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This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane:

Contents

Rosette groupsEdit

There are two families of discrete two-dimensional point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group.

Family Intl
(orbifold)
Schön. Geo [1]
Coxeter
Order Examples
Cyclic symmetry n
(n•)
Cn n
[n]+
   
n  
C1, [ ]+ (•)
 
C2, [2]+ (2•)
 
C3, [3]+ (3•)
 
C4, [4]+ (4•)
 
C5, [5]+ (5•)
 
C6, [6]+ (6•)
Dihedral symmetry nm
(*n•)
Dn n
[n]
   
2n  
D1, [ ] (*•)
 
D2, [2] (*2•)
 
D3, [3] (*3•)
 
D4, [4] (*4•)
 
D5, [5] (*5•)
 
D6, [6] (*6•)

Frieze groupsEdit

The 7 frieze groups, the two-dimensional line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups. The yellow regions represent the infinite fundamental domain in each.

[1,∞],      
IUC
(orbifold)
Geo Schönflies Coxeter Fundamental
domain
Example
p1
(∞•)
p1 C [1,∞]+
     
   
 
p1m1
(*∞•)
p1 C∞v [1,∞]
     
   
 
[2,∞+],       
IUC
(orbifold)
Geo Schönflies Coxeter Fundamental
domain
Example
p11g
(∞×)
p.g1 S2∞ [2+,∞+]
     
   
 
p11m
(∞*)
p. 1 C∞h [2,∞+]
     
   
 
[2,∞],      
IUC
(orbifold)
Geo Schönflies Coxeter Fundamental
domain
Example
p2
(22∞)
p2 D [2,∞]+
     
   
 
p2mg
(2*∞)
p2g D∞d [2+,∞]
     
   
 
p2mm
(*22∞)
p2 D∞h [2,∞]
     
   
 

Wallpaper groupsEdit

The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular).

The p1 and p2 groups, with no reflectional symmetry, are repeated in all classes. The related pure reflectional Coxeter group are given with all classes except oblique.

Square
[4,4],      
IUC
(Orb.)
Geo
Coxeter Fundamental
domain
p1
(°)
p1
 
p2
(2222)
p2
[4,1+,4]+
     
[1+,4,4,1+]+
     
 
pgg
(22×)
pg2g
[4+,4+]
     
 
pmm
(*2222)
p2
[4,1+,4]
     
[1+,4,4,1+]
     
 
cmm
(2*22)
c2
[(4,4,2+)]
    
 
p4
(442)
p4
[4,4]+
     
 
p4g
(4*2)
pg4
[4+,4]
     
 
p4m
(*442)
p4
[4,4]
     
 
Rectangular
[∞h,2,∞v],        
IUC
(Orb.)
Geo
Coxeter Fundamental
domain
p1
(°)
p1
[∞+,2,∞+]
     
 
p2
(2222)
p2
[∞,2,∞]+
       
 
pg(h)
(××)
pg1
h: [∞+,(2,∞)+]
       
 
pg(v)
(××)
pg1
v: [(∞,2)+,∞+]
       
 
pgm
(22*)
pg2
h: [(∞,2)+,∞]
       
 
pmg
(22*)
pg2
v: [∞,(2,∞)+]
       
 
pm(h)
(**)
p1
h: [∞+,2,∞]
       
 
pm(v)
(**)
p1
v: [∞,2,∞+]
       
 
pmm
(*2222)
p2
[∞,2,∞]
       
 
Rhombic
[∞h,2+,∞v],        
IUC
(Orb.)
Geo
Coxeter Fundamental
domain
p1
(°)
p1
[∞+,2+,∞+]
       
 
p2
(2222)
p2
[∞,2+,∞]+
       
 
cm(h)
(*×)
c1
h: [∞+,2+,∞]
       
 
cm(v)
(*×)
c1
v: [∞,2+,∞+]
       
 
pgg
(22×)
pg2g
[((∞,2)+)[2]]
     
 
cmm
(2*22)
c2
[∞,2+,∞]
       
 
Parallelogrammatic (oblique)
p1
(°)
p1
 
p2
(2222)
p2
 
Hexagonal/Triangular
[6,3],       / [3[3]],    
p1
(°)
p1
 
p2
(2222)
p2
[6,3]Δ  
cmm
(2*22)
c2
[6,3]  
p3
(333)
p3
[1+,6,3+]
     
[3[3]]+
   
 
p3m1
(*333)
p3
[1+,6,3]
     
[3[3]]
   
 
p31m
(3*3)
h3
[6,3+]
     
 
p6
(632)
p6
[6,3]+
     
 
p6m
(*632)
p6
[6,3]
     
 

Wallpaper subgroup relationshipsEdit

Subgroup relationships among the 17 wallpaper group[2]
o 2222 ×× ** 22× 22* *2222 2*22 442 4*2 *442 333 *333 3*3 632 *632
p1 p2 pg pm cm pgg pmg pmm cmm p4 p4g p4m p3 p3m1 p31m p6 p6m
o p1 2
2222 p2 2 2 2
×× pg 2 2
** pm 2 2 2 2
cm 2 2 2 3
22× pgg 4 2 2 3
22* pmg 4 2 2 2 4 2 3
*2222 pmm 4 2 4 2 4 4 2 2 2
2*22 cmm 4 2 4 4 2 2 2 2 4
442 p4 4 2 2
4*2 p4g 8 4 4 8 4 2 4 4 2 2 9
*442 p4m 8 4 8 4 4 4 4 2 2 2 2 2
333 p3 3 3
*333 p3m1 6 6 6 3 2 4 3
3*3 p31m 6 6 6 3 2 3 4
632 p6 6 3 2 4
*632 p6m 12 6 12 12 6 6 6 6 3 4 2 2 2 3

See alsoEdit

NotesEdit

  1. ^ The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF [1]
  2. ^ Coxeter, (1980), The 17 plane groups, Table 4

ReferencesEdit

  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 (Orbifold notation for polyhedra, Euclidean and hyperbolic tilings)
  • On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
  • 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 [2]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
  • Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9.
  • N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 12: Euclidean Symmetry Groups

External linksEdit