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Symmetry And Group

16-dimensional compact projective planes with a large group by Hahl H., Salzmann H. PDF

By Hahl H., Salzmann H.

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Thereafter the handle P1 Pq turns about so that P2 comes into contact with either I' or I"; call the one it contacts B, the other A. We now have a sort of rhomboid and roughly AP2 is the antero-posterior axis and BPI the dorsal-ventral one. Only the next division which along a plane perpendicular to the one separating A and B splits A as well as B into symmetric halves A = a a, B = b p, is that which determines left and right. A further slight shift of the configuration destroys this bilateral symmetry.

In the latter case translation is accompanied b y longitudinal slip reflection. Of course the pattern does not go on into infinity (nor docs a band ornament), but one may say that it is potentially infinite at least in one direction, as in the course of time ever new segmcrlts separated from each other i ~ ya l ~ u dconlc into being. Goethe saicl of the tails of vertebrates that they allude as it were to the potential infinity of organic cxistencc. Thc central part of the animal shown in this picture, a scolopendrid (Fig.

If so, wc have probal~lynot yet discovcrcd the appropriate mathematical tools. This ~vould not be so surprising. For aftcr all, the Egyptians excelled in the orna~uentalart four thousand years before the ~nathernaticiansdiscovcred in the group concept the proper trlathematical instl-urnent for the trcatnlerlt of ornarncnts and for the derivation of their possible symmetry classes. Andreas Speiser, who has taken a special interest in thc group-theoretic aspect of ornaments, tricd to apply cornl~inatorial principles of a ~ n a t h e ~ n a t i c anature l also to the formal problems of music.

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