By Kondratev A.S.
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Over the past 30 years the idea of finite teams has constructed dramatically. Our figuring out of finite basic teams has been more suitable by means of their class. many questions about arbitrary teams should be lowered to related questions about easy teams and purposes of the idea are starting to look in different branches of arithmetic.
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D. Birkhoff has to say on the mathematics of poetry and music in the two publications quoted in Lecture r, note 1. 8 metry, namely by one that is carried into itself by all rotations around a certain axis, for instance by a vase. Fig. 29 shows an attic vase of the geometric period which displays quite a number of simple ornaments of this type. The principle of symmetry is the same, although the style is no lo~lger"geometric," in this Khodian pitcher (Fig. G. Other illustrations are such capitals as these from early Egypt (Fig.
The rotations around a given center 0 form a group. The simplest type of congruences are the translations. ' The translations form a group; indeed the succession of the two translations -4 --+ 4 AR, RC results in the translation AC. What has all this to do with symmetry? It provides the adequate mathematical language to define it. Given a spatial configuration 8, those automorphisms of space which leave 8 unchanged form a group,'I and this group describts exactly the symmetry possessed by 3. Space itself has the full symmetry corresponding to the group of all autornorphisms, of all similarities.
For if i t contains an S that enlarges linear dimensions a t the ratio a : 1, a > 1, then all thc infinitely many iterations S1,S2,S3, . contained in the group would be different tlccause they enlarge at different scales a', a2, a" .... ationssuch as band ornaments a n d the like. After these general mathematical considerations let us now take u p sorne special groups of symmetry which are important in art or nature. T h e operation which defines bilateral symmetry, mirror reflection, is essentially a one-dirncnsional operation.