Coxeterjeva notacija

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Coxeterjeva notacija je v geometriji sistem, ki omogoča razvrščanje simetrijskih grup s tem, da opiše kote med osnovnimi zrcaljenji Coxeterjeve grupe. Uporablja se označevanje z oklepaji, kar lahko spremenimo za nekatere podgrupe.

Notacija se imenuje po kanadskem geometru Haroldu Scottu MacDonaldu Coxeterju (1907 – 2003). Točneje jo je definiral ameriški matematik Norman Johnson (rojen 1930).

Zrcalne grupe[uredi | uredi kodo]

Za Coxeterjeve grupe, ki so definirane s čistim zrcaljenjem, je neposredna povezava med notacijo z oklepaji in Coxeter-Dynkinovimi diagrami. Število v notaciji z oklepaji predstavlja red zrcaljenja v vejah Coxeterjevega grafa.

Coxeterjeva notacija se poenostavi s potencami, ki predstavljajo število položajev v oklepajih v vrstici za linearne grafe. Tako je grupa An predstavljena z [3n-1], kar pomeni n vozlov povezanih z n-1 vejami reda 3.

Nadaljnji razvejani grafi se pričnejo kot števila, ki so dana kot navpični položaji v oklepajih. Poenostavljeni so z večkratnimi nadpisanimi vrednostmi na dolžini oklepaja.

Coxeterjeve grupe, ki jih tvorijo ciklični grafi se prikažejo z oklepaji znotraj oklepaja. Zgled: [(a,b,c)] za trikotno grupo (a b c). Kadar so enaki, jih lahko grupiramo glede na dolžino cikla v oklepaju. Zgled: [(3,3,3,3)] = [3[4]].


Končne Coxeterjeve grupe
rang simbol
grupe
notacija z oklepaji Coxeterjev graf
2 A2 [3] CDel node.pngCDel 3.pngCDel node.png
2 BC2 [4] CDel node.pngCDel 4.pngCDel node.png
2 H2 [5] CDel node.pngCDel 5.pngCDel node.png
2 G2 [6] CDel node.pngCDel 6.pngCDel node.png
2 I2(p) [p] CDel node.pngCDel p.pngCDel node.png
3 H3 [5,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
3 A3 [3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3 BC3 [4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
4 A4 [3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4 BC4 [4,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4 D4 [31,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
4 F4 [3,4,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
4 H4 [5,3,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
n An [3n-1] CDel node.pngCDel 3.pngCDel node.pngCDel 3.png..CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
n BCn [4,3n-2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
n Dn [3n-3,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6 E6 [32,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
7 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 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
Afine Coxeterjeve grupe
simbol
grupe
notacija z oklepaji Coxeterjev graf
{\tilde{I}}_1 [∞] CDel node.pngCDel infin.pngCDel node.png
{\tilde{A}}_2 [3[3]] CDel node.pngCDel split1.pngCDel branch.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
{\tilde{A}}_3 [3[4]] CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{B}}_3 [4,31,1] CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{C}}_3 [4,3,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{A}}_4 [3[5]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
{\tilde{B}}_4 [4,3,31,1] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{C}}_4 [4,3,3,4] 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
{\tilde{A}}_n [3[n+1]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.png...CDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
or
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.png...CDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
{\tilde{B}}_n [4,3n-2,31,1] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{C}}_n [4,3n-1,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{D}}_n [ 31,1,3n-3,31,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{E}}_6 [32,2,2] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{\tilde{E}}_7 [33,3,1] CDel 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
{\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
Kompaktne hiperbolične Coxeterjeve grupe
simbol
grupe
notacija z oklepaji Coxeterjev graf
[p,q]
z 2(p+q)<pq
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
[(p,q,r)]
z p+q+r>9
CDel pqr.png
{\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
{\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

Razvrščanje po rangu[uredi | uredi kodo]

Coxeterjeve grupe lahko razvrstimo po njihovem rangu, kar je število vozlov v Coxeterjevem grafu. Struktura teh grup je dana tudi z vrstami tipov abstraktnih grup. Tukaj so abstraktne diederske grupe predstavljene z Dihn. Ciklične grupe pa so predstavljene z Zn tako, da je Dih1 = Z2.

Grupe z rangom ena[uredi | uredi kodo]

V eni razsežnosti dvostranska grupa (bilateralna grupa) [ ] predstavlja posamezno zrcalno simetrijo. To je Dih1 ali simetrija Z2, reda 2. Predstavljena je kot Coxeter-Dynkinov diagram s samo enim vozlom CDel node.png. Grupa identitete je direktna grupa [ ]+, Z1, s simetrijskim redom 1. Nadpis + kaže samo na to, da so izmenično zrcalni odboji zanemarjeni, kar pusti grupo identitete v tej najenostavnejši obliki.

grupa Coxeter Coxeterjev graf red opis
C1 [ ]+ 1 identiteta
D1 [ ] CDel node.png 2 zrcalna grupa

Grupe z rangom dva[uredi | uredi kodo]

V dveh razsežnostih se pravokotna grupa [2] Dih2 lahko prikaže kot direktni produkt [ ]×[ ] ali Z2×Z2 kot dvostranski grupi, ki ju predstavimo z dvema pravokotnima ogledaloma, pri tem pa je Coxeterjev graf CDel node.pngCDel 2.pngCDel node.png z redom 4

grupa intl orbifold notacija Coxeter red opis
Končne
Zn n nn [n]+ n ciklični: n-kratna vrtenja. Abstraktna grupa Zn, grupa celih števil pod seštevanjem po modulu n.
Dn nm *nn [n] 2n diederska: ciklična z zrcaljenji. Abstraktna grupa Dihn, diederska grupa.
afine
Z ∞∞ [∞]+ ciklično: apeirogonalna grupa. Abstraktna grupa Z, grupa celih števil pod seštevanjem.
Dih m *∞∞ [∞] diedersko: vzporedna zrcaljenja. Abstratna neskončna diederska grupa Dih.
hiperbolične
Z [πi/λ]+ psevdogonalna grupa
Dih [πi/λ] polna psevdogonalna grupa

Grupe z rangom tri[uredi | uredi kodo]

končne
intl* geo
[1]
orbifold Schönflies Conway Coxeter red
1 1 1 C1 C1 [ ]+ 1
1 22 ×1 Ci = S2 CC2 [2+,2+] 2
2 = m 1 *1 Cs = C1v = C1h ±C1 = CD2 [ ] 2
2
3
4
5
6
n
2
3
4
5
6
n
22
33
44
55
66
nn
C2
C3
C4
C5
C6
Cn
C2
C3
C4
C5
C6
Cn
[2]+
[3]+
[4]+
[5]+
[6]+
[n]+
2
3
4
5
6
n
2mm
3m
4mm
5m
6mm
nmm
nm
2
3
4
5
6
n
*22
*33
*44
*55
*66
*nn
C2v
C3v
C4v
C5v
C6v
Cnv
CD4
CD6
CD8
CD10
CD12
CD2n
[2]
[3]
[4]
[5]
[6]
[n]
4
6
8
10
12
2n
2/m
3/m
4/m
5/m
6/m
n/m
2 2
3 2
4 2
5 2
6 2
n 2
2*
3*
4*
5*
6*
n*
C2h
C3h
C4h
C5h
C6h
Cnh
±C2
CC6
±C4
CC10
±C6
±Cn / CC2n
[2,2+]
[2,3+]
[2,4+]
[2,5+]
[2,6+]
[2,n+]
4
6
8
10
12
2n
4
3
8
5
12
2n
n
4 2
6 2
8 2
10 2
12 2
2n 2





S4
S6
S8
S10
S12
S2n
CC4
±C3
CC8
±C5
CC12
CC2n / ±Cn
[2+,4+]
[2+,6+]
[2+,8+]
[2+,10+]
[2+,12+]
[2+,2n+]
4
6
8
10
12
2n
intl geo orbifold Schönflies Conway Coxeter red
222
32
422
52
622
n22
n2
2 2
3 2
4 2
5 2
6 2
n 2
222
223
224
225
226
22n
D2
D3
D4
D5
D6
Dn
D4
D6
D8
D10
D12
D2n
[2,2]+
[2,3]+
[2,4]+
[2,5]+
[2,6]+
[2,n]+
4
6
8
10
12
2n
mmm
6m2
4/mmm
10m2
6/mmm
n/mmm
2nm2
2 2
3 2
4 2
5 2
6 2
n 2
*222
*223
*224
*225
*226
*22n
D2h
D3h
D4h
D5h
D6h
Dnh
±D4
DD12
±D8
DD20
±D12
±D2n / DD4n
[2,2]
[2,3]
[2,4]
[2,5]
[2,6]
[2,n]
8
12
16
20
24
4n
42m
3m
82m
5m
122m
2n2m
nm
4 2
6 2
8 2
10 2
12 2
n 2
2*2
2*3
2*4
2*5
2*6
2*n
D2d
D3d
D4d
D5d
D6d
Dnd
±D4
±D6
DD16
±D10
DD24
DD4n / ±D2n
[2+,4]
[2+,6]
[2+,8]
[2+,10]
[2+,12]
[2+,2n]
8
12
16
20
24
4n
23 3 3 332 T T [3,3]+ 12
m3 4 3 3*2 Th ±T [3+,4] 24
43m 3 3 *332 Td TO [3,3] 24
432 4 3 432 O O [3,4]+ 24
m3m 4 3 *432 Oh ±O [3,4] 48
532 5 3 532 I I [3,5]+ 60
53m 5 3 *532 Ih ±I [3,5] 120
Polfine
intl (orbifold notacija) geo
Schönflies Coxeter red
n nn n Cn [1,n]+ n
nm *nn n Dn [1,n] 2n
IUC (orbifold) geo Schönflies Coxeter
p1 ∞∞ p1 C [1,∞]+
p1m1 *∞∞ p1 C∞v [1,∞]
IUC (orbifold) geo Schönflies Coxeter
p11g ∞x p.g1 S2∞ [∞+,2+]
p11m ∞* p.1 C∞h [∞+,2]
p2 22∞ p2 D [∞,2]+
p2mg 2*∞ p2g D∞d [∞,2+]
p2mm *22∞ p2 D∞h [∞,2]
Afine
IUC (orbifold) geometrijska Coxeter
p2 (2222) p2 [1+,4,4]+
p2gg (22x) pg2g [4+,4+]
p2mm (*2222) p2 [1+,4,4]
c2mm (2*22) c2 [[4+,4+]]
p4 (442) p4 [4,4]+
p4gm (4*2) pg4 [4+,4]
p4mm (*442) p4 [4,4]
IUC (orbifold) geometrijska Coxeter
p3 (333) p3 [1+,6,3+] = [3[3]]+
p3m1 (*333) p3 [1+,6,3] = [3[3]]
p31m (3*3) h3 [6,3+] = [3[3[3]]+]
p6 (632) p6 [6,3]+ = [3[3[3]]]+
p6mm (*632) p6 [6,3] = [3[3[3]]]

Grupe z rangom štiri[uredi | uredi kodo]

Točkovne grupe[uredi | uredi kodo]

Grupe ranga 4 definirajo štirirazsežne točkovne grupe:

Končne grupe
[ ]: CDel node.png
simbol red
[1]+ 1.1
[1] = [ ] 2.1
[2,1,1]: CDel node.pngCDel 2.pngCDel node.png
simbol red
[2,1,1]+ 2.1
[2,1,1] 4.1
[2,2,1]: CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
simbol red
[2+,2+,1] 2.1
[2,2,1]+ 4.1
[2+,2,1] 4.1
[2,2,1] 8.1
[2,2,2]: CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
simbol red
[2+,2+,2+] 2.1
[2+,2,2+] 4.1
[(2,2)+,2+] 4
[[2+,2+,2+]] 4
[2,2,2]+ 8
[2+,2,2] 8.1
[(2,2)+,2] 8
[[2+,2,2+]] 8.1
[2,2,2] 16.1
[[2,2,2]]+ 16
[[2,2+,2]] 16
[[2,2,2]] 32
[p,1,1]: CDel node.pngCDel p.pngCDel node.png
simbol red
[3,1,1]+ 3.1
[4,1,1]+ 4.2
[5,1,1]+ 5.1
[6,1,1]+ 6.1
[p,1,1]+ p
[3,1,1] 6.2
[4,1,1] 8.4
[5,1,1] 10.2
[6,1,1] 12.3
[p,1,1] 2p


[p,2,1]: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png
simbol red
[3,2,1]+ 6.1
[4,2,1]+ 8.3
[5,2,1]+ 10.2
[6,2,1]+ 12.3
[p,2,1]+ 2p
[3,2,1] 12.3
[4,2,1] 16.6
[5,2,1] 20.3
[6,2,1] 24.6
[p,2,1] 4p
[2p,2+,1]: CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2x.pngCDel node.png
simbol red
[6,2+,1] 12.3
[8,2+,1] 16.12
[10,2+,1] 20.3
[12,2+,1] 24.12
[2p,2+,1] 4p
[p,2,2]: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
simbol red
[3+,2,2+] 6
[4+,2,2+] 8
[5+,2,2+] 10
[6+,2,2+] 12
[p+,2,2+] 2p
[(3,2)+,2+] 6
[(4,2)+,2+] 8
[(5,2)+,2+] 10
[(6,2)+,2+] 12
[(p,2)+,2+] 2p
[3,2,2]+ 12
[4,2,2]+ 16
[5,2,2]+ 20
[6,2,2]+ 24
[p,2,2]+ 4p
[3,2,2+] 12
[4,2,2+] 16
[5,2,2+] 20
[6,2,2+] 24
[p,2,2+] 4p
[3+,2,2] 12
[4+,2,2] 16
[5+,2,2] 20
[6+,2,2] 24
[p+,2,2] 4p
[(3,2)+,2] 12
[(4,2)+,2] 16
[(5,2)+,2] 20
[(6,2)+,2] 24
[(p,2)+,2] 4p
[3,2,2] 24
[4,2,2] 32
[5,2,2] 40
[6,2,2] 48
[p,2,2] 8p
[2p,2+,2]: CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
simbol red
[6+,2+,2+] 6
[8+,2+,2+] 8
[10+,2+,2+] 10
[12+,2+,2+] 12
[2p+,2+,2+] 2p
[6+,2+,2] 12
[8+,2+,2] 16
[10+,2+,2] 20
[12+,2+,2] 24
[2p+,2+,2] 4p
[6+,(2,2)+] 12
[8+,(2,2)+] 16
[10+,(2,2)+] 20
[12+,(2,2)+] 24
[2p+,(2,2)+] 4p
[6,(2,2)+] 24
[8,(2,2)+] 32
[10,(2,2)+] 40
[12,(2,2)+] 48
[2p,(2,2)+] 8p
[6,2+,2] 24
[8,2+,2] 32
[10,2+,2] 40
[12,2+,2] 48
[2p,2+,2] 8p
[p,2,q]: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png
simbol red
[3+,2,3+] 9
[4+,2,3+] 12
[5+,2,3+] 15
[6+,2,3+] 18
[4+,2,4+] 16
[5+,2,4+] 20
[6+,2,4+] 24
[5+,2,5+] 25
[6+,2,5+] 30
[6+,2,6+] 36
[p+,2,q+] pq
[3,2,3]+ 18
[4,2,3]+ 24
[5,2,3]+ 30
[6,2,3]+ 36
[4,2,4]+ 32
[5,2,4]+ 40
[6,2,4]+ 48
[5,2,5]+ 50
[6,2,5]+ 60
[6,2,6]+ 72
[p,2,q]+ 2pq
[3+,2,3] 18
[4+,2,3] 24
[5+,2,3] 30
[6+,2,3] 36
[4+,2,4] 32
[5+,2,4] 40
[6+,2,4] 48
[5+,2,5] 50
[6+,2,5] 60
[6+,2,6] 72
[p+,2,q] 2pq
[3,2,3] 36
[4,2,3] 48
[5,2,3] 60
[6,2,3] 72
[4,2,4] 64
[5,2,4] 80
[6,2,4] 96
[5,2,5] 100
[6,2,5] 120
[6,2,6] 144
[p,2,q] 4pq
[(p,2)+,2q]: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node.png
simbol red
[(3,2)+,6+] 18
[(4,2)+,6+] 24
[(5,2)+,6+] 30
[(6,2)+,6+] 36
[(4,2)+,8+] 32
[(5,2)+,8+] 40
[(6,2)+,8+] 48
[(5,2)+,10+] 50
[(6,2)+,10+] 60
[(6,2)+,12+] 72
[(p,2)+,2q+] 2pq
[(3,2)+,6] 36
[(4,2)+,6] 48
[(5,2)+,6] 60
[(6,2)+,6] 72
[(4,2)+,8] 64
[(5,2)+,8] 80
[(6,2)+,8] 96
[(5,2)+,10] 100
[(6,2)+,10] 120
[(6,2)+,12] 144
[(p,2)+,2q] 4pq
[2p,2+,2q]: CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node.png
simbol red
[6+,2+,6+] 18
[8+,2+,6+] 24
[10+,2+,6+] 30
[12+,2+,6+] 36
[8+,2+,8+] 32
[10+,2+,8+] 40
[12+,2+,8+] 48
[10+,2+,10+] 50
[12+,2+,10+] 60
[12+,2+,12+] 72
[2p+,2+,2q+] 2pq
[6,2+,6+] 36
[8,2+,6+] 48
[10,2+,6+] 60
[12,2+,6+] 72
[8,2+,8+] 64
[10,2+,8+] 80
[12,2+,8+] 96
[10,2+,10+] 100
[12,2+,10+] 120
[12,2+,12+] 144
[2p,2+,2q+] 4pq
[6,2+,6] 72
[8,2+,6] 96
[10,2+,6] 120
[12,2+,6] 144
[8,2+,8] 128
[10,2+,8] 160
[12,2+,8] 192
[10,2+,10] 200
[12,2+,10] 240
[12,2+,12] 288
[2p,2+,2q] 8pq
[[p,2,p]]: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
simbol red
[[3+,2,3+]] 18
[[4+,2,4+]] 32
[[5+,2,5+]] 50
[[6+,2,6+]] 72
[[p+,2,p+]] 2p2
[[3,2,3]]+ 36
[[4,2,4]]+ 64
[[5,2,5]]+ 100
[[6,2,6]]+ 144
[[p,2,p]]+ 4p2
[[3,2,3]] 72
[[4,2,4]] 128
[[5,2,5]] 200
[[6,2,6]] 288
[[p,2,p]] 8p2
[[2p,2+,2p]]: CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel p.pngCDel node.png
simbol red
[[6+,2+,6+]] 36
[[8+,2+,8+]] 64
[[10+,2+,10+]] 100
[[12+,2+,12+]] 144
[[2p+,2+,2p+]] 4p2
[[6,2+,6]] 144
[[8,2+,8]] 256
[[10,2+,10]] 400
[[12,2+,12]] 576
[[2p,2+,2p]] 16p2
[3,3,1]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
simbol red
[3,3,1]+ 12.5
[3,3,1] 24
[4,3,1]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
simbol red
[4,3,1]+ 24.15
[3+,4,1] 24.10
[4,3,1] 48.36
[5,3,1]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
simbol red
[5,3,1]+ 60.13
[5,3,1] 120.2
[3,3,2]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
simbol red
[(3,3)+,2] 24
[3,3,2]+ 24
[3,3,2] 48
[4,3,2]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
simbol red
[(4,3)+,2+] 24
[4,(3,2)+] 48
[(4,3)+,2] 48
[4,3,2]+ 48
[4,3+,2] 48.22
[4,3,2] 96.5
[5,3,2]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
simbol red
[(5,3)+,2] 120.2
[5,3,2]+ 120.2
[5,3,2] 240
[31,1,1]: CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
simbol red
[31,1,1]+
= [1+,4,3,3]+
96.1
[31,1,1]
= [1+,4,3,3]
192
<[3,31,1]>
= [4,3,3]
384.1
[3[31,1,1]]
= [3,4,3]
1152.1
[3,3,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
simbol red
[3,3,3]+ 60
[3,3,3] 120.1
[[3,3,3]]+ 120.2
[[3,3,3]+] 120.1
[[3,3,3]] 240.1
[4,3,3]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
simbol red
[1+,4,3,3]+
= [3,31,1]+
96.1
[1+,4,3,3]
= [3,31,1]
192
[4,(3,3)+] 192
[4,3,3]+ 192
[4,3,3] 384
[3,4,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
simbol red
[3+,4,3+] 288.1
[3,4,3]+ 576.2
[3+,4,3] 576.1
[[3+,4,3+]] 576
[3,4,3] 1152.1
[[3,4,3]]+ 1152
[[3,4,3]+] 1152
[[3,4,3]] 2304
[5,3,3]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
simbol red
[5,3,3]+ 7200
[5,3,3] 14400

Prostorske grupe[uredi | uredi kodo]

Grupe ranga štiri definirajo tudi trirazsežne prostorske grupe:

ortorombski

[∞,2,∞,2,∞]
CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
simbol
[∞,2,∞,2,∞]
[∞,2,∞,2,∞]+
[∞+,2,∞,2,∞]
[∞+,2,∞+,2,∞]
[∞+,2,∞+,2,∞+]
[∞,2,∞,2+,∞]
[∞,2+,∞,2+,∞]
[(∞,2,∞)+,2,∞]
[(∞,2,∞)+,2,∞+]
trigonalni & heksagonalni
grupa simbol
[3,6,2,∞]
CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
[6,3,2,∞]
[6,3,2,∞]+
[6,3+,2,∞]
[(6,3)+,2,∞]
[6,3,2,∞+]
[6,3+,2,∞+]
[(6,3)+,2,∞+]
[1+,6,3,2,∞]
[1+,6,3,2,∞]+
[1+,6,3,2,∞+]
[(1+,6,3)+,2,∞]
[(1+,6,3)+,2,∞+]
[3[3],2,∞]
CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
[3[3],2,∞]
[3[3],2,∞]+
[3[3],2,∞+]
[(3[3])+,2,∞]
[(3[3])+,2,∞+]

tetragonalni

[4,4,2,∞]
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
[4,4,2,∞]
[4,4,2,∞]+
[(4,4)+,2,∞]
[4,4,2+,∞]
[4,4,2,∞+]
[(4,4)+,2+,∞]
[(4,4)+,2,∞+]
[4,4,2+,∞+]
[(4,4)+,2+,∞+]
[4+,4+,2+,∞]
[4,4+,2,∞]
[4,4+,2+,∞]
[4,4+,2,∞+]
[4,4+,2+,∞+]
kubični
grupa Coxeter prostorska grupa indeks
[4,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[4,3,4] (221) Pm3m 1
[4,3,4]+ (222) Pn3n 2
[4,3+,4] (223) Pm3n 2
[4,(3,4)+] (224) Pn3m 2
[4,3,4,1+] (225) Fm3m 2
[4,(3,4,1+)+] (226) Fm3c 4
[1+,4,3,4,1+] (227) Fd3m 4
[4,3,4,1+]+ (228) Fd3c 4
[[4,3,4]] [[4,3,4]] (229) Im3m
[[4,3,4]]+ (230) Ia3d
[[4,3+,4]]
[[4,3,4]]+
[4,31,1]
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
[4,31,1] = [4,3,4,1+] 2
[4,(31,1)+] = [4,(3,4,1+)+] 4
[1+,4,31,1] = [1+,4,3,4,1+] 4
[4,31,1]+ = [4,3,4,1+]+ 4
[1+,4,31,1]+ = [1+,4,3,4,1+]+ 2
<[4,31,1]> = [4,3,4] 1
[(3[4])]
CDel branch.pngCDel 3ab.pngCDel branch.png
[(3[4])] = [1+,4,31,1] 4
[(3[4])]+ = [1+,4,31,1]+ 2
<[(3,3,3,3)]> = [4,31,1] 2
<<[(3[4])]>> = [4,3,4] 1
[[(3[4])]]
[4[(3[4])]] = [[4,3,4]]

Grupa na premici[uredi | uredi kodo]

Rang štiri definira tudi trirazsežne grupe na premici:

Polfine (3D)
točkovna grupa grupa premic
Hermann-Mauguin Schönflies Hermann-Mauguin Offset type Wallpaper Coxeter
[∞h,2,pv]
paren n neparen n paren n neparen n IUC orbifold diagram
n Cn Pnq Helical: q p1 o Wallpaper group diagram p1 rect.svg [∞+,2,n+]
2n n S2n P2n Pn None p11g, pg(h) xx Wallpaper group diagram pg.svg [(∞,2)+,2n+]
n/m 2n Cnh Pn/m P2n None p11m, pm(h) ** Wallpaper group diagram pm.svg [∞+,2,n]
2n/m C2nh P2nn/m Zigzag c11m, cm(h) *x Wallpaper group diagram cm.svg [∞+,2+,2n]
nmm nm Cnv Pnmm Pnm None p1m1, pm(v) ** Wallpaper group diagram pm rotated.svg [∞,2,n+]
Pncc Pnc Planar reflection p1g1, pg(v) xx Wallpaper group diagram pg rotated.svg [∞+,(2,n)+]
2nmm C2nv P2nnmc Zigzag c1m1, cm(v) *x Wallpaper group diagram cm rotated.svg [∞,2+,2n+]
n22 n2 Dn Pnq22 Pnq2 Helical: q p2 2222 Wallpaper group diagram p2.svg [∞,2,n]+
2n2m nm Dnd P2n2m Pnm None p2mg, pmg(h) 22* Wallpaper group diagram pmg.svg [(∞,2)+,2n]
P2n2c Pnc Planar reflection p2gg, pgg 22x Wallpaper group diagram pgg rhombic.svg [∞+,2+,2n+]
n/mmm 2n2m Dnh Pn/mmm P2n2m None p2mm, pmm *2222 Wallpaper group diagram pmm.svg [∞,2,n]
Pn/mcc P2n2c Planar reflection p2mg, pmg(v) 22* Wallpaper group diagram pmg rotated.svg [∞,(2,n)+]
2n/mmm D2nh P2nn/mcm cikcak c2mm, cmm 2*22 Wallpaper group diagram cmm.svg [∞,2+,2n]

Tapetne grupe[uredi | uredi kodo]

Grupe z rangom štiri definirajo tudi nekatere dvorazsežne tapetne grupe:

Afine (2D ravnina)
IUC (orbifold) geo Coxeter
p1 (o) p1 [∞+,2,∞+]
p2 (2222) p2 [∞,2,∞]+
c2mm (2*22) c2 [∞,2+,∞]
p11g (xx) pg1 h: [∞+,(2,∞)+]
p1g1 (xx) pg1 v: [(∞,2)+,∞+]
p2gm (22*) pg2 h: [(∞,2)+,∞]
p2mg (22*) pg2 v: [∞,(2,∞)+]
IUC (Orbifold) Geo Coxeter
p11m (**) p1 h: [∞+,2,∞]
p1m1 (**) p1 v: [∞,2,∞+]
p2mm (*2222) p2 [∞,2,∞]
c11m (*x) c1 h: [∞+,2+,∞]
c1m1 (*x) c1 v: [∞,2+,∞+]
p2gg (22x) pg2g [∞+,2+,∞+]
c2mm (2*22) c2 [∞,2+,∞]

Opombe in sklici[uredi | uredi kodo]

  1. ^ The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF [1]