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Books 249 This book is concerned mainly with topological complexes having V vertices, Eedges,F polygonal faces and C solidcells. Each face serves as the interface between two of the cells. The possibility of a finite complex is included by regarding the peripheral facesas belongingto an extra cell coveringthe whole outsidespace;inotherwords,thecomplexcoversa topological3sphere .With this restriction, the numericalpropertiessatisfythe Euler-Schllfli formula V -E +F - C = 0. The familiarform of Euler’s fomula (V- E+ F= 2) is included by regarding the surface of a single polyhedron as the common boundary of 2 cells, the ‘inside’ and ‘outside’, so that C=2. Such complexes play a fundamental role in crystallography, architecture and many branches of physics and of biology. Appropriately, a Foreword has been written by the eminent metallurgist Cyril Stanley Smith, who sees their ‘immediateapplicability both to the symmetry of most alloycrystalsand to the non-symmetrical space-filling aggregates of crystalline grains and other microconstituents’. Especially relevant to such considerations are the chapterson StatisticalSymmetryand on Random.TwodimensionalNets . This work is reminiscent of a beautifullyillustratedpape’r by Helmut Emde entitled Homogene Polytope [Bayerische Akad.Wiss., Math. - Naturwiss. Klasse, Abhandlungen 89, 1 (1958)l.Likeh i m ,Loebmissedthe opportunityto simplifymany of the algebraicmanipulationsby employingthe Schlaflisymbol {n, 2 / m , s}: three elementary ‘valencies’ that, like chemical valencies, usually range over the values 2, 3, 4, 5, 6. In that notation,thefust 12figuresinchapter9arerespectively {b.2,2}, {2,3,2}, {3,3,2}, {4,3,2}, {5,3,2}, {6,3,2}, {2,4,2}, {3,4,2}, (4, 4, 2}, {3. 5, 2}, {3, 6, 2}, {2, 3, 3); then, on page 58, he describes{2,4,3}, {2,5,3}, {2,6,3), {n.2, s}; on page 59, {3,3, 3}, {4,3,3}; and on page 60there is a splendidphotograph of a skeletalmodel of (3, 4, 3). In order to explain the role of these three small numbers, it is perhaps worthwhile to mention that the author’s eight parameters k. I, m. n, P. q. s are the RFp, s, 2s q, p, 1. 2rlr. r of the reviewer’s Regular Complex Polytopes [(Cambridge: Cambridge Univ. Press, 1974) p. 301. In Chapter 13, the concept of Dirichletdomains is illustrated by a fascinating map of the schools in Cambridge, Massachusetts ,each surroundedby a polygonalregion that minimizes the distancethat the children must walk from their homes. The next five chapters’constitute a very readable introduction to geometricalcrystallography. Finally, the ‘Coda’describes(in a ‘photographicessay’) two ways in which two congruent cubes canbe cutup and reassembledto makea singleconvexsolid.The first way istocut onecubeintosixsquarepyramidsand stickone pyramidonto eachfaceof the other cube.Thesecond(andmore exciting) way is ‘toturn eachcubeinsideout’after cuttingit into four irregular bipyramids of the kind that Buckminster Fuller callsoc-tets;then the twojaggedlookingobjectsfit together asif by magic. Certainly this elegant little book should find a place on the shelves of the world’s Philomorphs (‘brought together by a common interest in the underlying patterns of interaction between things’). Elementary Probability Theory with Stochastic Processes. Kai Lai Chung. Springer, New York, 1974. 325 pp., illus. $12.00. Reviewed by Lawrence E. Jerome, This is an excellent mathematics text, both in terms of its approach and its readability; the writing style is light, almost ‘breezy’at times. There are many exampleiand problems with solutions, interspersed with interesting historical asides and insights.I would have appreciateda morecompletetreatmentof thehistoricalliterature,whichissohard totrackdownin thefield ofmathematics. The advantage of Chung’s approach is that he uses more powerful mathematical techniques (set theory, distribution functions,transforms,etc.)tooperatein trialspaceasopposedto the traditional sample space. Trial space, as the term implies, *I0295 Menhart Lane, Cupertino, CA 95014, U.S.A. refers to representations of real, experimental space; sample space is the combinatorial representation of all possible outcomesand hence is much larger and cumbersome.The key is Chung’s use of indicatorfunctions that allow one to operate in trial space because ‘the expectation of an indicator random variable is just the probability of the corresponding event: E(IA) = P(A)’ (p. 163). Strangelyenough,Chungdoesnot make this advantageof his approach very clear. He does not show how much easierit is to operate in trial space instead of samplespace. For instance, his derivationof the binomialformulafor...

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