There are 17 ways to arrange a motif regularly in a plane.
Figure 1: the seventeen possible plane symmetries
Already thousands of years these repetition schemes are applied in decorative art. Yet it was not until the 19th century, with the rise of crystallography, that a scientific classification and description (Federov) of the seventeen toke place. Each of the seventeen arrangements is composed in a specific way out of one or more of four types of 'spatial transformation': shift, rotation, reflection and slide-reflection. These transformations can be conceived as 'symmetrical operators' which are 'active' in de plane in the plane. The seventeen can be classified in terms of the occurence of (combinations of) these four operators.
Figure 2: Spatial transformations
The most simple type is that in which occur shifts only. The regular repetition of the shift in two directions results in a 'stacking' of parallellograms, each of which can be seen as the "repeat-unit" containing all the information about the pattern (motif).
Figure 3: Stacking of parallellograms
the form of the repeatunit
When certain symmetrical operators are active, the form of the repeatunit 'grades up' from a parallellogram to a rhomb, a rectangle or a square.
The shiftpattern and the 'repeatunits' which are generated by it, also lies at the base of the remaining sixteen types of symmetry. But these sixteen contain - besides the shift- one or more of the other three 'symmetrical operators', which makes there structure more complex (independant of the characteristics of the motif used). For example in the symmetrical arrangement beneath, shift is combined with rotation. It is customary to indicate the additional symmetrical operators within the parrallelogram which represents the repeatunit.
Figure 4: symmetrical operators
number of orientations
In most types of plansymmetrie there is a repetition of the basic 'motif' within each repeatunit (see right picture figure 4). In these cases the 'motif' occurs in different orientations in space, within a repeatunit. Together these motifs in different orientations contibute to a higher order motif. So the repatunit can be conceived as divided in sub-areas, in each of which is concnetrated all the structural information about the motie, exept the repetionpattern which ' rules in the repeatunit. The sub-areas are named 'generating regio' on the symmetrical operators which are active in the plane symmtrie.
The following scheme gives an overview of the characteristics of the seventeen types of plane-symmetries.For every type the repeatunit is also represented in a grafical way.
(Click on the type-indications.)
of the motif
p1 1 no yes 1 pg 1 no yes 2 pm 1 yes no 2 cm 1 yes yes 2 p2 2 no no 2 pgg 2 no yes 4 pmg 2 yes yes 4 pmm 1 yes no 4 cmm 2 yes yes 4 p3 3 no no 3 p31m 3 yes yes 6 p3m1 3 yes yes 6 p4 4 no no 4 p4g 4 yes yes 8 p4m 4 yes yes 8 p6 6 no no 6 p6m 6 yes yes 12
Cultural historical notes
Through the ages, there have been cultures in which this wondrous world appealed more to the imagination then in the current culture and brought artists to great artistic achievement. Already the craftsmen at the time of the great Pharaoh's applied all 17 in the affixing of decorations within the tombs of the elite of their time. Perhaps not so mathematically elaborate, they still were able to create beautiful patterns on ground of combinations of shift, rotation, reflection en slide refelction. Often the spiral was a basic element in the motifs they used. In fact these ancient Egyptians practiced already a form of geometrical art, long before the cultures that we associate with such an art form - Greek and Islamic - had their heyday.
Absolute masters at playing with the laws of symmetry,were the Islamic decoration artists during the prime time of the Islamic culture Their game with lines within symmetry laws, resulting in a colorful splendor of stars, rosettes and polygons, is often literally dizzying. Never reached geometric decoration art greater beauty.
And then there was in the middle of the 20th century the Dutchman Maurits Escher, who on his own caused a culture flow and astonished the world with his masterpieces in the area of regular plane division. Worldwide one is acquinted with the around each other winding reptiles legs, sliding mirroring horses and the swarming of interlocking insect legs.
All these famous ancestors were in some way playing with the 'regimes' wich reign in the different plane symmetries. in plane symmetries.