1. Periodic patterns

2. Periodic patterns
Periodic patterns are infinite by definition, but only a limited part can be shown.
The repetitive unit of the periodic pattern is called motif.
Motif
Motif
Motif (carbon atom)
Pattern = layer in graphite

3. Translation lattice
By translating a motif in 1, 2 or 3 dimension an infinite 1-,2- or 3-D periodic pattern is created.
Motif
Translation vectors
Start/Endpoints of
translation vectors =
Lattice points
shortest possible translations
shown. The start point of the
first two translation vectors
can be freely choosen
Translations move the individual motifs into coincidence with an adjacent motif (remember the
pattern is infinite!). The start situation can not be distinguished from the end situation =>
translations are symmetry elements! The lattice points are sets of points with identical surroundings.

4. Unit cell
unit cell
A box delimited by four lattice points is called a unit cell. An infinite number of unit cells are possible. Usually the cell bounded by the shortest translations is chosen.
unit cell with motif content
The periodic pattern can be created by translating the unit cell content by the translation vectors delimiting the unit cell.

5. Structure + = Motif Lattice Periodic Pattern, structure
Overall content:
The origin of the lattice can be chosen arbitrarily. Changing the origin of the lattice will not change the overall content of the unit cell, only its arrangement.

6. Glide mirror I
A 2-D periodic pattern can have all proper rotation axes, mirrors and the inversion center as symmetry operations. New symmetry operations are translations and the combination of mirrors with translations = glide mirrors.
Translations => combination with mirror planes = new symmetry element: glide mirror planes
Translation
Mirror operation
periodicity in this direction
graphic symbol
g written symbol

7. Glide mirror II
A periodic pattern has an infinite number of symmetry elements, because it is infinite. The symmetry elements are automatically multiplied by the translation which is also a symmetry element. 2-D periodic patterns have rotation axes (2,3,4,6), mirrors and glide mirrors as symmetry elements. The symmetry content can be separated into the symmetry of the lattice and the symmetry of the motif. The symmetry of the pattern can be equal or lower than the symmetry of the lattice/motif.

8. The 5 planar lattices I  ≠ 90°  = 90°  = 90°
Only 5 symmetrically distinct planar lattices can be distinguished: a  b
a ≠ b
≠ 90° p2 a  b
a ≠b
= 90°
p2mm a  b
a ≠ b
= 90°
c2mm
The left "primitive" unit cell is the smallest cell, but the mirrors have an odd orientation relative to the orientation of the cell edges. The "centered" cell on the right reflects better the symmetry of the lattice and is preferred.

9. The 5 planar lattices II  = 60°  = 90° a b a = b p6mm a b a = b

10. Motif symmetry The 17 plane groups 1 2 m 2mm 4 4mm 3 3m 6 6mm
Plane lattices + motif symmetry = 17 possible symmetry combinations
=> 17 plane groups
The 5 lattices have the highest symmetry for each group. The addition of
the motif to the lattice can only reduce the symmetry, but not increase it.

11. The plane group cm Lattice symmetry Motif symmetry Pattern symmetry m
compared to the lattice symmetry,
all 2-fold axes and the vertical
mirror planes are lost
c2mm

12. The 17 plane groups I

13. The 17 plane groups II