Coxeterjeva notacija

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 (1930 – 2017).

Zrcalne grupe

Več informacij: točkovna grupa

Za Coxeterjeve grupe, ki so definirane s čistim zrcaljenjem, je neposredna povezava med notacijo z oklepaji in Coxeter-Dinkinovimi 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 A_n predstavljena z [3^n-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 trikotniško 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] 2 BC2 [4] 2 H2 [5] 2 G2 [6] 2 I2(p) [p] 3 H3 [5,3] 3 A3 [3,3] 3 BC3 [4,3] 4 A4 [3,3,3] 4 BC4 [4,3,3] 4 D4 [31,1,1] 4 F4 [3,4,3] 4 H4 [5,3,3] n An [3n-1] .. n BCn [4,3n-2] ... n Dn [3n-3,1,1] ... 6 E6 [32,2,1] 7 E7 [33,2,1] 8 E8 [34,2,1]

Afine Coxeterjeve grupe simbol grupe notacija z oklepaji Coxeterjev graf ${\displaystyle {\tilde {I}}_{1}}$ [∞] ${\displaystyle {\tilde {A}}_{2}}$ [3[3]] ${\displaystyle {\tilde {C}}_{2}}$ [4,4] ${\displaystyle {\tilde {G}}_{2}}$ [6,3] ${\displaystyle {\tilde {A}}_{3}}$ [3[4]] ${\displaystyle {\tilde {B}}_{3}}$ [4,31,1] ${\displaystyle {\tilde {C}}_{3}}$ [4,3,4] ${\displaystyle {\tilde {A}}_{4}}$ [3[5]] ${\displaystyle {\tilde {B}}_{4}}$ [4,3,31,1] ${\displaystyle {\tilde {C}}_{4}}$ [4,3,3,4] ${\displaystyle {\tilde {D}}_{4}}$ [ 31,1,1,1] ${\displaystyle {\tilde {F}}_{4}}$ [3,4,3,3] ${\displaystyle {\tilde {A}}_{n}}$ [3[n+1]] ... or ... ${\displaystyle {\tilde {B}}_{n}}$ [4,3n-2,31,1] ... ${\displaystyle {\tilde {C}}_{n}}$ [4,3n-1,4] ... ${\displaystyle {\tilde {D}}_{n}}$ [ 31,1,3n-3,31,1] ... ${\displaystyle {\tilde {E}}_{6}}$ [32,2,2] ${\displaystyle {\tilde {E}}_{7}}$ [33,3,1] ${\displaystyle {\tilde {E}}_{8}}$ [35,2,1]

Kompaktne hiperbolične Coxeterjeve grupe simbol grupe notacija z oklepaji Coxeterjev graf [p,q] z 2(p+q)<pq [(p,q,r)] z p+q+r>9 ${\displaystyle {\bar {BH}}_{3}}$ [4,3,5] ${\displaystyle {\bar {K}}_{3}}$ [5,3,5] ${\displaystyle {\bar {J}}_{3}}$ [3,5,3] ${\displaystyle {\bar {DH}}_{3}}$ [5,31,1] ${\displaystyle {\widehat {AB}}_{3}}$ [(3,3,3,4)]   ${\displaystyle {\widehat {AH}}_{3}}$ [(3,3,3,5)]   ${\displaystyle {\widehat {BB}}_{3}}$ [(3,4,3,4)] ${\displaystyle {\widehat {BH}}_{3}}$ [(3,4,3,5)] ${\displaystyle {\widehat {HH}}_{3}}$ [(3,5,3,5)] ${\displaystyle {\bar {H}}_{4}}$ [3,3,3,5] ${\displaystyle {\bar {BH}}_{4}}$ [4,3,3,5] ${\displaystyle {\bar {K}}_{4}}$ [5,3,3,5] ${\displaystyle {\bar {DH}}_{4}}$ [5,3,31,1] ${\displaystyle {\widehat {AF}}_{4}}$ [(3,3,3,3,4)]

Razvrščanje po rangu

Coxeterjeve grupe se lahko razvrstijo po njihovem rangu, kar je število vozlov v Coxeterjevem grafu. Struktura teh grup je dana tudi z vrstami tipov abstraktnih grup. Tukaj so abstraktne diedrske grupe predstavljene z Dih_n. Ciklične grupe pa so predstavljene z Z_n tako, da je Dih₁ = Z₂.

Grupe z rangom ena

V eni razsežnosti dvostranska grupa (bilateralna grupa) [ ] predstavlja posamezno zrcalno simetrijo. To je Dih₁ ali simetrija Z₂, reda 2. Predstavljena je kot Coxeter-Dinkinov diagram s samo enim vozlom . 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	[ ]		2	zrcalna grupa

Grupe z rangom dva

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 z redom 4

grupa	intl	orbiterost	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	diedrska: ciklična z zrcaljenji. Abstraktna grupa Dihn, diedrska grupa.
afine
Z∞	∞	∞∞	[∞]+	∞	ciklično: apeirogonalna grupa. Abstraktna grupa Z∞, grupa celih števil pod seštevanjem.
Dih∞	∞m	*∞∞	[∞]	∞	diedrsko: vzporedna zrcaljenja. Abstratna neskončna diedrska grupa Dih∞.
hiperbolične
Z∞			[πi/λ]+	∞	psevdogonalna grupa
Dih∞			[πi/λ]	∞	polna psevdogonalna grupa

Grupe z rangom tri

Več informacij: Seznam grup sferne simetrije‎ in Seznam grup ravninske simetrije

končne
intl* geo [1] orbiterost 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 2× 3× 4× 5× 6× n× 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 orbiterost 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 (orbiterost) geo Schönflies Coxeter red n nn n Cn [1,n]+ n nm *nn n Dn [1,n] 2n IUC (orbiterost) geo Schönflies Coxeter p1 ∞∞ p1 C∞ [1,∞]+ p1m1 *∞∞ p1 C∞v [1,∞]	IUC (orbiterost) 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 (orbiterost) 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 (orbiterost) 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

Točkovne grupe

Grupe ranga 4 definirajo štirirazsežne točkovne grupe:

Končne grupe

[ ]: simbol red [1]+ 1.1 [1] = [ ] 2.1 [2,1,1]: simbol red [2,1,1]+ 2.1 [2,1,1] 4.1 [2,2,1]: 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]: 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]: 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]: 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]: 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]: 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]: 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]: 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]: 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]: 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]]: 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]]: 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]: simbol red [3,3,1]+ 12.5 [3,3,1] 24 [4,3,1]: simbol red [4,3,1]+ 24.15 [3+,4,1] 24.10 [4,3,1] 48.36 [5,3,1]: simbol red [5,3,1]+ 60.13 [5,3,1] 120.2 [3,3,2]: simbol red [(3,3)+,2] 24 [3,3,2]+ 24 [3,3,2] 48 [4,3,2]: 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]: simbol red [(5,3)+,2] 120.2 [5,3,2]+ 120.2 [5,3,2] 240 [31,1,1]: 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]: 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]: 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]: 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]: simbol red [5,3,3]+ 7200 [5,3,3] 14400

Prostorske grupe

Grupe ranga štiri definirajo tudi trirazsežne prostorske grupe:

ortorombski [∞,2,∞,2,∞] 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,∞] [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,∞] [3[3],2,∞] [3[3],2,∞]+ [3[3],2,∞+] [(3[3])+,2,∞] [(3[3])+,2,∞+]

tetragonalni [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,∞+] [4,4+,2+,∞+]

kubični grupa Coxeter prostorska grupa indeks [4,3,4] [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] [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])] [(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

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	orbiterost	diagram
n		Cn	Pnq		Helical: q	p1	o		[∞+,2,n+]
2n	n	S2n	P2n	Pn	None	p11g, pg(h)	xx		[(∞,2)+,2n+]
n/m	2n	Cnh	Pn/m	P2n	None	p11m, pm(h)	**		[∞+,2,n]
2n/m		C2nh	P2nn/m		Zigzag	c11m, cm(h)	*x		[∞+,2+,2n]
nmm	nm	Cnv	Pnmm	Pnm	None	p1m1, pm(v)	**		[∞,2,n+]
			Pncc	Pnc	Planar reflection	p1g1, pg(v)	xx		[∞+,(2,n)+]
2nmm		C2nv	P2nnmc		Zigzag	c1m1, cm(v)	*x		[∞,2+,2n+]
n22	n2	Dn	Pnq22	Pnq2	Helical: q	p2	2222		[∞,2,n]+
2n2m	nm	Dnd	P2n2m	Pnm	None	p2mg, pmg(h)	22*		[(∞,2)+,2n]
			P2n2c	Pnc	Planar reflection	p2gg, pgg	22x		[∞+,2+,2n+]
n/mmm	2n2m	Dnh	Pn/mmm	P2n2m	None	p2mm, pmm	*2222		[∞,2,n]
			Pn/mcc	P2n2c	Planar reflection	p2mg, pmg(v)	22*		[∞,(2,n)+]
2n/mmm		D2nh	P2nn/mcm		cikcak	c2mm, cmm	2*22		[∞,2+,2n]

Tapetne grupe

Grupe z rangom štiri definirajo tudi nekatere dvorazsežne tapetne grupe:

Afine (2D ravnina)

IUC (orbiterost) 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 (orbiterost) 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+,∞]