Ogled vira Coxeter-Dinkinov diagram

[[Slika:Finite coxeter.svg|sličica|desno|300px|Coxeter-Dinkinovi diagrami za osnovne končne Coxeterjeve grupe.]]
[[Slika:Affine_coxeter.svg|sličica|desno|300px|Coxeter-Dinkinovi diagrami za osnovne afine Coxeterjeve grupe.]]
'''Coxeter-Dinkinov diagram''' (tudi Coxeterjev diagram ali Coxeterjev graf) je [[Graf (matematika)|graf]], ki ima s številkami označene [[stranica|stranice]] (imenujejo se '''veje''') s katerimi se prikaže prostorske odnose med zbirko [[zrcalo|zrcal]] oziroma odbojnih [[hiperravnina|hiperravnin]]. Opisujejo [[kalejdoskop]]sko konstrukcijo: vsak vozel grafa predstavlja ogledalo (v domeni [[faceta (matematika)|facete]]). Oznaka pri vsaki veji določa stopnjo [[diedrski kot|diedrskega kota]] dveh ogledal (v domeni [[greben (matematika)|grebena]]). Neoznačene veje pomenijo red 3.

Vsak diagram predstavlja [[Coxeterjeva grupa|Coxeterjevo grupo]] in tudi Coxeterjeve grupe so razvrščene po pripadajočih diagramih.

Podobni so [[Dinkinov diagram|Dinkinovi diagrami]]. Ti se od Coxeterjevih diagramov razlikujejo samo v tem, da so v Dinkinovih diagramih veje, ki imajo oznako 4 ali več, [[usmerjeni graf|usmerjene]]. Coxeterjevi diagrami so neusmerjeni. Razen tega morajo Dinkinovi diagrami zadoščati še dodatni [[kristalografski omejitveni izrek|kristalografski]] omejitvi, ki zahteva, da so dovoljene veje samo 2, 3, 4 in 6.

== Opis diagramov ==

Veje Coxeter-Dinkinovih diagramov so označene z [[racionalno število|racionalnimi števili]] <math> p \,</math>, kar predstavlja diedrski kot v velikosti 180°/''p''. Če je p enako 2, je kot 90° in se lahko v diagramu veja izpusti. Kadar je veja neoznačena, to pomeni, da zanjo velja <math> p = 3 \,</math>, kar pomeni kot 60º. Vzporedni zrcali imata oznako "∞"

== Geometrijska ponazoritev ==

Coxeter-Dinkinov diagram se lahko prikaže kot domena ogledal. Ogledalo v tem primeru predstavlja [[hiperravnina|hiperravnino]] s pomočjo sfernega ali [[evklidski prostor|evklidskega]] ali hiperboličnega prostora z dano razsežnostjo.

Takšna ponazoritev kaže osnovne domene za dvo in trirazsežne [[evklidska grupa|evklidske grupe]] in dvorazsežne [[sferna grupa|sferne grupe]].

{| class=wikitable width=720
|colspan=2|[[Slika:Coxeter-dynkin plane groups_sl.png|720px]]<BR>Coxeterjeve grupe v ravnini s pripadajočimi diagrami. Ogledala domen so označena kot veje ''m''1, ''m''2  itd. Oglišča so obarvana v skladu z zaporedjem odboja. Prizmatske grupe <math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math> so prikazane kot podvojitev <math>{\tilde{C}}_2</math>, toda nastale bi lahko tudi kot pravokotne domene iz podvojitev <math>{\tilde{G}}_2</math> trikotnikov.  <math>{\tilde{A}}_2</math> je podvojitev  <math>{\tilde{G}}_2</math> trikotnika.
|- valign=top
|[[Slika:Coxeter-Dynkin 3-space groups_sl.png|360px]]<BR>Coxeterjeve grupe v trirazsežnem  prostoru z diagrami. Zrcala (stranice trikotnika) so označena  z nasprotnim ogliščem 0..3. Veje grafa so obarvane z zaporedjem odboja.<BR><math>{\tilde{C}}_3</math> izpolni 1/48 kocke. <math>{\tilde{B}}_3</math> izpolni 1/24 kocke. <math>{\tilde{A}}_3</math> izpolni1/12 kocke.
|[[Slika:Coxeter-Dynkin sphere groups_sl.png|360px]]<BR>Coxeterjeve grupe na sferi s pripadajočimi diagrami. Osnovna domena je prikazana v rumeni barvi. Oglišča domen (in veje grafa) so obarvane v zaporedju zrcaljenja.
|}

== Uporaba v uniformnih politopih ==

Coxeter-Dinkinovi diagrami lahko opišejo skoraj vse vrste [[uniformni politop|uniformnih politopov]] in [[uniformna teselacija|uniformnih teselacij]]

== Cartanove matrike ==

Vsakemu Coxeterjevemu diagramu pripada odgovarjajoča [[Cartanova matrika]]. Vse Cartanove matrike [[Coxeterjeva grupa|Coxeterjevih grup]] so simetrične. Elementi Cartanove matrike so a<sub>i,j</sub> = a<sub>j,i</sub> = -2*cos(π/p),
kjer je:
* <math> p \,</math> red veje med pari zrcal.

[[Determinanta]] Cartanove matrike določa ali je grupa končna (pozitivna), afina (nič) ali hiperbolična (negativna). Hiperbolična grupa ja kompaktna, če so vse njene podgrupe končne.

{| class=wikitable  width=720
|+ rang 2 Coxeterjeve grupe
|-
!rowspan=2|red<BR>simetrije<BR>''p''
!rowspan=2|ime<BR>grupe
!rowspan=2|Coxeterjev diagram
!colspan=2|Cartanova matrika
|-
!<math>\left [\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}\right ]</math>
!determinanta<P>(4-a<sub>21</sub>*a<sub>12</sub>)
|- align=center
!colspan=7|[[Dinkinov diagram#Končni Dinkinovi diagrami|končne]] (determinanta>0)
|- align=center
!2
!I<sub>2</sub>(2) = A<sub>1</sub>xA<sub>1</sub>
|{{CDD|node|2|node}}
|<math>\left [\begin{smallmatrix}2&0\\0&2\end{smallmatrix}\right ]</math>
|4
|- align=center
!3
!I<sub>2</sub>(3) = A<sub>2</sub>
|{{CDD|node|3|node}}
|<math>\left [\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}\right ]</math>
|3
|- align=center
!4
!I<sub>2</sub>(4) = BC<sub>2</sub>
|{{CDD||node|4|node}}
|<math>\left [\begin{smallmatrix}2&-\sqrt{2}\\-\sqrt{2}&2\end{smallmatrix}\right ]</math>
|2
|- align=center
!5
!I<sub>2</sub>(5) = H<sub>2</sub>
|{{CDD|node|5|node}}
|<math>\left [\begin{smallmatrix}2&-\phi\\-\phi&2\end{smallmatrix}\right ]</math>
|<math>4\sin^2(\pi/5)</math><P>
=<math>(5-\sqrt{5})/2</math><P>
~1,38196601125
|- align=center
!6
!I<sub>2</sub>(6) = G<sub>2</sub>
|{{CDD|node|6|node}}
|<math>\left [\begin{smallmatrix}2&-\sqrt{3}\\-\sqrt{3}&2\end{smallmatrix}\right ]</math>
|1
|- align=center
!8
!I<sub>2</sub>(8)
|{{CDD|node|8|node}}
|<math>\left [\begin{smallmatrix}2&-2\cos(\pi/8)\\-2\cos(\pi/8)&2\end{smallmatrix}\right ]</math>
|<math>2-\sqrt{2}</math><BR>
~0,58578643763
|- align=center
!10
!I<sub>2</sub>(10)
|{{CDD|node|10|node}}
|<math>\left [\begin{smallmatrix}2&-2\cos(\pi/10)\\-2\cos(\pi/10)&2\end{smallmatrix}\right ]</math>
|<math>4\sin^2(\pi/10)</math><P>
=<math>(3-\sqrt{5})/2</math><P>
~0,38196601125
|- align=center
!12
!I<sub>2</sub>(12)
|{{CDD|node|12|node}}
|<math>\left [\begin{smallmatrix}2&-2\cos(\pi/12)\\-2\cos(\pi/12)&2\end{smallmatrix}\right ]</math>
|<math>2-\sqrt{3}</math><BR>
~0,26794919243
|- align=center
!''p''
!I<sub>2</sub>(p)
|{{CDD|node|p|node}}
|<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|[[Dinkinov diagram#Afini Dinkinovi diagrami|afine]] (determinanta=0)
|- align=center
!∞
!I<sub>2</sub>(∞) = <math>{\tilde{I}}_1</math> = <math>{\tilde{A}}_1</math>
|{{CDD|node|infin|node}}
|<math>\left [\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}\right ]</math>
|0
|}

== Končne Coxeterjeve grupe ==

{| class="wikitable"
|+ Povezani končni  Dinkinovi grafi  za  rang 1 do 9
|- align=center
!rowspan=2|rang
!colspan=3|[[enostavna Liejeva grupa|enostavne Lijeve grupe]]
!colspan=2|[[posebna liejeva grupa|posebne Liejeve grupe]]
!colspan=2|&nbsp;
|-
!height=35|<math>{A}_{1+}</math>
!<math>{BC}_{2+}</math>
!<math>{D}_{2+}</math>
![[En (Liejeva algebra)|<math>{E}_{3-8}</math>]]
![[F4 (matematika)|<math>{F}_{4}</math>]] / [[G2 (matematika)|<math>{G}_{2}</math>]]
!<math>{H}_{2-4}</math>
!<math>{I}_{2}(p)</math>
|- align=center
!1
|A<sub>1</sub>=[]<P>{{CDD|node}}
|&nbsp;
|&nbsp;
|&nbsp;
|&nbsp;
|- align=center
!2
|A<sub>2</sub>=[3]<P>{{CDD|node|3|node}}
|BC<sub>2</sub>=[4]<P>{{CDD|node|4|node}}
|D<sub>2</sub>=A<sub>1</sub>xA<sub>1</sub><P>{{CDD|nodes}}
|&nbsp;
|G<sub>2</sub>=[6]<P>{{CDD|node|6|node}}
|H<sub>2</sub>=[6]<P>{{CDD|node|5|node}}
|I<sub>2</sub>[p]<P>{{CDD|node|p|node}}
|- align=center
!3
|A<sub>3</sub>=[3<sup>2</sup>]<P>{{CDD|node|3|node|3|node}}
|BC<sub>3</sub>=[3,4]<P>{{CDD|node|4|node|3|node}}
|D<sub>3</sub>=A<sub>3</sub><P>{{CDD|nodes|split2|node}}
|E<sub>3</sub>=A<sub>2</sub>xA<sub>1</sub><P>{{CDD|nodea|3a|nodea}}&nbsp;{{CDD|nodeb}}
|&nbsp;
|H<sub>3</sub>&nbsp;<P>{{CDD|node|5|node|3|node}}
|- align=center
!4
|A<sub>4</sub>=[3<sup>3</sup>]<P>{{CDD|node|3|node|3|node|3|node}}
|BC<sub>4</sub>=[3<sup>2</sup>,4]<P>{{CDD|node|4|node|3|node|3|node}}
|D<sub>4</sub>=[3<sup>1,1,1</sup>]<P>{{CDD|nodes|split2|node|3|node}}
|E<sub>4</sub>=A<sub>4</sub><P>{{CDD|nodea|3a|nodea|3a|branch}}
|F<sub>4</sub><P>{{CDD|node|3|node|4|node|3|node}}
|H<sub>4</sub>&nbsp;<P>{{CDD|node|5|node|3|node|3|node}}
|- align=center
!5
|A<sub>5</sub>=[3<sup>4</sup>]<P>{{CDD|node|3|node|3|node|3|node|3|node}}
|BC<sub>5</sub>=[3<sup>3</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|3|node}}
|D<sub>5</sub>=[3<sup>2,1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node}}
|E<sub>5</sub>=D<sub>5</sub><P>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea}}
|rowspan=6|&nbsp;
|- align=center
!6
|A<sub>6</sub>=[3<sup>5</sup>]<P>{{CDD|node|3|node|3|node|3|node|3|node|3|node}}
|BC<sub>6</sub>=[3<sup>4</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|3|node|3|node}}
|D<sub>6</sub>=[3<sup>3,1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node}}
|E<sub>6</sub>=[3<sup>2,2,1</sup>]<P>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|- align=center
!7
|A<sub>7</sub>=[3<sup>6</sup>]<P>{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|BC<sub>7</sub>=[3<sup>5</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node}}
|D<sub>7</sub>=[3<sup>4,1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node}}
|E<sub>7</sub>=[3<sup>3,2,1</sup>]<P>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
|- align=center
!8
|A<sub>8</sub>=[3<sup>7</sup>]<P>{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|BC<sub>8</sub>=[3<sup>6</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|D<sub>8</sub>=[3<sup>5,1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node}}
|E<sub>8</sub>=[3<sup>4,2,1</sup>]<P>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|- align=center
!9
|A<sub>9</sub>=[3<sup>8</sup>]<P>{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|BC<sub>9</sub>=[3<sup>7</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|D<sub>9</sub>=[3<sup>6,1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|rowspan|&nbsp;
|- align=center
!10+
|..
|..
|..
|..
|}
== Afine Coxeterjeve grupe ==

{| class="wikitable"
|+ Afini Dinkinovi grafi  od 2 do10 vozlov
!rang
!<math>{\tilde{A}}_{1+}</math> (P<sub>2+</sub>)
!<math>{\tilde{B}}_{3+}</math> (S<sub>4+</sub>)
!<math>{\tilde{C}}_{1+}</math> (R<sub>2+</sub>)
!<math>{\tilde{D}}_{4+}</math> (Q<sub>5+</sub>)
! <math>{\tilde{E}}_{n}</math> (T<sub>n+1</sub>) / <math>{\tilde{F}}_{4}</math> (U<sub>5</sub>) / <math>{\tilde{G}}_{2}</math> (V<sub>3</sub>)
|- align=center valign=top
!2
|<math>{\tilde{A}}_{1}</math>=[∞]<P>{{CDD|node|infin|node}}
|rowspan=2|&nbsp;
|<math>{\tilde{C}}_{1}</math>=[∞]<P>{{CDD|node|infin|node}}
|rowspan=3|&nbsp;
|&nbsp;
|- align=center valign=top
!3
|<math>{\tilde{A}}_{2}</math>=[3<sup>[3]</sup>]<P>{{CDD|branch|split2|node}}
|<math>{\tilde{C}}_{2}</math>=[4,4]<P>{{CDD|node|4|node|4|node}}
|<math>{\tilde{G}}_{2}</math>=[6,3]<P>{{CDD|node|6|node|3|node}}
|- align=center valign=top
!4
|<math>{\tilde{A}}_{3}</math>=[3<sup>[4]</sup>]<P>{{CDD|node|split1|nodes|split2|node}}
|<math>{\tilde{B}}_{3}</math>=[4,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|4|node}}
|<math>{\tilde{C}}_{3}</math>=[4,3,4]<P>{{CDD|node|4|node|3|node|4|node}}
|&nbsp;
|- align=center valign=top
!5
|<math>{\tilde{A}}_{4}</math>=[3<sup>[5]</sup>]<P>{{CDD|branch|3ab|nodes|split2|node}}
|<math>{\tilde{B}}_{4}</math>=[4,3,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|4|node}}
|<math>{\tilde{C}}_{4}</math>=[4,3<sup>2</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|4|node}}
|<math>{\tilde{D}}_{4}</math>=[3<sup>1,1,1,1</sup></sup>]<P>{{CDD|nodes|split2|node|split1|nodes}}
|<math>{\tilde{F}}_{4}</math>=[3,4,3,3]<P>{{CDD|node|3|node|4|node|3|node|3|node}}
|- align=center valign=top
!6
|<math>{\tilde{A}}_{5}</math>=[3<sup>[6]</sup>]<P>{{CDD|node|split1|nodes|3ab|nodes|split2|node}}
|<math>{\tilde{B}}_{5}</math>=[4,3<sup>2</sup>,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|4|node}}
|<math>{\tilde{C}}_{5}</math>=[4,3<sup>3</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|3|node|4|node}}
|<math>{\tilde{D}}_{5}</math>=[3<sup>1,1</sup>,3,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|split1|nodes}}
|&nbsp;
|- align=center valign=top
!7
|<math>{\tilde{A}}_{6}</math>=[3<sup>[7]</sup>]<P>{{CDD|branch|3ab|nodes|3ab|nodes|split2|node}}
|<math>{\tilde{B}}_{6}</math>=[4,3<sup>3</sup>,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node|4|node}}
|<math>{\tilde{C}}_{6}</math>=[4,3<sup>4</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|3|node|3|node|4|node}}
|<math>{\tilde{D}}_{6}</math>=[3<sup>1,1</sup>,3<sup>2</sup>,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|split1|nodes}}
|<math>{\tilde{E}}_{6}</math>=[3<sup>2,2,2</sup>]<P>{{CDD|nodes|3ab|nodes|split2|node|3|node|3|node}}
|- align=center valign=top
!8
|<math>{\tilde{A}}_{7}</math>=[3<sup>[8]</sup>]<P>{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}
|<math>{\tilde{B}}_{7}</math>=[4,3<sup>4</sup>,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|4|node}}
|<math>{\tilde{C}}_{7}</math>=[4,3<sup>5</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|4|node}}
|<math>{\tilde{D}}_{7}</math>=[3<sup>1,1</sup>,3<sup>3</sup>,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node|split1|nodes}}
|<math>{\tilde{E}}_{7}</math>=[3<sup>3,3,1</sup>]<P>{{CDD|nodes|3ab|nodes|3ab|nodes|split2|node|3|node}}
|- align=center valign=top
!9
|<math>{\tilde{A}}_{8}</math>=[3<sup>[9]</sup>]<P>{{CDD|branch|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}
|<math>{\tilde{B}}_{8}</math>=[4,3<sup>5</sup>,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|4|node}}
|<math>{\tilde{C}}_{8}</math>=[4,3<sup>6</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
|<math>{\tilde{D}}_{8}</math>=[3<sup>1,1</sup>,3<sup>4</sup>,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|split1|nodes}}
|<math>{\tilde{E}}_{8}</math>=[3<sup>5,2,1</sup>]<P>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|- align=center valign=top
!10
|<math>{\tilde{A}}_{9}</math>=[3<sup>[10]</sup>]<P>{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}
|<math>{\tilde{B}}_{9}</math>=[4,3<sup>6</sup>,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
|<math>{\tilde{C}}_{9}</math>=[4,3<sup>7</sup>,4]<P>{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
|<math>{\tilde{D}}_{9}</math>=[3<sup>1,1</sup>,3<sup>5</sup>,3<sup>1,1</sup>]<P>{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}}
|- align=center valign=top
!11
|...
|...
|...
|...
|}

== Hiperbolične Coxeterjeve grupe ==

=== Kompaktne ===

==== Rang 3 ====

{| class="wikitable"
|+ Kompaktne hiperbolične Coxeterjeve grupe
|-
! valign=top|linearne
! valign=top|ciklične
|-
| valign=top|[[Seznam pravilnih politopov#Hiperbolično tlakovanje|'''∞:''']] [p,q], 2(p+q)<pq<BR>
{{CDD|node|7|node|3|node}}<BR>
{{CDD|node|8|node|3|node}}<BR>
{{CDD|node|9|node|3|node}}<BR>
...<BR>
{{CDD|node|5|node|4|node}}<BR>
{{CDD|node|6|node|4|node}}<BR>
...<BR>
{{CDD|node|5|node|5|node}}<BR>
{{CDD|node|6|node|5|node}}<BR>
...<BR>
| valign=top|∞ [(p,q,r)], p+q+r>9
{| 
|- valign=top
| 
{{CDD|3|node|4|node|3|node|3}}<BR>
{{CDD|3|node|4|node|4|node|3}}<BR>
{{CDD|3|node|4|node|4|node|4}}<BR>
|
{{CDD|3|node|5|node|3|node|3}}<BR>
{{CDD|3|node|5|node|4|node|3}}<BR>
{{CDD|3|node|5|node|4|node|4}}<BR>
{{CDD|3|node|5|node|5|node|3}}<BR>
{{CDD|3|node|5|node|5|node|4}}<BR>
{{CDD|3|node|5|node|5|node|5}}
|
{{CDD|3|node|6|node|3|node|3}}<BR>
{{CDD|3|node|6|node|4|node|3}}<BR>
{{CDD|3|node|6|node|4|node|4}}<BR>
{{CDD|3|node|6|node|5|node|3}}<BR>
{{CDD|3|node|6|node|5|node|4}}<BR>
{{CDD|3|node|6|node|5|node|5}}<BR>
{{CDD|3|node|6|node|6|node|3}}<BR>
{{CDD|3|node|6|node|6|node|4}}<BR>
{{CDD|3|node|6|node|6|node|5}}<BR>
{{CDD|3|node|6|node|6|node|6}}
|
{{CDD|3|node|7|node|3|node|3}}<BR>
...
|}
|}
==== Rangi od 4 do 5 ====

{| class="wikitable"
|+ Kompaktne hiperbolične Coxeterjeve grupe
|-
! valign=top|razsežnost<BR>'''H<sup>d</sup>'''
! valign=top|rang
! valign=top|skupno število
! valign=top|linearne
! valign=top|razcepljene
! valign=top|ciklične
|-
!'''H<sup>3</sup>'''
!4
!9
| valign=top align=right|<div  style="text-align: center;">[[Seznam pravilnih politopov#Tessellations of hyperbolic 3-space|'''3:''']]</div>
<math>{\bar{BH}}_3</math> = [4,3,5]: {{CDD|node|4|node|3|node|5|node}}<BR>
<math>{\bar{K}}_3</math> = [5,3,5]: {{CDD|node|5|node|3|node|5|node}}<BR>
<math>{\bar{J}}_3</math> = [3,5,3]: {{CDD|node|3|node|5|node|3|node}}
| valign=top align=right|
<math>{\bar{DH}}_3</math> = [5,3<sup>1,1</sup>]: {{CDD|node|5|node|split1|nodes}}
| valign=top align=right|
<math>{\widehat{AB}}_3</math> = [(3,3,3,4)]: {{CDD|label4|branch|3ab|branch}}&nbsp;<BR>
<math>{\widehat{AH}}_3</math> = [(3,3,3,5)]: {{CDD|label5|branch|3ab|branch}}&nbsp;<BR>
<math>{\widehat{BB}}_3</math> = [(3,4,3,4)]: {{CDD|label4|branch|3ab|branch|label4}}<BR>
<math>{\widehat{BH}}_3</math> = [(3,4,3,5)]: {{CDD|label4|branch|3ab|branch|label5}}<BR>
<math>{\widehat{HH}}_3</math> = [(3,5,3,5)]: {{CDD|label5|branch|3ab|branch|label5}}
|-
!'''H<sup>4</sup>'''
!5
!5
| valign=top align=right|<div  style="text-align: center;">[[Seznam pravilnih politopov#Teselacije hiperboličnega 4-prostora|'''3:''']]</div>
<math>{\bar{H}}_4</math> = [3,3,3,5]: {{CDD|node|3|node|3|node|3|node|5|node}}<BR>
<math>{\bar{BH}}_4</math> = [4,3,3,5]: {{CDD|node|4|node|3|node|3|node|5|node}}<BR>
<math>{\bar{K}}_4</math> = [5,3,3,5]: {{CDD|node|5|node|3|node|3|node|5|node}}
| valign=top align=right|
<math>{\bar{DH}}_4</math> = [5,3,3<sup>1,1</sup>]: {{CDD|node|5|node|3|node|split1|nodes}}
| valign=top align=right|
<math>{\widehat{AF}}_4</math> = [(3,3,3,3,4)]: {{CDD|label4|branch|3ab|nodes|split2|node}}
|}

=== Nekompaktni ===

==== rang 3 ====
{| class=wikitable
!linearni grafi
!ciklični grafi
|- valign=top
|
* [p,∞] {{CDD|node|p|node|infin|node}}
* [∞,∞] {{CDD|node|infin|node|infin|node}}
|
* [(p,q,∞)] {{CDD|3|node|p|node|q|node|infin|3}}
* [(p,∞,∞)] {{CDD|3|node|p|node|infin|node|infin|3}}
* [(∞,∞,∞)] {{CDD|3|node|infin|node|infin|node|infin|3}}
|}

==== Rangi od 4 do 10 ====

Znanih je skupno 48 nekompaktnih hiperboličnih Coxeterjevih grup z rangom od 4 do 10. V naslednji preglednici je vseh 58 razvrščenih v pet skupin.

{| class=wikitable
|+ Hiperbolične nekompaktne grupe 
|- align=center
!rang
!skupno<BR>število
!colspan=4|grupe
|- align=center
!4
!23
|align=right|
<math>{\widehat{BR}}_3</math> = [(3,3,4,4)]: {{CDD|label4|branch|4-3|branch|2}}<BR>
<math>{\widehat{CR}}_3</math> = [(3,4,4,4)]: {{CDD|label4|branch|4-3|branch|label4}}<BR>
<math>{\widehat{RR}}_3</math> = [(4,4,4,4)]: {{CDD|label4|branch|4-4|branch|label4}}<BR>
<math>{\widehat{AV}}_3</math> = [(3,3,3,6)]: {{CDD|label6|branch|3ab|branch|2}}<BR>
<math>{\widehat{BV}}_3</math> = [(3,4,3,6)]: {{CDD|label6|branch|3ab|branch|label4}}<BR>
<math>{\widehat{HV}}_3</math> = [(3,5,3,6)]: {{CDD|label6|branch|3ab|branch|label5}}<BR>
<math>{\widehat{VV}}_3</math> = [(3,6,3,6)]: {{CDD|label6|branch|3ab|branch|label6}}

|align=right|
<math>{\bar{P}}_3</math> = [3,3<sup>[3]</sup>]: {{CDD|branch|split2|node|3|node}}<BR>
<math>{\bar{BP}}_3</math> = [4,3<sup>[3]</sup>]: {{CDD|branch|split2|node|4|node}}<BR>
<math>{\bar{HP}}_3</math> = [5,3<sup>[3]</sup>]: {{CDD|branch|split2|node|5|node}}<BR>
<math>{\bar{VP}}_3</math> = [6,3<sup>[3]</sup>]: {{CDD|branch|split2|node|6|node}}<BR>
<math>{\bar{DV}}_3</math> = [6,3<sup>1,1</sup>]: {{CDD|nodes|split2|node|6|node}}<BR>
<math>{\bar{O}}_3</math> = [6,4<sup>1,1</sup>]: {{CDD|nodes|split2-44|node|3|node}}<BR>
<math>{\bar{M}}_3</math> = [4,4<sup>1,1</sup>]: {{CDD|nodes|split2-44|node|4|node}}

|align=right|
<math>{\bar{R}}_3</math> = [3,4,4]: {{CDD|node|4|node|4|node|3|node}}<BR>
<math>{\bar{N}}_3</math> = [4,4,4]: {{CDD|node|4|node|4|node|4|node}}<BR>
<math>{\bar{V}}_3</math> = [3,3,6]: {{CDD|node|3|node|3|node|6|node}}<BR>
<math>{\bar{BV}}_3</math> = [4,3,6]: {{CDD|node|4|node|3|node|6|node}}<BR>
<math>{\bar{HV}}_3</math> = [5,3,6]: {{CDD|node|5|node|3|node|6|node}}<BR>
<math>{\bar{Y}}_3</math> = [3,6,3]: {{CDD|node|3|node|6|node|3|node}}<BR>
<math>{\bar{Z}}_3</math> = [6,3,6]: {{CDD|node|6|node|3|node|6|node}}

|align=right|
<math>{\bar{DP}}_3</math> = [3<sup>[ ]x[ ]</sup>]: {{CDD|node|split1|branch|split2|node}}<BR>
<math>{\bar{PP}}_3</math> = [3<sup>[3,3]</sup>]: {{CDD|tet}}

|- align=center
!5
!9
|align=right|
<math>{\bar{P}}_4</math> = [3,3<sup>[4]</sup>]: {{CDD|node|split1|nodes|split2|node|3|node}}<P>
<math>{\bar{BP}}_4</math> = [4,3<sup>[4]</sup>]: {{CDD|node|split1|nodes|split2|node|4|node}}<BR>
<math>{\widehat{FR}}_4</math> = [(3,3,4,3,4)]: {{CDD|branch|4-4|nodes|split2|node}}<BR>
<math>{\bar{DP}}_4</math> = [3<sup>[3]x[ ]</sup>]: {{CDD|node|split1|branchbranch|split2|node}}

|align=right|
<math>{\bar{N}}_4</math> = [4,/3\,3,4]: {{CDD|nodes|split2-43|node|3|node|4|node}}<BR>
<math>{\bar{O}}_4</math> = [3,4,3<sup>1,1</sup>]: {{CDD|nodes|split2|node|4|node|3|node}}<BR>
<math>{\bar{S}}_4</math> = [4,3<sup>2,1</sup>]: {{CDD|nodes|split2-43|node|3|node|3|node}}

|align=right|
<math>{\bar{R}}_4</math> = [3,4,3,4]: {{CDD|node|4|node|3|node|4|node|3|node}}
|align=right|<math>{\bar{M}}_4</math> = [4,3<sup>1,1,1</sup>]: {{CDD|nodes|split2-43|node|split1|nodes}}<BR>

|- align=center
!6
!12
|align=right|
<math>{\bar{P}}_5</math> = [3,3<sup>[5]</sup>]: {{CDD|branch|3ab|nodes|split2|node|3|node}}<BR>
<math>{\widehat{AU}}_5</math> = [(3,3,3,3,3,4)]: {{CDD|label4|branch|3ab|nodes|3ab|branch}}<P>
<math>{\widehat{AR}}_5</math> = [(3,3,4,3,3,4)]: {{CDD|label4|branch|3ab|nodes|3ab|branch|label4}}

|align=right|
<math>{\bar{S}}_5</math> = [4,3,3<sup>2,1</sup>]: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|4a|nodea}}<BR>
<math>{\bar{O}}_5</math> = [3,4,3<sup>1,1</sup>]: {{CDD|nodes|split2|node|3|node|4|node|3|node}}<BR>
<math>{\bar{N}}_5</math> = [4,3,/3\,3,4]: {{CDD|nodea|4a|nodea|3a|branch|3a|nodea|4a|nodea}}

|align=right| 
<math>{\bar{U}}_5</math> = [3,3,3,4,3]: {{CDD|node|3|node|3|node|3|node|4|node|3|node}}<BR>
<math>{\bar{X}}_5</math> = [3,3,4,3,3]: {{CDD|node|3|node|3|node|4|node|3|node|3|node}}<BR>
<math>{\bar{R}}_5</math> = [3,4,3,3,4]: {{CDD|node|3|node|4|node|3|node|3|node|4|node}}

|align=right|<math>{\bar{Q}}_5</math> = [3<sup>2,1,1,1</sup>]: {{CDD|nodea|3a|nodes|split2|node|split1|nodes}}<BR>
<math>{\bar{M}}_5</math> = [4,3,3<sup>1,1,1</sup>]: {{CDD|nodea|4a|nodes|split2|node|split1|nodes}}<BR>
<math>{\bar{L}}_5</math> = [3<sup>1,1,1,1,1</sup>]: {{CDD|star5}}
|- align=center
!7
!3
|align=right|
<math>{\bar{P}}_6</math> = [3,3<sup>[6]</sup>]:<BR>{{CDD|node|split1|nodes|3ab|nodes|split2|node|3|node}}
|align=right|<math>{\bar{Q}}_6</math> = [3<sup>1,1</sup>,3,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|branch|3a|branch|3a|nodea|3a|nodea}}

|align=right|<math>{\bar{S}}_6</math> = [4,3,3,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|4a|nodea}}
|
|- align=center
!8
!4
|align=right|<math>{\bar{P}}_7</math> = [3,3<sup>[7]</sup>]:<BR>{{CDD|branch|3ab|nodes|3ab|nodes|split2|node|3|node}}
|align=right|<math>{\bar{Q}}_7</math> = [3<sup>1,1</sup>,3<sup>2</sup>,3<sup>2,1</sup>]:<BR> {{CDD|nodea|3a|branch|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|align=right|<math>{\bar{S}}_7</math> = [4,3<sup>3</sup>,3<sup>2,1</sup>]:<BR> {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|4a|nodea}}
|align=right|<math>{\bar{T}}_7</math> = [3<sup>3,2,2</sup>]:<BR>{{CDD|nodes|3ab|nodes|split2|node|3|node|3|node|3|node}}
|- align=center
!9
!4
|align=right|<math>{\bar{P}}_8</math> = [3,3<sup>[8]</sup>]:<BR>{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node|3|node}}
|align=right|<math>{\bar{Q}}_8</math> = [3<sup>1,1</sup>,3<sup>3</sup>,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|align=right|<math>{\bar{S}}_8</math> = [4,3<sup>4</sup>,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|4a|nodea}}
|align=right|<math>{\bar{T}}_8</math> = [3<sup>4,3,1</sup>]:<BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
|- align=center
!10
!3
|
|align=right|<math>{\bar{Q}}_9</math> = [3<sup>1,1</sup>,3<sup>4</sup>,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|align=right|<math>{\bar{S}}_9</math> = [4,3<sup>5</sup>,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|4a|nodea}}
|align=right|<math>{\bar{T}}_9</math> = [3<sup>6,2,1</sup>]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|}

== Zunanje povezave ==

* {{MathWorld|id=Coxeter-DynkinDiagram|title=Coxeter-Dynkin Diagram}}

{{-}}

{{poliedri}}
{{normativna kontrola}}

[[Kategorija:Coxeterjeve grupe]]