Coxeter-Dinkinov diagram

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Coxeter-Dinkinovi diagrami za osnovne končne Coxeterjeve grupe.
Coxeter-Dinkinovi diagrami za osnovne afine Coxeterjeve grupe.

Coxeter-Dinkinov diagram (tudi Coxeterjev diagram ali Coxeterjev graf) je graf, ki ima s številkami označene stranice (imenujemo jih veje) s katerimi prikažemo prostorske odnose med zbirko zrcal oziroma odbojnih hiperravnin. Opisujejo kalejdoskopsko konstrukcijo: vsak vozel grafa predstavlja ogledalo (v domeni facete). Oznaka pri vsaki veji določa stopnjo diederskega kota dveh ogledal (v domeni grebena). Neoznačene veje pomenijo red 3.

Vsak diagram predstavlja Coxeterjevo grupo in tudi Coxeterjeve grupe so razvrščene po pripadajočih diagramih.

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

Opis diagramov[uredi | uredi kodo]

Veje Coxeter-Dinkinovih diagramov so označene z racionalnimi števili  p \,, kar predstavlja diederski 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  p = 3 \, kar pomeni kot 60º. Vzporedni zrcali imata oznako "∞"

Geometrijska ponazoritev[uredi | uredi kodo]

Coxeter-Dinkinov diagram se lahko prikaže kot domena ogledal. Ogledalo v tem primeru predstavlja hiperravnino s pomočjo sfernega ali Evklidskega ali hiperboličnega prostora z dano razsežnostjo.

Takšna ponazoritev kaže osnovne domene za dvo in trirazsežne Evklidske grupe in dvorazsežne sferne grupe.

Coxeter-dynkin plane groups sl.png
Coxeterjeve grupe v ravnini s pripadajočimi diagrami. Ogledala domen so označena kot veje m1, m2 itd. Oglišča so obarvana v skladu z zaporedjem odboja. Prizmatske grupe {\tilde{I}}_1x{\tilde{I}}_1 so prikazane kot podvojitev {\tilde{C}}_2, toda nastale bi lahko tudi kot pravokotne domene iz podvojitev {\tilde{G}}_2 trikotnikov. {\tilde{A}}_2 je podvojitev {\tilde{G}}_2 trikotnika.
Coxeter-Dynkin 3-space groups sl.png
Coxeterjeve grupe v trirazsežnem prostoru z diagrami. Zrcala (stranice trikotnika) so označena z nasprotnim ogliščem 0..3. Veje grafa so obarvane z zaporedjem odboja.
{\tilde{C}}_3 izpolni 1/48 kocke. {\tilde{B}}_3 izpolni 1/24 kocke. {\tilde{A}}_3 izpolni1/12 kocke.
Coxeter-Dynkin sphere groups sl.png
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[uredi | uredi kodo]

Coxeter-Dinkinovi diagrami lahko opišejo skoraj vse vrste uniformnih politopov in uniformnih teselacij

Cartanove matrike[uredi | uredi kodo]

Vsakenu Coxeterjevemu diagramu pripada odgovarjajoča Cartanova matrika. Vse Cartanove matrike Coxeterjevih grup so simetrične. Elementi Cartanove matrike so ai,j = aj,i = -2*cos(π/p) kjer je

  •  p \, red veje med pari zrcal.

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

rang 2 Coxeterjeve grupe
red
simetrije
p
ime
grupe
Coxeterjev diagram Cartanova matrika
\left [\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}\right ] determinanta

(4-a21*a12)

končne (determinanta>0)
2 I2(2) = A1xA1 CDel node.pngCDel 2.pngCDel node.png \left [\begin{smallmatrix}2&0\\0&2\end{smallmatrix}\right ] 4
3 I2(3) = A2 CDel node.pngCDel 3.pngCDel node.png \left [\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}\right ] 3
4 I2(4) = BC2 CDel node.pngCDel 4.pngCDel node.png \left [\begin{smallmatrix}2&-\sqrt{2}\\-\sqrt{2}&2\end{smallmatrix}\right ] 2
5 I2(5) = H2 CDel node.pngCDel 5.pngCDel node.png \left [\begin{smallmatrix}2&-\phi\\-\phi&2\end{smallmatrix}\right ] 4\sin^2(\pi/5)

=(5-\sqrt{5})/2

~1,38196601125

6 I2(6) = G2 CDel node.pngCDel 6.pngCDel node.png \left [\begin{smallmatrix}2&-\sqrt{3}\\-\sqrt{3}&2\end{smallmatrix}\right ] 1
8 I2(8) CDel node.pngCDel 8.pngCDel node.png \left [\begin{smallmatrix}2&-2\cos(\pi/8)\\-2\cos(\pi/8)&2\end{smallmatrix}\right ] 2-\sqrt{2}

~0,58578643763

10 I2(10) CDel node.pngCDel 10.pngCDel node.png \left [\begin{smallmatrix}2&-2\cos(\pi/10)\\-2\cos(\pi/10)&2\end{smallmatrix}\right ] 4\sin^2(\pi/10)

=(3-\sqrt{5})/2

~0,38196601125

12 I2(12) CDel node.pngCDel 12.pngCDel node.png \left [\begin{smallmatrix}2&-2\cos(\pi/12)\\-2\cos(\pi/12)&2\end{smallmatrix}\right ] 2-\sqrt{3}

~0,26794919243

p I2(p) CDel node.pngCDel p.pngCDel node.png \left [\begin{smallmatrix}2&-2\cos(\pi/p)\\-2\cos(\pi/p)&2\end{smallmatrix}\right ] 4\sin^2(\pi/p)
afine (determinanta=0)
I2(∞) = {\tilde{I}}_1 = {\tilde{A}}_1 CDel node.pngCDel infin.pngCDel node.png \left [\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}\right ] 0

Končne Coxeterjeve grupe[uredi | uredi kodo]

Povezani končni Dinkinovi grafi za rang 1 do 9
rang enostavne Lijeve grupe posebne Liejeve grupe  
{A}_{1+} {BC}_{2+} {D}_{2+} {E}_{3-8} {F}_{4} / {G}_{2} {H}_{2-4} {I}_{2}(p)
1 A1=[]

CDel node.png

       
2 A2=[3]

CDel node.pngCDel 3.pngCDel node.png

BC2=[4]

CDel node.pngCDel 4.pngCDel node.png

D2=A1xA1

CDel nodes.png

  G2=[6]

CDel node.pngCDel 6.pngCDel node.png

H2=[6]

CDel node.pngCDel 5.pngCDel node.png

I2[p]

CDel node.pngCDel p.pngCDel node.png

3 A3=[32]

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

BC3=[3,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png

D3=A3

CDel nodes.pngCDel split2.pngCDel node.png

E3=A2xA1

CDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodeb.png

  H3 

CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png

4 A4=[33]

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

BC4=[32,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

D4=[31,1,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png

E4=A4

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png

F4

CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png

H4 

CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

5 A5=[34]

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

BC5=[33,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

D5=[32,1,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

E5=D5

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png

 
6 A6=[35]

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

BC6=[34,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

D6=[33,1,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

E6=[32,2,1]

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

7 A7=[36]

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

BC7=[35,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

D7=[34,1,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

E7=[33,2,1]

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

8 A8=[37]

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

BC8=[36,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

D8=[35,1,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

E8=[34,2,1]

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

9 A9=[38]

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

BC9=[37,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

D9=[36,1,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

 
10+ .. .. .. ..

Afine Coxeterjeve grupe[uredi | uredi kodo]

Afini Dinkinovi grafi od 2 do10 vozlov
rang {\tilde{A}}_{1+} (P2+) {\tilde{B}}_{3+} (S4+) {\tilde{C}}_{1+} (R2+) {\tilde{D}}_{4+} (Q5+) {\tilde{E}}_{n} (Tn+1) / {\tilde{F}}_{4} (U5) / {\tilde{G}}_{2} (V3)
2 {\tilde{A}}_{1}=[∞]

CDel node.pngCDel infin.pngCDel node.png

  {\tilde{C}}_{1}=[∞]

CDel node.pngCDel infin.pngCDel node.png

   
3 {\tilde{A}}_{2}=[3[3]]

CDel branch.pngCDel split2.pngCDel node.png

{\tilde{C}}_{2}=[4,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{G}}_{2}=[6,3]

CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

4 {\tilde{A}}_{3}=[3[4]]

CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png

{\tilde{B}}_{3}=[4,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{C}}_{3}=[4,3,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

 
5 {\tilde{A}}_{4}=[3[5]]

CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

{\tilde{B}}_{4}=[4,3,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{C}}_{4}=[4,32,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{D}}_{4}=[31,1,1,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png

{\tilde{F}}_{4}=[3,4,3,3]

CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

6 {\tilde{A}}_{5}=[3[6]]

CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

{\tilde{B}}_{5}=[4,32,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{C}}_{5}=[4,33,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{D}}_{5}=[31,1,3,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

 
7 {\tilde{A}}_{6}=[3[7]]

CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

{\tilde{B}}_{6}=[4,33,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{C}}_{6}=[4,34,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{D}}_{6}=[31,1,32,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

{\tilde{E}}_{6}=[32,2,2]

CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

8 {\tilde{A}}_{7}=[3[8]]

CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

{\tilde{B}}_{7}=[4,34,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{C}}_{7}=[4,35,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{D}}_{7}=[31,1,33,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

{\tilde{E}}_{7}=[33,3,1]

CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png

9 {\tilde{A}}_{8}=[3[9]]

CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

{\tilde{B}}_{8}=[4,35,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{C}}_{8}=[4,36,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{D}}_{8}=[31,1,34,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

{\tilde{E}}_{8}=[35,2,1]

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

10 {\tilde{A}}_{9}=[3[10]]

CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

{\tilde{B}}_{9}=[4,36,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{C}}_{9}=[4,37,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\tilde{D}}_{9}=[31,1,35,31,1]

CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

11 ... ... ... ...

Hiperbolične Coxeterjeve grupe[uredi | uredi kodo]

Kompaktne[uredi | uredi kodo]

Rang 3[uredi | uredi kodo]

Kompaktne hiperbolične Coxeterjeve grupe
linearne ciklične
∞: [p,q], 2(p+q)<pq

CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 9.pngCDel node.pngCDel 3.pngCDel node.png
...
CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
...
CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png
CDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
...

∞ [(p,q,r)], p+q+r>9

CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png

CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.png

CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 5.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.png

CDel 3.pngCDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png
...

Rangi od 4 do 5[uredi | uredi kodo]

Kompaktne hiperbolične Coxeterjeve grupe
razsežnost
Hd
rang skupno število linearne razcepljene ciklične
H3 4 9
3:

{\bar{BH}}_3 = [4,3,5]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{K}}_3 = [5,3,5]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{J}}_3 = [3,5,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{DH}}_3 = [5,31,1]: CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png

{\widehat{AB}}_3 = [(3,3,3,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png 
{\widehat{AH}}_3 = [(3,3,3,5)]: CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png 
{\widehat{BB}}_3 = [(3,4,3,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
{\widehat{BH}}_3 = [(3,4,3,5)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
{\widehat{HH}}_3 = [(3,5,3,5)]: CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png

H4 5 5
3:

{\bar{H}}_4 = [3,3,3,5]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{BH}}_4 = [4,3,3,5]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{K}}_4 = [5,3,3,5]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png

{\bar{DH}}_4 = [5,3,31,1]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

{\widehat{AF}}_4 = [(3,3,3,3,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

Nekompaktni[uredi | uredi kodo]

rang 3[uredi | uredi kodo]

linearni grafi ciklični grafi
  • [p,∞] CDel node.pngCDel p.pngCDel node.pngCDel infin.pngCDel node.png
  • [∞,∞] CDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png
  • [(p,q,∞)] CDel 3.pngCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel infin.pngCDel 3.png
  • [(p,∞,∞)] CDel 3.pngCDel node.pngCDel p.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel 3.png
  • [(∞,∞,∞)] CDel 3.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel 3.png

Rangi od 4 do 10[uredi | uredi kodo]

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

Hiperbolične nekompaktne grupe
rang skupno
število
grupe
4 23

{\widehat{BR}}_3 = [(3,3,4,4)]: CDel label4.pngCDel branch.pngCDel 4-3.pngCDel branch.pngCDel 2.png
{\widehat{CR}}_3 = [(3,4,4,4)]: CDel label4.pngCDel branch.pngCDel 4-3.pngCDel branch.pngCDel label4.png
{\widehat{RR}}_3 = [(4,4,4,4)]: CDel label4.pngCDel branch.pngCdel 4-4.pngCDel branch.pngCDel label4.png
{\widehat{AV}}_3 = [(3,3,3,6)]: CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.png
{\widehat{BV}}_3 = [(3,4,3,6)]: CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
{\widehat{HV}}_3 = [(3,5,3,6)]: CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
{\widehat{VV}}_3 = [(3,6,3,6)]: CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label6.png

{\bar{P}}_3 = [3,3[3]]: CDel branch.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{BP}}_3 = [4,3[3]]: CDel branch.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
{\bar{HP}}_3 = [5,3[3]]: CDel branch.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{VP}}_3 = [6,3[3]]: CDel branch.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png
{\bar{DV}}_3 = [6,31,1]: CDel nodes.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png
{\bar{O}}_3 = [6,41,1]: CDel nodes.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{M}}_3 = [4,41,1]: CDel nodes.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node.png

{\bar{R}}_3 = [3,4,4]: CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{N}}_3 = [4,4,4]: CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
{\bar{V}}_3 = [3,3,6]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
{\bar{BV}}_3 = [4,3,6]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
{\bar{HV}}_3 = [5,3,6]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
{\bar{Y}}_3 = [3,6,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{Z}}_3 = [6,3,6]: CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png

{\bar{DP}}_3 = [3[ ]x[ ]]: CDel node.pngCDel split1.pngCDel branch.pngCDel split2.pngCDel node.png
{\bar{PP}}_3 = [3[3,3]]: CDel tet.png

5 9 {\bar{P}}_4 = [3,3[4]]: CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{BP}}_4 = [4,3[4]]: CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
{\widehat{FR}}_4 = [(3,3,4,3,4)]: CDel branch.pngCdel 4-4.pngCDel nodes.pngCDel split2.pngCDel node.png
{\bar{DP}}_4 = [3[3]x[ ]]: CDel node.pngCDel split1.pngCDel branchbranch.pngCDel split2.pngCDel node.png

{\bar{N}}_4 = [4,/3\,3,4]: CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\bar{O}}_4 = [3,4,31,1]: CDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{S}}_4 = [4,32,1]: CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{R}}_4 = [3,4,3,4]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{M}}_4 = [4,31,1,1]: CDel nodes.pngCDel split2-43.pngCDel node.pngCDel split1.pngCDel nodes.png
6 12

{\bar{P}}_5 = [3,3[5]]: CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png

{\widehat{AU}}_5 = [(3,3,3,3,3,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png

{\widehat{AR}}_5 = [(3,3,4,3,3,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel label4.png

{\bar{S}}_5 = [4,3,32,1]: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
{\bar{O}}_5 = [3,4,31,1]: CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{N}}_5 = [4,3,/3\,3,4]: CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png

{\bar{U}}_5 = [3,3,3,4,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{X}}_5 = [3,3,4,3,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{R}}_5 = [3,4,3,3,4]: CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\bar{Q}}_5 = [32,1,1,1]: CDel nodea.pngCDel 3a.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png

{\bar{M}}_5 = [4,3,31,1,1]: CDel nodea.pngCDel 4a.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png
{\bar{L}}_5 = [31,1,1,1,1]: CDel star5.png

7 3

{\bar{P}}_6 = [3,3[6]]:
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{Q}}_6 = [31,1,3,32,1]:
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{\bar{S}}_6 = [4,3,3,32,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
8 4 {\bar{P}}_7 = [3,3[7]]:
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{Q}}_7 = [31,1,32,32,1]:
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{\bar{S}}_7 = [4,33,32,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
{\bar{T}}_7 = [33,2,2]:
CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
9 4 {\bar{P}}_8 = [3,3[8]]:
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{Q}}_8 = [31,1,33,32,1]:
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{\bar{S}}_8 = [4,34,32,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
{\bar{T}}_8 = [34,3,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
10 3 {\bar{Q}}_9 = [31,1,34,32,1]:
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{\bar{S}}_9 = [4,35,32,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
{\bar{T}}_9 = [36,2,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Zunanje povezave[uredi | uredi kodo]