discrete plane symmetry groups

isometries of the plane

An isometry is a transformation which preserves distances (thus it also preserves angles, orthogonality and parallelism).
There are four different plane isometries:
    • the translations: no fixed point (identity if the vector is null)
    • the rotations: one fixed point, the center (identity if the angle is null, half turn if the angle is π)
    • the (pure) reflections: a line of fixed points (the axis)
    • the glide reflections: compositions of a reflection and a translation
       (they can be expressed as commutative compositions of a reflection and a translation parallel to the axis)
In the complex plane, with a,b∈ℂ and |a|=1, the isometries are defined by the two transformations:
z → az+b translation iff a=1, else rotation with center b/(1-a) and angle arg(a)
z → az+b pure reflection iff ab+b=0 (axis with argument arg(a)/2 and going through the point b/2),
else glide reflection (axis with vector (ab+b)/2 and going through b/2)

An isometry of the plane is the composition of at most three reflection; thus the translations (compositions of two reflections with parallel axis) and the rotations (compositions of two reflections with non parallel axis) preserve the orientation, and the pure and glide reflections change it (a glide reflection is the composition of three reflections with at least two non parallel axes).
The isometries of the plane form a non commutative group for the composition; the translations form a normal sub-group. Any isometry is the unique composition of a rotation or reflection with the origin as fixed point and a translation; applying a translation and then a rotation or a reflection corresponds to applying the rotation or the reflection first and then the translation by the rotated or reflected translation vector.

Below we'll focus on the groups G  of isometries without translations (rosette) or whose translations built a normal sub-group TG  generated over  by one element (frieze) or two independent elements (wallpaper). The lattice  is the orbit of the origin under TG  (set of images of the origin by the translations); it is symmetric with respect to the origin.
For any isometry f of G  we define the reduced isometry γ  with the origin as fixed point by
      if f(z)=az+b, then γ(z)=az,   and else f(z)=az+b, then γ(z)=az.
These reduced isometries  preserve the lattice  and form the reduced group ΓG  isomorph to the quotient group G/TG.

the two infinite families of rosette groups

A rosette group R  doesn't contain any translation (and thus no glide reflection):  TR ={id}.  Its lattice is reduced to the origin.
It is isomorphic to a cyclic or a dihedral group (examples: n=5).
gr r5 rn = <r> = ΓR  Cn  
only n rotations
r5m rnm = <r,m> = ΓR  Dn  
n rotations and n reflections

the seven frieze groups

frieze group F  contains a sub-group of translations generated by one translation:  TF =<t>=ℤ•t.  Its lattice is a set of points regularly spaced on a line Δ, and the reduced group ΓF  is a sub-group of the Klein's group {id,-id,m,μ} where Δ is the axis of μ.
•   Crystallographic notation:  fxy  where f means "frieze",  x=1,m  without/with vertical mirror reflections,  y=1,m,g,2  no other symmetry, a pure (m) or glide (g) reflection with an horizontal  axis, half turns (order 2 rotations r).
•   Orbifold notation (J.H. Conway): for an "infinite rotational symmetry around a line", 2 for a set of half-turns (the half-turns after * have their centers on intersections of reflection axes) and x for a glide reflection.
•   Notation of L. Fejes Tóth: Fnp
•   The names given by John Conway to these moves.

C1 D1 C2 D2
spinning hop
spinning jump
spinning sidle
no rotations half turns

group f11 f11 = <t>  C 
ΓF = {id}  C1
group f1m f1m = <t,μ>  C×D1 
ΓF = {id,μ}  D1
group fm1 fm1 = <t,m>=<m',m">  D 
ΓF = {id,m}  D1
group f1g f1g = <g>  C 
ΓF = {id,μ}  D1
group f12 f12 = <t,r> = <r',r">  D 
ΓF = {id,-id}  C2
group fmm fmm = <t,r,μ> = <t,m,μ>  D×D1 
ΓF = {id,-id,m,μ}  D2
group fmg fmg = <m,r> = <r,g> = <m,g>  D 
ΓF = {id,-id,m,μ}  D2
    The group fmg is the one of the sinusoid.
    The seven frameworks (symmetry axis and centers) of the friezes groups.

Above r, r' and r" denote central symmetries,  m, m' and m" vertical mirror  reflections,  μ and g the mirror reflection and a glide reflection with axis Δ.

the seventeen wallpaper groups

wallpaper group W  contains a sub-group of translations generated by two independent translations:  TW =<t1,t2>=ℤ•t1+ℤ•t2.
The reduced group ΓW  is finite with maximum order 12 and only the rotations of orders 2, 3 ,4 and 6 preserve the lattice.
The possible reduced groups  are thus the dihedral groups D6 and D4 and their sub-groups.
Important!  To a rotation of ΓW  corresponds, in W , a family of rotations of same order; to a reflection of ΓW  corresponds a family of reflections (whether all pure or whether all glide or whether both alternated) with parallel axes.

•   Crystallographic notation: p or c for the lattice (primitive or centered) and 1,m,g,n as for the frieze groups (rotation rn f highest order).
•   Orbifold notation(J.H. Conway): integer n for a set of rn rotations, * for a set of reflections, x for glide reflections (the half-turns after * have their centers on intersections of reflection axes).
•   Notation of L. Fejes Tóth: Wnp

C1 D1 C2 D2 C4 D4 C3 D3 C6 D6
no rotations only half-turns quarter turns only thirds of turns sixths of turns

The key role played by the "fixed lines for W " associated to the axes of the reflections of the reduced group ΓW, and thus by the families of reflections in W,  suggests a more simple and concise and more explicit notation (last two lines of the table above): p, g and a for each family of reflections (pure, glide, alternate), cn for the highest rotation's order n of the groups without reflections (if there are reflections n is the number of families, namely the sum of the exponents); we may add a sign - or + to specify that there are no rotations or rotations not centered on a reflection axis.

The name of each group ("pga" notation) is a link which opens a pop-up window with a tiling (table above) or a big sample of the pattern (drawings below).
On the graphics, in grey, the small polygons show the rotation's centers and orders, the full/dashed lines are the axes of the pure/glide reflections. In blue, a base of the lattice; in green, a generator set of the group.

wp group pm
p- = <two pll m, one pll t>
rectangular lattice, ΓW = D1
wp group pg
g- = <two pll g>
rectangular lattice, ΓW = D1
wp group cm
a- = <one m, one g pll>
rhombic lattice, ΓW = D1
wp group p1
c1=TW = <two t>
oblique lattice, Γw = C1
wp group pmm
p2 = <four m (rectangle)>
rectangular lattice, ΓW = D2
wp group pmg
pg = <one m, two r2>
rectangular lattice, ΓP = D2
wp group p2
c2= <three r2>
oblique lattice, ΓW = C2
wp group pgg
g2= <two ppd g>
rectangular lattice, ΓW = D2
wp group cmm
a2 = <two ppd m, one r2>
rhombic lattice, ΓW = D2
wp group p4
c4= <one r2, one r4>
square lattice, ΓW = C4
wp group p4m
p2a2 = <three m (triangle 45,45,90)>
square lattice, ΓW = D4
wp group p4g
g2a2 = <one m, one r4>
square lattice, ΓW = D4
wp group p3
c3= <deux r3>
hexagonal lattice, ΓW = C3
wp group p3m1
a3 = <three m (equilateral triangle)>
hexagonal lattice, ΓW = D3
wp group p31m
a3= <one m, one r3>
hexagonal lattice, ΓW = D3
wp group p6
c6= <one r2, one r3>
hexagonal lattice, ΓW = C6
wp group p6m
a6 = <three m (triangle 30,60,90)>
hexagonal lattice, ΓW = D6
The 17 frameworks (reflection's axes and rotation's centers) of the tiling's groups.

The method and the new results presented on this page are widely inspired from a research work by a dear friend, Georges Lion (1936-2014):
DISCRETE GROUPS OF PLANE ISOMETRIES - A new classification and their representations as Wallpaper groups. (Georges Lion, 2010)

Evgraf S. Fedorov described the 17 groups in 1891.
The Egyptian artists knew 12 of these types of pavings; the five missing groups are those which exhibit 3-fold symmetry.
Examples of 13 of these 17 types of paving appear on the mosaics of the Alhambra palace in Spain, near Granada (architecture of the Islamic Middle Ages where animal and human patterns where forbidden); the four missing groups are g-, c2+, g²+ and a³.
There are also many examples of paving with figurative drawings in Maurits Cornelis Escher's work.

Concerning the isohedral  or tile-transitive  tilings (for each pair of tiles there is an isometry in the group which sends one onto the other, the tiles are thus all identical), it's only in 1968 that H. Heesch described 28 types of tiles which pave the plane. This result, "forgotten" during about half a century, has been "cleaned" by John Conway and Xavier Hubaut (see last reference): the 28 types have been reduced to 19 because somes types are particular cases of others. For 15 of the 17 groups only one type of tile is associated to the group, and for each of the two groups with only glide reflections there are two.
You may view examples of tilings in the pop-up windows accessible from the table (a marked tile is necessary for a³).

the tilings with polygons

Which polygons tile the plane? All the triangles, the quadrilaterals (except the crossed), three families of hexagons, but also some types of convex pentagons. A fifteenth type of pentagon (bottom right) has just been discovered (August 2015); are there others? No! (Michael Rao - ENS Lyon, France - August 2017)
Only the fourteenth is unique (up to similarity); the others belong to families (at least one parameter).
With polygons with seven or more sides it's impossible.

pavings with polygons
these pictures come from here and here
a nice applet to explore the fifteen types

exercises: •  Draw different pavings built with 2×1 rectangles, then establish the group of each paving (take the pleasure to search before referring to a few examples).
•  There are 11 (semi)regular tilings with regular polygons: 3 regular with only one polygon, 6 with two polygons and 2 with three polygons. The corresponding groups are among the richest: some are obvious, can you find them all?
•  Which groups correspond to the 15 tilings with pentagons?
video: classer les pavages  found on Youtube (in French): Deux minutes pour... (19' - 65 Mb)
algorithms: tiling recognition  by Brian Sanderson  -  reconnaissance des pavages  by Nicolas Hannachi
references: •  DISCRETE GROUPS OF PLANE ISOMETRIES - A new classification and their representations as Wallpaper groups.
      by Georges Lion (2010)
•  The plane isometries, the frieze groups, the wallpaper groups ... on www.answers.com
•  A complete study of the plane isometries  (450 Mb PDF, in French)
•  Conway's orbifold notation by J.H.Conway (English and Italian)
•  The Discontinuous Groups of Rotation and Translation in the Plane  complete web pages by Xah Lee
•  Isometrica  by George Baloglou
•  Subgroups lattices for crystallographic groups  by Raymond F. Tennant
•  Tess is a program which allows you to achieve all types of symmetric drawings (used to generate the figures on this page)
•  with the applet Kali  one can use all the above groups (downloadable program)
•  collections of patterns and tilings arranged by their symmetry groups, and much more!
•  Brian Sanderson's Pattern Recognition Algorithm  - PDF version
•  Isohedrally compatible tilings  by Philip M. Maynard
•  Tiling the plane with congruent pentagons  by Doris Schattschneider
•  M.C. Escher: Vision of Symmetry  by Doris Schattschneider, Abrams, New-york, 1990-2004
•  Parcelles d'infini - Promenade au jardin d'Escher  the figurative tilings of Alain Nicolas (in French)
•  the tessellation world  of Makoto Nakamura deserves a visit
Groupes cristallographiques du plan  (in French) by Xavier Hubaut (an other approach with the rotation's centers) - copy of the pdf version
•  Les pavages du plan  (in French) by Xavier Hubaut: examples with drawings by M.C. Escher - copy of the pdf version

an aperiodic tiling by John Conway (1937-2020)

aperiodic tiling
Divide a right-angle triangle of sides 1, 2 and √5 into five isometric triangles (and similar to the big triangle). Enlarge (factor 2) and divide each of these five triangles in the same way; iterate the process.
We thus obtain a nice aperiodic tiling (no invariance by translation) whose triangles can take an infinity of different directions. Simplicity and beauty! Thank you John Conway.
    reference: deux minutes pour Jonh Conway (youtube, in French)

The Penrose tilings, based on the golden triangles, form another family of nice aperiodic tilings.
triangle 1-2-√5

tilings by Maurits Cornelis Escher

pavage Caire pavage Caire pavage Caire pavage Caire
The Cairo tiling in two drawings by M.C. Escher (1967)
poissons1 poissons2 poissons3
Fishes by M.C. Escher (1938, 1940 & 1943)

Circle Limit III  (woodcut by M.C. Escher - 1959)

This drawing also depicts a tiling, but here we are in the Poincaré disk, a model of the hyperbolic plane. Another world...

• Hyperbolic tessellations  by David E. Joyce
•  Hyperbolic planar tessellations  by Don Hatch

See also the non periodic Penrose tilings.

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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects February 2007
updated 01-06-2020