discrete plane symmetry groups

isometries of the plane

Below we'll focus on the groups G  of isometries without translations (rosette) or whose translations built a normal sub-group T_G  generated over ℤ by one element (frieze) or two independent elements (wallpaper). The lattice  is the orbit of the origin under T_G  (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 Γ_G  isomorph to the quotient group G/T_G.

the two infinite families of rosette groups

rn = <r> = ΓR ≡ Cn   only n rotations

rnm = <r,m> = ΓR ≡ Dn   n rotations and n reflections

the seven frieze groups

	f11 = <t> ≡ C∞  ΓF = {id} ≡ C1		f1m = <t,μ> ≡ C∞×D1  ΓF = {id,μ} ≡ D1
	fm1 = <t,m>=<m',m"> ≡ D∞  ΓF = {id,m} ≡ D1		f1g = <g> ≡ C∞  ΓF = {id,μ} ≡ D1
	f12 = <t,r> = <r',r"> ≡ D∞  ΓF = {id,-id} ≡ C2		fmm = <t,r,μ> = <t,m,μ> ≡ D∞×D1  ΓF = {id,-id,m,μ} ≡ D2
	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

•   Crystallographic notation: p or c for the lattice (primitive or centered) and 1,m,g,n as for the frieze groups (rotation r_n f highest order).
•   Orbifold notation(J.H. Conway): integer n for a set of r_n 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: W_n^p

The key role played by the "fixed lines for W " associated to the axes of the reflections of the reduced group Γ_W, 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), c_n 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.

p- = <two pll m, one pll t> rectangular lattice, ΓW = D1	g- = <two pll g> rectangular lattice, ΓW = D1	a- = <one m, one g pll> rhombic lattice, ΓW = D1
c1- =TW = <two t> oblique lattice, Γw = C1	p2 = <four m (rectangle)> rectangular lattice, ΓW = D2	pg = <one m, two r2> rectangular lattice, ΓP = D2
c2+ = <three r2> oblique lattice, ΓW = C2	g2+ = <two ppd g> rectangular lattice, ΓW = D2	a2 = <two ppd m, one r2> rhombic lattice, ΓW = D2
c4+ = <one r2, one r4> square lattice, ΓW = C4	p2a2 = <three m (triangle 45,45,90)> square lattice, ΓW = D4	g2a2 = <one m, one r4> square lattice, ΓW = D4
c3+ = <deux r3> hexagonal lattice, ΓW = C3	a3 = <three m (equilateral triangle)> hexagonal lattice, ΓW = D3	a3+ = <one m, one r3> hexagonal lattice, ΓW = D3
c6+ = <one r2, one r3> hexagonal lattice, ΓW = C6	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 convex polygons

these pictures come from here and here
a nice applet to explore the fifteen types
Wikipedia is also a good reference

other tilings of the plane    

home page

convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects

February 2007 updated 18-06-2023