By Hermann Weyl
Defines symmetry via a dialogue of its many makes use of in a big range of fields either educational and natural.
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Closer investigation has shown that, while there are 10 possibilities for F, there are exactly 17 essentially different possibilities for the full group of congruences A. Thus there are 17 essentially different kinds of symmetry possible for a two-dimensional ornament with double infinite rapport. Examples for all 17 groups of symmetry are found among the decorative patterns of antiquity, in particular among; the Egyptian ornaments. One can hardly overestimate the depth of geometric imagination and inventiveness reflected in these patterns.
This has never been proved. Let us now return from the three-dimensional space to the two-dimensional plane and engage in a more systematic investigation of symmetry with double infinite rapport. First we have to make this notion precise. As was mentioned before, the translations, the parallel displacements of a plane form a group. A translation a can be completely described by fixing the point A' into which it carries a given point A. The translation or vector BB' is the same as the translation --* ---t --4 AA' if BB' is parallel to AA' and of the samc length.
Of Leonardo's table ( I ) . The simplest figures which have the corresponding symmetry are the regular polygons: the regular triangle, the square, the regular pentagon, etc. The fact that there is for every number n = 3, 4, 5, - . a regular polygon of n sides is closely related to the existence for every n of a rotational group of order n in plane geometry. Both facts are far from trivial. Indeed, the situation in three dimensions is altogether different: there do not exist infinitely many regular polyhedra in 3-space, but not more than five, often called the Platonic solids because they play an eminent role in Plato's natural philosophy.