

![4,3,3,4,3^{''n''&minus;5}]

![3^{''n''&minus;4,1,1,1}]

![3,4,3^{''n''&minus;3}]

![3^[6],3^{''n''&minus;6}]

![4,3,3,3^{''n''&minus;5,1}]

![3^1,1,3,3^{''n''&minus;5,1}]

|-

!3

|

|[4,3^−1,1]<br>B₂A₁<br>{{CDD|nodea|4a|nodea|2|nodeb}}

|[4,3]<br>B₃<br>{{CDD|node|4|node|3|node}}

|[3^−1,1,1,1]<br>A₁³<br>{{CDD|nodeabc}}

|[3,4]<br>B₃<br>{{CDD|node|3|node|4|node}}

|

|[4,3,3]<br>C₃<br>{{CDD|node|4|node|3|node}}

|-

!4

| style="background:#ffffe0;"|[3³]<br>A₄<br>{{CDD|branch|3ab|nodes}}

| style="background:#ffffe0;"|[4,3,3]<br>B₄<br>{{CDD|nodea|4a|nodea|3a|branch}}

| style="background:#ffffe0;"|[4,3,3]<br>C₄<br>{{CDD|node|4|node|3|node|3|node}}

| style="background:#ffffe0;"|[3^0,1,1,1]<br>D₄<br>{{CDD|node|branch3|splitsplit2|node}}

| style="background:#ffffe0;"|[3,4,3]<br>F₄<br>{{CDD|node|3|node|4|node|3|node}}

|

|[4,3,3,3^−1,1]<br>B₃A₁<br>{{CDD|nodea|4a|nodea|3a|nodea|2|nodeb}}

|[3^1,1,3,3^−1,1]<br>A₃A₁<br>{{CDD|nodea|3a|branch|2|nodeb}}

|-

!5

| style="background:#ffe0e0;"|[3^[5]]<br><math>A_{4}^{+}={\tilde{A}}_{4}</math><br>{{CDD|branch|3ab|nodes|split2|node_c1}}

| style="background:#ffe0e0;"|[4,3,3^1,1]<br><math>B_{4}^{+}={\tilde{B}}_{4}</math><br>{{CDD|nodea|4a|nodea|3a|branch|3a|nodea_c1}}

| style="background:#ffe0e0;"|[4,3,3,4]<br><math>C_{4}^{+}={\tilde{C}}_{4}</math><br>{{CDD|node|4|node|3|node|3|node|4|node}}

| style="background:#ffe0e0;"|[3^1,1,1,1]<br><math>D_{4}^{+}={\tilde{D}}_{4}</math><br>{{CDD|node|branch3|splitsplit2|node|3|node_c1}}

| style="background:#ffe0e0;"|[3,4,3,3]<br><math>F_{4}^{+}={\tilde{F}}_{4}</math><br>{{CDD|node|3|node|4|node|3|node|3|node_c1}}

| style="background:#ffffe0;"|[3⁴]<br>A₅<br>{{CDD|node|split1|nodes|3ab|nodes}}

| style="background:#ffffe0;"|[4,3,3,3,3]<br>B₅<br>{{CDD|nodea|4a|nodea|3a|nodea|3a|branch}}

| style="background:#ffffe0;"|[3^1,1,3,3]<br>D₅<br>{{CDD|nodea|3a|branch|3a|branch}}

|-

!6

| style="background:#e0ffe0;"|[3^[5],3]<br><math>A_{4}^{++}={\overline{P}}_5</math><br>{{CDD|branch|3ab|nodes|split2|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[4,3,3^2,1]<br><math>B_{4}^{++}={\overline{S}}_5</math><br>{{CDD|nodea|4a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2}}

| style="background:#e0ffe0;"|[4,3,3,4,3]<br><math>C_{4}^{++}={\overline{R}}_5</math><br>{{CDD|node|4|node|3|node|3|node|4|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[3^2,1,1,1]<br><math>D_{4}^{++}={\overline{Q}}_5</math><br>{{CDD|node|branch3|splitsplit2|node|3|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[3,4,3³]<br><math>F_{4}^{++}={\overline{U}}_5</math><br>{{CDD|node|3|node|4|node|3|node|3|node_c1|3|node_c2}}

| style="background:#ffe0e0;"|[3^[6]]<br><math>A_{5}^{+}={\tilde{A}}_5</math><br>{{CDD|node|split1|nodes|3ab|nodes|split2|node_c1}}

| style="background:#ffe0e0;"|[4,3,3,3^1,1]<br><math>B_{5}^{+}={\tilde{B}}_{5}</math><br>{{CDD|nodea|4a|nodea|3a|nodea|3a|branch|3a|nodea_c1}}

| style="background:#ffe0e0;"|[3^1,1,3,3^1,1]<br><math>D_{5}^{+}={\tilde{D}}_5</math><br>{{CDD|nodea|3a|branch|3a|branch|3a|nodea_c1}}

|-

!7

| style="background:#e0e0ff;"|[3^[5],3,3]<br>A₄⁺⁺⁺<br>{{CDD|branch|3ab|nodes|split2|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[4,3,3^3,1]<br>B₄⁺⁺⁺<br>{{CDD|nodea|4a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2|3a|nodea_c3}}

| style="background:#e0e0ff;"|[4,3,3,4,3,3]<br>C₄⁺⁺⁺<br>{{CDD|node|4|node|3|node|3|node|4|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[3^3,1,1,1]<br>D₄⁺⁺⁺<br>{{CDD|node|branch3|splitsplit2|node|3|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[3,4,3⁴]<br>F₄⁺⁺⁺<br>{{CDD|node|3|node|4|node|3|node|3|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0ffe0;"|[3^[6],3]<br><math>A_{5}^{++}={\overline{P}}_6</math><br>{{CDD|node|split1|nodes|3ab|nodes|split2|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[4,3,3,3^2,1]<br><math>B_{5}^{++}={\overline{S}}_6</math><br>{{CDD|nodea|4a|nodea|3a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2}}

| style="background:#e0ffe0;"|[3^1,1,3,3^2,1]<br><math>D_{5}^{++}={\overline{Q}}_6</math><br>{{CDD|nodea|3a|branch|3a|branch|3a|nodea_c1|3a|nodea_c2}}

|-

!8

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|[3^[6],3,3]<br>A₅⁺⁺⁺<br>{{CDD|node|split1|nodes|3ab|nodes|split2|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[4,3,3,3^3,1]<br>B₅⁺⁺⁺<br>{{CDD|nodea|4a|nodea|3a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2|3a|nodea_c3}}

| style="background:#e0e0ff;"|[3^1,1,3,3^3,1]<br>D₅⁺⁺⁺<br>{{CDD|nodea|3a|branch|3a|branch|3a|nodea_c1|3a|nodea_c2|3a|nodea_c3}}

|- align=center

!Det({{mvar|M_n}})

|5(5&minus;''n'')

|colspan=2|2(5&minus;''n'')

|4(5&minus;''n'')

|5&minus;''n''

|6(6&minus;''n'')

|4(6&minus;''n'')

|}

{| class=wikitable

|+ Some higher extended series

|-

!Finite

!<math>A_6</math>

!<math>B_6</math>

!<math>D_6</math>

!<math>E_6</math>

!<math>A_7</math>

!<math>B_7</math>

!<math>D_7</math>

!<math>E_7</math>

![[En (Lie algebra)|<math>E_8</math>]]

|-

!Rank {{mvar|n}}

![3^[7],3^{''n''&minus;7}]

![4,3³,3^{''n''&minus;6,1}]

![3^1,1,3,3,3^{''n''&minus;6,1}]

![3^{''n''&minus;5,2,2}]

![3^[8],3^{''n''&minus;8}]

![4,3⁴,3^{''n''&minus;7,1}]

![3^1,1,3,3,3,3^{''n''&minus;7,1}]

![3^{''n''&minus;5,3,1}]

!''E_n''=[3^{''n''&minus;4,2,1}]

|-

!3

|

|

|

|

|

|

|

|

|[3^−1,2,1]<br>E₃=A₂A₁<br>{{CDD|nodea|3a|nodea|2|nodeb}}

|-

!4

|

|

|

|[3^−1,2,2]<br>A₂²<br>{{CDD|nodes|3ab|nodes}}

|

|

|

|[3^−1,3,1]<br>A₃A₁<br>{{CDD|nodea|3a|nodea|3a|nodea|2|nodeb}}

|[3^0,2,1]<br>E₄=A₄<br>{{CDD|nodea|3a|nodea|3a|branch}}

|-

!5

|

|[4,3,3,3,3^−1,1]<br>B₄A₁<br>{{CDD|nodea|4a|nodea|3a|nodea|3a|nodea|2|nodeb}}

|[3^1,1,3,3,3^−1,1]<br>D₄A₁<br>{{CDD|nodea|3a|branch|3a|nodea|2|nodeb}}

|[3^0,2,2]<br>A₅<br>{{CDD|nodes|3ab|nodes|split2|node}}

|

|

|

|[3^0,3,1]<br>A₅<br>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch}}

|[3^1,2,1]<br>E₅=D₅<br>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea}}

|-

!6

| style="background:#ffffe0;"|[3⁵]<br>A₆<br>{{CDD|branch|3ab|nodes|3ab|nodes}}

| style="background:#ffffe0;"|[4,3⁴]<br>B₆<br>{{CDD|nodea|4a|nodea|3a|nodea|3a|nodea|3a|branch}}

| style="background:#ffffe0;"|[3^1,1,3,3,3]<br>D₆<br>{{CDD|nodea|3a|branch|3a|nodea|3a|branch}}

| style="background:#ffffe0;"|[3^1,2,2]<br>E₆<br>{{CDD|nodes|3ab|nodes|split2|node|3|node}}

|

|[4,3,3,3,3,3^−1,1]<br>B₅A₁<br>{{CDD|nodea|4a|nodea|3a|nodea|3a|nodea|3a|nodea|2|nodeb}}

|[3^1,1,3,3,3,3^−1,1]<br>D₅A₁<br>{{CDD|nodea|3a|branch|3a|nodea|3a|nodea|2|nodeb}}

|[3^1,3,1]<br>D₆<br>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea}}

|[3^2,2,1]<br>E₆ [[:File:DynkinE6Full.svg|*]]<br>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}

|-

!7

| style="background:#ffe0e0;"|[3^[7]]<br><math>A_{6}^{+}={\tilde{A}}_6</math><br>{{CDD|branch|3ab|nodes|3ab|nodes|split2|node_c1}}

| style="background:#ffe0e0;"|[4,3³,3^1,1]<br><math>B_{6}^{+}={\tilde{B}}_{6}</math><br>{{CDD|nodea|4a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_c1}}

| style="background:#ffe0e0;"|[3^1,1,3,3,3^1,1]<br><math>D_{6}^{+}={\tilde{D}}_6</math><br>{{CDD|nodea|3a|branch|3a|nodea|3a|branch|3a|nodea_c1}}

| style="background:#ffe0e0;"|[3^2,2,2]<br><math>E_{6}^{+}={\tilde{E}}_6</math><br>{{CDD|nodes|3ab|nodes|split2|node|3|node|3|node_c1}}

| style="background:#ffffe0;"|[3⁶]<br>A₇<br>{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes}}

| style="background:#ffffe0;"|[4,3⁵]<br>B₇<br>{{CDD|nodea|4a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch}}

| style="background:#ffffe0;"|[3^1,1,3,3,3,3^0,1]<br>D₇<br>{{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|branch}}

| style="background:#ffffe0;"|[3^2,3,1]<br>E₇ [[:File:DynkinE7Full.svg|*]]<br>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}

|[3^3,2,1]<br>E₇ [[:File:DynkinE7Full.svg|*]]<br>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}

|-

!8

| style="background:#e0ffe0;"|[3^[7],3]<br><math>A_{6}^{++}={\overline{P}}_7</math><br>{{CDD|branch|3ab|nodes|3ab|nodes|split2|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[4,3³,3^2,1]<br><math>B_{6}^{++}={\overline{S}}_7</math><br>{{CDD|nodea|4a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2}}

| style="background:#e0ffe0;"|[3^1,1,3,3,3^2,1]<br><math>D_{6}^{++}={\overline{Q}}_7</math><br>{{CDD|nodea|3a|branch|3a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2}}

| style="background:#e0ffe0;"|[3^3,2,2]<br><math>E_{6}^{++}={\overline{T}}_7</math><br>{{CDD|nodes|3ab|nodes|split2|node|3|node|3|node_c1|3|node_c2}}

| style="background:#ffe0e0;"|[3^[8]]<br><math>A_{7}^{+}={\tilde{A}}_7</math> [[:File:AffineA7.svg|*]]<br>{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node_c1}}

| style="background:#ffe0e0;"|[4,3⁴,3^1,1]<br><math>B_{7}^{+}={\tilde{B}}_{7}</math> [[:File:AffineB7.svg|*]]<br>{{CDD|nodea|4a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_c1}}

| style="background:#ffe0e0;"|[3^1,1,3,3,3,3^1,1]<br><math>D_{7}^{+}={\tilde{D}}_7</math> [[:File:AffineD7.svg|*]]<br>{{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|branch|3a|nodea_c1}}

| style="background:#ffe0e0;"|[3^3,3,1]<br><math>E_{7}^{+}={\tilde{E}}_{7}</math> [[:File:AffineE7.svg|*]]<br>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_c1}}

| style="background:#ffffe0;"|[3^4,2,1]<br>E₈ [[:File:DynkinE8Full.svg|*]]<br>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

|-

!9

| style="background:#e0e0ff;"|[3^[7],3,3]<br>A₆⁺⁺⁺<br>{{CDD|branch|3ab|nodes|3ab|nodes|split2|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[4,3³,3^3,1]<br>B₆⁺⁺⁺<br>{{CDD|nodea|4a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2|3a|nodea_c3}}

| style="background:#e0e0ff;"|[3^1,1,3,3,3^3,1]<br>D₆⁺⁺⁺<br>{{CDD|nodea|3a|branch|3a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2|3a|nodea_c3}}

| style="background:#e0e0ff;"|[3^4,2,2]<br>E₆⁺⁺⁺<br>{{CDD|nodes|3ab|nodes|split2|node|3|node|3|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0ffe0;"|[3^[8],3]<br><math>A_{7}^{++}={\overline{P}}_8</math> [[:File:HyberbolicAffineA7.svg|*]]<br>{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node_c1|3|node_c2}}

| style="background:#e0ffe0;"|[4,3⁴,3^2,1]<br><math>B_{7}^{++}={\overline{S}}_8</math> [[:File:HyberbolicAffineB7.svg|*]]<br>{{CDD|nodea|4a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2}}

| style="background:#e0ffe0;"|[3^1,1,3,3,3,3^2,1]<br><math>D_{7}^{++}={\overline{Q}}_8</math> [[:File:HyberbolicAffineD7.svg|*]]<br>{{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2}}

| style="background:#e0ffe0;"|[3^4,3,1]<br><math>E_{7}^{++}={\overline{T}}_8</math> [[:File:HyberbolicAffineE7.svg|*]]<br>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_c1|3a|nodea_c2}}

| style="background:#ffe0e0;"|[3^5,2,1]<br>E₉=<math>E_{8}^{+}={\tilde{E}}_{8}</math> [[:File:E9-AffineE8.svg|*]]<br>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_c1}}

|-

!10

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|[3^[8],3,3]<br>A₇⁺⁺⁺ [[:File:VeryExtendedAffineA7.svg|*]]<br>{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node_c1|3|node_c2|3|node_c3}}

| style="background:#e0e0ff;"|[4,3⁴,3^3,1]<br>B₇⁺⁺⁺ [[:File:VeryExtendedAffineB7.svg|*]]<br>{{CDD|nodea|4a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2|3a|nodea_c3}}

| style="background:#e0e0ff;"|[3^1,1,3,3,3,3^3,1]<br>D₇⁺⁺⁺ [[:File:VeryExtendedAffineD7.svg|*]]<br>{{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|branch|3a|nodea_c1|3a|nodea_c2|3a|nodea_c3}}

| style="background:#e0e0ff;"|[3^5,3,1]<br>E₇⁺⁺⁺ [[:File:VeryExtendedAffineE7.svg|*]]<br>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_c1|3a|nodea_c2|3a|nodea_c3}}

| style="background:#e0ffe0;"|[3^6,2,1]<br>E₁₀=<math>E_{8}^{++}={\overline{T}}_9</math> [[:File:E10-HyperbolicAffineE8.svg|*]]<br>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_c1|3a|nodea_c2}}

|-

!11

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|

| style="background:#e0e0ff;"|[3^7,2,1]<br>E₁₁=E₈⁺⁺⁺ [[:File:E11-VeryExtendedAffineE8.svg|*]]<br>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_c1|3a|nodea_c2|3a|nodea_c3}}

|- align=center

!Det({{mvar|M_n}})

|7(7&minus;''n'')

|2(7&minus;''n'')

|4(7&minus;''n'')

|3(7&minus;''n'')

|8(8&minus;''n'')

|2(8&minus;''n'')

|4(8&minus;''n'')

|2(8&minus;''n'')

|9&minus;''n''

|}

== Geometric folding ==

{| class="wikitable" style="float:right;"

|+ Finite and affine foldings<ref>''John Crisp'', '[[Injective map]]s between [[Artin group]]s', in Down under group theory, Proceedings of the Special Year on Geometric Group Theory, (Australian National University, Canberra, Australia, 1996), [http://math.u-bourgogne.fr/IMB/crisp/injart.ps Postscript] {{webarchive|url=https://web.archive.org/web/20051016143215/http://math.u-bourgogne.fr/IMB/crisp/injart.ps |date=2005-10-16 }}, pp. 13-14, and [https://books.google.com/books?id=RpgZuF7do8kC&q=folding+description&pg=PP1 googlebook, Geometric group theory down under, p. 131]</ref>

|-

!colspan=4|φ_A : A_Γ &rarr; A_Γ' for finite types

|- valign=top

!Γ

!Γ'

!Folding description

!Coxeter–Dynkin diagrams

|- align=center valign=top

|I₂([[Coxeter number|h]])

|Γ(h)

|Dihedral folding

|rowspan=9|[[File:Geometric folding Coxeter graphs.png|201px]]

|- align=center

|rowspan=2|B_n

|A_2n

|(I,s_n)

|- align=center

|D_n+1, A_2n-1

|(A₃,±ε)

|- align=center

|F₄

|E₆

|(A₃,±ε)

|- align=center

|H₄

|E₈

|rowspan=3|(A₄,±ε)

|- align=center

|H₃

|D₆

|- align=center

|H₂

|A₄

|- align=center

|rowspan=2|G₂

|A₅

|(A₅,±ε)

|- align=center

|D₄

|(D₄,±ε)

|-

!colspan=4|φ: A_Γ⁺ &rarr; A_Γ'⁺ for affine types

|- align=center valign=top

|<math>{\tilde{A}}_{n-1}</math>

|<math>{\tilde{A}}_{kn-1}</math>

|Locally trivial

|rowspan=14|[[File:Geometric folding Coxeter graphs affine.png|212px]]

|- align=center

|rowspan=2|<math>{\tilde{B}}_{n}</math>

|<math>{\tilde{D}}_{2n+1}</math>

|(I,s_n)

|- align=center

|<math>{\tilde{D}}_{n+1}</math>, <math>{\tilde{D}}_{2n}</math>

|(A₃,±ε)

|- align=center

|rowspan=2|<math>{\tilde{C}}_{n}</math>

|<math>{\tilde{B}}_{n+1}</math>, <math>{\tilde{C}}_{2n}</math>

|(A₃,±ε)

|- align=center

|<math>{\tilde{C}}_{2n+1}</math>

|(I,s_n)

|- align=center

|rowspan=3|<math>{\tilde{C}}_{n}</math>

|<math>{\tilde{A}}_{2n+1}</math>

|(I,s_n) & (I,s₀)

|- align=center

|<math>{\tilde{A}}_{2n}</math>

|(A₃,ε) & (I,s₀)

|- align=center

|<math>{\tilde{A}}_{2n-1}</math>

|(A₃,ε) & (A₃,ε')

|- align=center

|<math>{\tilde{C}}_{n}</math>

|<math>{\tilde{D}}_{n+2}</math>

|(A₃,&minus;ε) & (A₃,&minus;ε')

|- align=center

|<math>{\tilde{C}}_{2}</math>

|<math>{\tilde{D}}_{5}</math>

|(I,s₁)

|- align=center

|<math>{\tilde{F}}_{4}</math>

|<math>{\tilde{E}}_{6}</math>, <math>{\tilde{E}}_{7}</math>

|(A₃,±ε)

|- align=center

|rowspan=3|<math>{\tilde{G}}_{2}</math>

|<math>{\tilde{D}}_{6}</math>, <math>{\tilde{E}}_{7}</math>

|(A₅,±ε)

|- align=center

|<math>{\tilde{B}}_{3}</math>, <math>{\tilde{F}}_{4}</math>

|(B₃,±ε)

|- align=center

|<math>{\tilde{D}}_{4}</math>, <math>{\tilde{E}}_{6}</math>

|(D₄,±ε)

|}

{{See also|Dynkin diagram#Folding}}

A (simply-laced) Coxeter–Dynkin diagram (finite, [[#Affine Coxeter groups|affine]], or hyperbolic) that has a symmetry (satisfying one condition, below) can be quotiented by the symmetry, yielding a new, generally multiply laced diagram, with the process called "folding".<ref>{{cite journal | title = Generalized Dynkin diagrams and root systems and their folding | journal = Topological Field Theory | citeseerx = 10.1.1.54.3122 | first = Jean-Bernard | last = Zuber | pages = 28–30 | bibcode = 1998tftp.conf..453Z | year = 1998 | arxiv = hep-th/9707046 }}</ref><ref>{{Cite journal| arxiv = 1110.5228 | title = Affine extensions of non-crystallographic Coxeter groups induced by projection | journal = Journal of Mathematical Physics | volume = 54 | issue = 9 | pages = 093508 | first1 = Pierre-Philippe | last1 = Dechant | first2 = Celine | last2 = Boehm | first3 = Reidun | last3 = Twarock | year = 2013 | doi = 10.1063/1.4820441 | bibcode = 2013JMP....54i3508D | s2cid = 59469917 }}</ref>

For example, in D₄ folding to G₂, the edge in G₂ points from the class of the 3 outer nodes (valence 1), to the class of the central node (valence 3). And E₈ folds into 2 copies of H₄, the second copy scaled by [[Golden ratio|&tau;]].<ref>{{cite journal | doi=10.1007/s00006-016-0675-9 | title=The E ₈ Geometry from a Clifford Perspective | date=2017 | last1=Dechant | first1=Pierre-Philippe | journal=Advances in Applied Clifford Algebras | volume=27 | pages=397–421 | doi-access=free | arxiv=1603.04805 }}</ref>

Geometrically this corresponds to [[orthogonal projection]]s of [[uniform polytope]]s and tessellations. Notably, any finite simply-laced Coxeter–Dynkin diagram can be folded to I₂(''h''), where ''h'' is the [[Coxeter number]], which corresponds geometrically to a projection to the [[Coxeter plane]].

{| class=wikitable

|-

|[[File:Geometric folding Coxeter graphs hyperbolic.png]]<br>A few hyperbolic foldings

|}

{{Clear}}

== Complex reflections ==

Coxeter–Dynkin diagrams have been extended to [[Complex vector space|complex space]], Cⁿ where nodes are [[unitary reflection]]s of period greater than 2. Nodes are labeled by an index, assumed to be 2 for ordinary real reflection if suppressed. Coxeter writes the complex group, p[q]r, as diagram {{CDD|pnode|3|q|3|rnode}}.<ref>Coxeter, ''Complex Regular Polytopes'', second edition, (1991)</ref>

A 1-dimensional ''regular [[complex polytope]]'' in <math>\mathbb{C}^1</math> is represented as {{CDD|pnode_1}}, having ''p'' vertices. Its real representation is a [[regular polygon]], {''p''}. Its symmetry is _''p''[] or {{CDD|pnode}}, order ''p''. A [[unitary operator]] generator for {{CDD|pnode}} is seen as a rotation in <math>\mathbb{R}^2</math> by 2{{pi}}/''p'' radians [[counter clockwise]], and a {{CDD|pnode_1}} edge is created by sequential applications of a single unitary reflection. A unitary reflection generator for a 1-polytope with ''p'' vertices is {{math| ''e''^{2{{pi}}''i''/''p''} {{=}} cos(2{{pi}}/''p'') + ''i'' sin(2{{pi}}/''p'')}}. When ''p'' = 2, the generator is ''e''^{{pi}}''i'' = −1, the same as a [[point reflection]] in the real plane.

In a higher polytope, _''p''{} or {{CDD|pnode_1}} represents a ''p''-edge element, with a 2-edge, {} or {{CDD|node_1}}, representing an ordinary real edge between two vertices.

{| class=wikitable width=640

|+ Regular complex 1-polytopes

|[[File:Complex 1-topes as k-edges.png|640px]]<BR>Complex 1-polytopes, {{CDD|pnode_1}}, represented in the [[Argand plane]] as regular polygons for ''p'' = 2, 3, 4, 5, and 6, with black vertices. The centroid of the ''p'' vertices is shown seen in red. The sides of the polygons represent one application of the symmetry generator, mapping each vertex to the next counterclockwise copy. These polygonal sides are not edge elements of the polytope, as a complex 1-polytope can have no edges (it often ''is'' a complex edge) and only contains vertex elements.

|}

{| class=wikitable width=100 align=right

|- valign=top

|[[File:Rank2_shephard_subgroups.png|240px]]<BR>12 irreducible Shephard groups with their subgroup index relations.<ref>Coxeter, Complex Regular Polytopes, p. 177, Table III</ref> Subgroups index 2 relate by removing a real reflection:<BR>_''p''[2''q'']₂ &rarr; _''p''[''q'']_''p'', index 2.<BR>_''p''[4]_''q'' &rarr; _''p''[''q'']_''p'', index ''q''.

|[[File:Rank2_shephard_subgroups2_series.png|203px]]<BR>_''p''[4]₂ subgroups: p=2,3,4...<BR>_''p''[4]₂ &rarr; [''p''], index ''p''<BR>_''p''[4]₂ &rarr; _''p''[]×_''p''[], index 2

|}

A [[regular complex polytope|regular complex polygon]] in <math>\mathbb{C}^2</math>, has the form _''p''{''q''}_''r'' or Coxeter diagram {{CDD|pnode_1|3|q|3|rnode}}. The symmetry group of a regular complex polygon {{CDD|pnode|3|q|3|rnode}} is not called a [[Coxeter group]], but instead a [[Shephard group]], a type of [[Complex reflection group]]. The order of _''p''[''q'']_''r'' is <math>8/q \cdot (1/p+2/q+1/r-1)^{-2}</math>.<ref>''Unitary Reflection Groups'', p.87</ref>

The rank 2 Shephard groups are: ₂[''q'']₂, _''p''[4]₂, ₃[3]₃, ₃[6]₂, ₃[4]₃, ₄[3]₄, ₃[8]₂, ₄[6]₂, ₄[4]₃, ₃[5]₃, ₅[3]₅, ₃[10]₂, ₅[6]₂, and ₅[4]₃ or {{CDD|node|q|node}}, {{CDD|pnode|4|node}}, {{CDD|3node|3|3node}}, {{CDD|3node|6|node}}, {{CDD|3node|4|3node}}, {{CDD|4node|3|4node}}, {{CDD|3node|8|node}}, {{CDD|4node|6|node}}, {{CDD|4node|4|3node}}, {{CDD|3node|5|3node}}, {{CDD|5node|3|5node}}, {{CDD|3node|10|node}}, {{CDD|5node|6|node}}, {{CDD|5node|4|3node}} of order 2''q'', 2''p''², 24, 48, 72, 96, 144, 192, 288, 360, 600, 1200, and 1800 respectively.

The symmetry group _''p''₁[''q'']_''p''₂ is represented by 2 generators R₁, R₂, where:

:{{math|1=R₁^''p''₁ = R₂^''p''₂ = I}}.

If ''q'' is even, (R₂R₁)^''q''/2 = (R₁R₂)^''q''/2. If ''q'' is odd, (R₂R₁)^(q-1)/2R₂ = (R₁R₂)^(''q''-1)/2R₁. When ''q'' is odd, ''p''₁=''p''₂.

The <math>\mathbb{C}^3</math> group {{CDD|node|psplit1|branch}} or [1 1 1]^p is defined by 3 period 2 unitary reflections {R₁, R₂, R₃}:

:R₁² = R₁² = R₃² = (R₁R₂)³ = (R₂R₃)³ = (R₃R₁)³ = (R₁R₂R₃R₁)^''p'' = 1.

The period ''p'' can be seen as a [[double rotation]] in real <math>\mathbb{R}^4</math>.

A similar <math>\mathbb{C}^3</math> group {{CDD|node|antipsplit1|branch}} or [1 1 1]^(p) is defined by 3 period 2 unitary reflections {R₁, R₂, R₃}:

:R₁² = R₁² = R₃² = (R₁R₂)³ = (R₂R₃)³ = (R₃R₁)³ = (R₁R₂R₃R₂)^''p'' = 1.

== See also ==

* [[Coxeter group]]

* [[Schwarz triangle]]

* [[Goursat tetrahedron]]

* [[Dynkin diagram]]

* [[Uniform polytope]]

** [[Wythoff constructionsymbol]]

** [[Uniform polyhedron]]

** [[List of uniform polyhedra]]

** [[List of uniform planar tilings]]

** [[Uniform 4-polytope]]

** [[Convex uniform honeycomb]]

** [[Convex uniform honeycombs in hyperbolic space]]

* [[Wythoff construction]] and [[Wythoff symbol]]

== References ==

{{Reflist}}

* [[Harold Scott MacDonald Coxeter|Coxeter]] ''Regular Polytopes'' (1963), Macmillian Company

{{Reflist|group=note}}

** ''Regular Polytopes'', Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Section 11.3 Representation by graphs)

{{notelist}}

* [[Harold Scott MacDonald Coxeter|Coxeter]] ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, ISBN 99-35678 (Chapter 3: Wythoff's Construction for Uniform Polytopes)

== Further reading ==

* [[James E. Humphreys]], ''Reflection Groups and Coxeter Groups'', Cambridge studies in advanced mathematics, 29 (1990)

* '''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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html], Googlebooks [https://books.google.com/books?id=fUm5Mwfx8rAC&dq=Kaleidoscopes%20Coxeter&pg=PP1]

** (Paper 17) [[Harold Scott MacDonald Coxeter|Coxeter]], ''The Evolution of Coxeter-Dynkin diagrams'', [Nieuw Archief voor Wiskunde 9 (1991) 233-248]

* [[Harold Scott MacDonald Coxeter|Coxeter]], ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, {{ISBN|978-0-486-40919-1}} (Chapter 3: Wythoff's Construction for Uniform Polytopes)

* [[Harold Scott MacDonald Coxeter|Coxeter]], ''Regular Polytopes'' (1963), Macmillan Company

** [[Regular Polytopes (book)|''Regular Polytopes'']], Third edition, (1973), Dover edition, {{ISBN|0-486-61480-8}} (Chapter 5: The Kaleidoscope, and Section 11.3 Representation by graphs)

* H.S.M. Coxeter and W. O. J. Moser, ''Generators and Relations for Discrete Groups'' 4th ed, Springer-Verlag, New York, 1980

* [[Norman Johnson (mathematician)|Norman Johnson]], ''Geometries and Transformations'', Chapters 11,12,13, preprint 2011

* [[Norman Johnson (mathematician)|N. W. Johnson]], [[Ruth Kellerhals|R. Kellerhals]], J. G. Ratcliffe, S. T. Tschantz, ''The size of a hyperbolic Coxeter simplex'', Transformation Groups, 1999, Volume 4, Issue 4, pp. 329–353 [https://link.springer.com/article/10.1007%2FBF01238563] [https://web.archive.org/web/20140223225217/http://homeweb1.unifr.ch/kellerha/pub/TGarticle.pdf]

* Norman W. Johnson and Asia Ivic Weiss [https://cms.math.ca/cjm/v51/weisscox8.pdf Quadratic Integers and Coxeter Groups] {{Webarchive|url=https://web.archive.org/web/20230326121528/http://cms.math.ca/cjm/v51/weisscox8.pdf |date=2023-03-26 }} [[PDF]] Can. J. Math. Vol. 51 (6), 1999, pp.&nbsp;1307–1336

== External links ==

{{Commons category|Coxeter-Dynkin diagrams}}

* [http://mathworld.wolfram.com/Coxeter-DynkinDiagram.html MathWorld]

* {{MathWorld |urlname=Coxeter-DynkinDiagram |title=Coxeter–Dynkin diagram}}

* [https://ecommons.cornell.edu/handle/1813/17339 October 1978 discussion on the history of the Coxeter diagrams] by Coxeter and Dynkin in [[Toronto]], [[Canada]]; Eugene Dynkin Collection of Mathematics Interviews, [[Cornell University Library]].

{{Authority control}}

{{geometry-stub}}

{{DEFAULTSORT:Coxeter-Dynkin diagram}}

[[Category:polytopes]]

[[Category:Coxeter groups]]

[[Category:Polytope notation systems]]