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Automorphisms of Order 2 of an Abelian Group by Miller G. A.

By Miller G. A.

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As a shoot grows it translates, one might say, a slow teinporal into a spatial rhythm. ReThis and the next picture are taken from Studium Centrole, p. 249 and p. 241 (art~cleby W. Troll, "Symmetriebetrachtung in der Biologie"). FIG 27 FIG 28 flection, inversion in time, plays a far less important part in music than rhythm does. A inclody changes its character to a consitlerable degree if played backward, and I, who a m a poor musician, find it hard to recugnize reflection when it is used in the construction of a fugue; it certainly has no such spontaneous effect as r h y t h n ~ .

Quite obviously some accidents hard to control decide whether at a spot of the solution a laevo- or dextro-crystal comes into being; and thus in agreement with the symmetric and optically inactive character of the solution as a whole and with the law of chance the amounts of substance deposited in the one and the other form at any moment of the process of crystallization are equal or very nearly equal. O n the other hand nature, in giving us the wonderful gift of grapes so much enjoyed by Noah, produced only one of the forms, and it remained for Pasteur to produce the other!

I t seems best to fix the underlying general concept with some precision beforehand, and to that end a little mathematics is needed, for which I ask your patience. I have spoken of transformations. A mapping S of space associates with every space point p a point p' as its image. A special such mapping is the identity I carrying every point p into itself. Given two mappings S, 'I,one can perform one after the other: if S carries p into p' and T carries pt into p" then the resulting mapping, which we denote by ST, carries p into p".

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