We can build all the possible types of pattern on a two dimensional surface using just four symmetry operations:
- Translation: there will exist two directions in which it is possible to pick up the entire surface, move it by a repeat length along this direction and drop it to find that the pattern is unchanged. This operation (often denoted by the symbol "t") can be repeated as many times as we like in either direction with the same result. We can therefore always draw on our pattern a primitive cell representing the way a pattern can be constructed by repeating the cell. The primitive cell and the two types of translation symmetry have to be present in every wallpaper pattern.
- Rotation: We may also define a subset of wallpaper patterns with rotation centres. We can pick up and rotate the entire pattern around a rotation centre by an angle then drop it to find exactly the same pattern. It can be shown that rotation symmetries can only appear with rotation angles of 60o, 90o, 120o or 180o (that is, six-fold, four-fold, three-fold or two-fold rotations). If a rotation centre is present in one primitive cell it has to appear in the same place in all the primitive cells.
- Mirror reflections: any pattern is reflected across the mirror plane.
- Glides: - special type of translation. The name is based on the marks made by the action of ice skaters - push forwards (a translation) and the foot goes left, push forward and then the foot goes right. (For those familiar with chess, this is rather like the "knights move".) On the whole, I find that glides are the operations hardest to visualise or recognise.
It is important to understand that the group to which a pattern belongs is determined by the collection of symmetry operations that are possible on that pattern. Some of the symmetry groups can be expressed as patterns in more than one Bravais lattice - for example, the "p2" group (which has a single two-fold rotation operation in addition to the translations of the lattice) can be expressed on all the Bravais lattices, while "p4" (which has a four-fold rotation) can only be expressed in the square primitive cell. It can be shown, as a fairly straightforward, but not in any way trivial, exercise in Group Theory that there are exactly 17 possible combinations of symmetry operations in the plane. These are illustrated below (as I get round to adding the relevant illustration for each group!).
The conventions for naming groups are not entirely easy to explain and in fact there are several different conventions See this Wikipedia article for an explanation and further references, such as they are. In practice one can just accept the label without explanation.
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The least symmetric type of primitive cell might therefore look something like the example on the right: it is a a regular parallelogram (so it can be repeated indefinitely in either the horizontal direction or a direction a little clockwise of vertical). The unsymmetrical motif, however, means that there are no possible reflection planes or rotational symmetry planes. This is the "P1" group (in an Oblique Bravais cell). Note that it is possible to create examples of P1 patterns in all the other Bravais cells. |
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Without the triangle motif we could, of course, find two-fold rotation centres either at any vertex, or the centre of mass of the cell. We can also add this symmetry by reproducing, inverting and carefully positioning the motif,s as in this second example. This is and example of the P2 group (in an oblique Bravais cell). One again, we can create examples of P2 patterns in all the other Bravais Cells. |
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The pg group (in a rectangular primitive cell only). There is, however, an alternative example in which the glide axis is parallel to the short side of the rectangle. |
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The pmg group always appears in a rectangular primitive cell. There is, however, an alternative example in which the reflection axes are along the long side of the rectangle and the glide planes are parallel to the short side. |
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The pm group always appears in a rectangular primitive cell. There is, however, an alternative example in which the reflection axes are along the long side of the rectangle and the glide planes are parallel to the short side. |
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The pmm group has two mirror planes at 90 degrees to each other in a rectangular primitive cell. t is also possible to represent this group in the square primitive cell. |
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The pgg group: always appears in a rectangular primitive cell. There is, however, an alternative example is possible in which the reflection axes are parallel to the short side of the rectangle and the glide planes parallel to the long side. It is also possible to represent this group in the square primitive cell. |
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The p3 group has three-fold rotation centres, and is only expressible in the hexagonal Bravais cell. |
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| The p3m1 group introduces mirror planes through the three-fold rotation centres. |  |
| The p31m group has fewer mirror planes. |  |
| The p4 group has only a four fold rotation axis. |  |
| The p4g group also introduces glide planes. |  |
| The p4m group has a number of mirror planes. |  |
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The cmm group (in a square cell) has two mirror planes at right angles to each other. This group can also be expressed in the hexagonal unit cell. |
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The cm group is expressed in a hexagonal unit cell. It has one mirror plane and two glide planes. There is, however, an alternative example in which the reflection axes, glide planes and motifs are rotated 90 degrees (i.e. running parallel to the short diagonal). |
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The example on the right, the p6 group, on the other hand, is a cell with rather a large amount of symmetry, including six-fold rotations. Rotation centres are denoted by the red polygons (hexagonal for six-fold rotation, triangle for three-fold and parallelogram for two-fold). Mirror planes are show by dotted lines. (It is called "hexagonal" because of the six-fold rotation centres - and one could build hexagonal tiles from combinations of this primitive cell. |
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The example on the right, the p6m group, on the other hand, is a cell with rather a large amount of symmetry, including six-fold rotations. Rotation centres are denoted by the red polygons (hexagonal for six-fold rotation, triangle for three-fold and parallelogram for two-fold). Mirror planes are show by dotted lines. (It is called "hexagonal" because of the six-fold rotation centres - and one could build hexagonal tiles from combinations of this primitive cell. Patterns showing P6 symmetry can only be created on the Hexagonal Bravais cell. |
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Symmetry arises from placing restrictions on the dimensions and angles of the primitive cell, and also the location of pattern motifs within the cell. This is extremely well explained in the book Creating Symmetry by Frank Farris. (See my Books page.) There are a number of ways of classifying the symmetry of patterns on a 2D surface. (See, for example, the Wallpaper Group page in Wikipedia.) I am labelling my images with the scheme explained by Farris, in which we first look at the number of rotational symmetries and then the mirror planes. So, the first example pattern in the table above is a "p1" pattern, the second a "p2", the third a "p6" - but we can also create "p3" and "p4" patterns, which in practice frequently turn out to be the most visually interesting, and this website contains many examples.