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SYMMETRIE,BAUPLAN DERWELT by Henning Genz. Piper, Munich, Germany, 1987.465 pp. DM 56.00. Reviewed by RudolfArnheim, 1200Earhart Rd.,No. 537,Ann Arbor, MI 48105, U.S.A. Henning Genz, a professor of theoretical physics at the University of Karlsruhe, has written an extensive treatise on symmetry as the building principle of the physical world. Although thoroughlysystematicand quite technical , the book iswritten with close reference to daily experience. It does so within the limitsof its subject and without mathematicalformulas. Each chapter is enjoyablysimple at its beginning, but later when the author proceeds to the latest refinements of quantum mechanics the going can be rough. Over all, the book has a breathtaking philosophical range, of considerable interest to the general reader, but so far not availablein English. For our practical purposes, especially in the arts,we commonlylimit symmetry to mirror reflections around central points, lines, surfacesor bodies in the three dimensionsof space. Natural science includesthree further dimensions of space,which Genz summarizes by saying that an object is called symmetrical when it can be subjectedto spatialmanipulationswithout being changed (p. 44).These manipulations are displacementsin space or time, changesof scale by enlargement or diminution, and rotations. Excluded are changesof an object into its mirror reflection,since this cannot be accomp lished physically, at least not in the macroscopicworld one cannot transform a right hand into a left hand without remodeling it. Displacementwithout change is, of course,one of the mostcommon manip ulations in practicallife, aswe move objectsfrom one place to another. In the arts,multiplicationof an object’sshape makes for ornaments.The concept becomesmostinterestingwhen displacement is applied to the persistence of objectsor other definitepatterns in time. Even before time and spacecameabout, total symmetrycan be assumed to have existedin the sense that the universe was uniform in all directions-a situation that survivesin the background radiation reaching us with the sameintensityfrom all directions (p. 383). After the Big Bang, however,space and time emerge through the creation of structure, which discontinuesuniversal symmetry.Now spatial directions make for differencesin shape, and the course of time may produce changes in objects.What we lose in total symmetry, we gain by obtaining the world of objects!Avital exception, however,persists in this otherwiseand suddenly confusing complexity of the world. The laws of nature preserve the equallynecessary order, without which the world would be chaos. The laws of nature preserve symmetriesin the complexityand, says Genz, can be considered eternal. What actually happens in our world results from the interaction between the persistentlyvalid laws of nature and the accidentalcontingenciesby which the laws happen to apply (p. 439). When Kepler found that the planets move around the sun not in circular orbits and at a steadypace, but in eccentric ellipsesand at changingvelocity , he led us to understand how the invariant law of gravitation interacts with the particular “initialconditions” that gave the planets their first push. So much for the first parameter of symmetry,displacementsin space and time. When combined with scale and rotation, symmetry carriesus once more from the obvious to the aweinspiring .Change of scale is handy in supplyingus with models and reproductions of manageablesize,smaller or larger than the originals.This is true for models of chemistry and physics as well as for the lantern slides and reproductions of works of art. Beyond thisvisual facility,however, the laws of nature prevent us from simply varying the size of objects.As iswell known, “whenthe size of a macroscopic system is doubled, it contains eight times as many molecules and atoms”because by doubling height, breadth, and length, one increases the volume by a factor of eight (p. 149).Thus, as D’Arcy Thompson has quoted, in his beautiful chapter on magnitude, from Goethe, “carehas been taken to prevent the trees from reaching the sky” [11. In the purely visual realm, a set of concentric circleshas symmetry by enlargement of scale. By displacing the circlesstepwise,one obtains two infinitelylong series, one toward ever larger circles,the other toward ever smaller ones. Add rotation and you approach the shape of fractals.As long as fractals are obtained by mathematical or mechanical construction, their internal symmetriesare perfectly realized, but when they are sought in nature, their perfection is modified by...


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