Wallpaper group

The planar crystallographic groups are the symmetry groups of periodic patterns or tilings of the Euclidean plane. There are exactly 17 such groups. You meet in three-dimensional space, the 230 crystallographic space groups.

In terms of group theory, the groups consist of the set of all Kongruenzabbildungen that map the pattern onto itself, together with the composition of mappings as the group operation.

View all 17 symmetry forms occur in the arabesques in the Alhambra in Spain.

  • 4.1 Group p1
  • 4.2 group p2
  • 4.3 Group pm
  • 4.4 pg group
  • 4.5 cm group
  • 4.6 Group pmm
  • 4.7 Group pmg
  • 4.8 pgg group
  • 4.9 cmm group
  • 4:10 group p4
  • 4:11 Group P4M
  • 4:12 Group P4g
  • 4:13 group p3
  • 4:14 Group p3m1
  • 4:15 Group p31m
  • 4:16 group p6
  • 4:17 Group P6M

Symmetry elements

A periodic pattern may include combinations of the following basic elements of symmetry:

  • 2- merous, ie a rotation of 180 ° and a point reflection
  • 3- merous, ie a rotation of 120 °
  • 4- merous, ie a rotation of 90 °
  • 6 trifoliate, so rotation through 60 °

Other rotations than those listed are impossible. The reason is that (apart from the twofold rotation) for each symmetry group, a periodic tiling of the plane is one with regular polygons corresponding Zähligkeiten. And a tiling with pentagons, for example, is impossible, because due to the interior angle sum of a interior angle of 108 ° results, so that such a tiling would not go up at the corners. In non-Euclidean geometries, however, also symmetry groups with other Zähligkeiten are possible.

Of course, a 4 -fold rotational symmetry implies a two -fold, as well as implied a 6 ​​-fold both a 3 -fold and a 2 -fold. It is usually given only the highest value for each center of rotation.

Any periodic pattern can be generated by these operations are repeatedly applied to a restricted unit cell to the full level is cradled. By definition, the symmetry group of a periodic pattern always contains two linearly independent translations. This also makes it possible to generate a translational displacement merely by repeated the whole cell sample. The translatory cell case includes one or more copies of the elementary cell.

Notation

Orbifold notation

The properties of a symmetry group can also be described by the so-called orbifold notation.

  • N number (2, 3, 4, 6 ) denote an n- zähliges center of rotation.
  • A * is a mirror axis. Digits that are faced with a *, are off the mirror axes
  • Digits that follow an * are on the mirror axis

Quick Overview

List

The elements specified in the structure diagrams are marked as follows:

Different elements of the equivalence classes are characterized by different colors and rotations.

The yellow shaded area indicates a unit cell, the entire area depicted a translational cell.

Group p1

  • Orbifold notation: ∘ 1
  • This group has only shift as the only form of symmetry.

Group p2

  • Orbifold notation: 2222.
  • This group has four classes of point mirror centers. This twofold rotation is in addition to the translational symmetry the only form.

Group pm

  • Orbifold notation: **.
  • This group has two parallel mirror lines. There is no rotational symmetry.

Group pg

  • Orbifold notation: × ×.
  • This group has two parallel Gleitspiegelachsen. There is no rotational symmetry.

Group cm

  • Orbifold notation: * ×.
  • This group has alternately parallel to each other and mirror axes Gleitspiegelachsen.

Group pmm

  • Orbifold notation: * 2222.
  • This group is characterized by mutually perpendicular axes of reflection. Is two-fold rotation centers arise at the intersection of two mirror axes. There are four classes of turning centers and four classes of mirror axes.

Group pmg

  • Orbifold notation: 22 *.
  • Here there is a single class of mirror axes and perpendicular to two different classes of Gleitspiegelachsen on which two-fold rotation centers arise.

Pgg group

  • Orbifold notation: 22 ×.
  • This group has no easy axis of symmetry, however, two mutually perpendicular Gleitspiegelachsen, as well as two classes of point mirror centers.

Cmm group

  • Orbifold notation: 2 * 22
  • This group contains two classes of mirror axes which are perpendicular to each other, with the two-fold rotational centers at the intersections. An additional class of two-fold rotational centers located off the mirror axes. This also leads to two classes of Gleitspiegelachsen.

Group p4

  • Orbifold notation: 442
  • This group does not have any form of axial symmetry. Distinguishing are fourfold rotations for which there are two classes of centers. Between arise twofold axis centers.

Group P4M

  • Orbifold notation: * 442
  • This group is also referred to as p4mm.

Group P4g

  • Orbifold notation: 4 * 2
  • This group is also referred to as p4gm.

Group p3

  • Orbifold notation: 333

Group p3m1

  • Orbifold notation: * 333

Group p31m

  • Orbifold notation: 3 * 3

Group p6

  • Orbifold notation: 632

Group P6M

  • Orbifold notation: * 632
  • This group is also referred to as p6mm.
Translation (geometry) Reflection (mathematics) Glide reflection Internal and external angle Parallelogram
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