Regular Polytopes (book)

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Regular Polytopes
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Cover of the Dover edition, 1973
AuthorHarold Scott MacDonald Coxeter
LanguageEnglish
SubjectGeometry
Published1947, 1963, 1973
PublisherMethuen, Pitman, Macmillan, Dover
Pages321
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OCLC798003

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Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948,[1]Template:R/superscript[2]Template:R/superscript[3]Template:R/superscript[4]Template:R/superscript[5]Template:R/superscript[6]Template:R/superscript[7]Template:R/superscript[8]Template:R/superscript with a second edition published by Macmillan in 1963[9]Template:R/superscript[10]Template:R/superscript[11]Template:R/superscript[12]Template:R/superscript and a third edition by Dover Publications in 1973.[13]Template:R/superscript[14]Template:R/superscript[15]Template:R/superscript The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries.[15]Template:R/superscript

Overview

The main topics of the book are the Platonic solids (regular convex polyhedra), related polyhedra, and their higher-dimensional generalizations.[1]Template:R/superscript[2]Template:R/superscript It has 14 chapters, along with multiple appendices,[3]Template:R/superscript providing a more complete treatment of the subject than any earlier work, and incorporating material from 18 of Coxeter's own previous papers.[1]Template:R/superscript It includes many figures (both photographs of models by Paul Donchian and drawings), tables of numerical values, and historical remarks on the subject.[1]Template:R/superscript[2]Template:R/superscript

The first chapter discusses regular polygons, regular polyhedra, basic concepts of graph theory, and the Euler characteristic.[3]Template:R/superscript Using the Euler characteristic, Coxeter derives a Diophantine equation whose integer solutions describe and classify the regular polyhedra. The second chapter uses combinations of regular polyhedra and their duals to generate related polyhedra,[1]Template:R/superscript including the semiregular polyhedra, and discusses zonohedra and Petrie polygons.[3]Template:R/superscript Here and throughout the book, the shapes it discusses are identified and classified by their Schläfli symbols.[1]Template:R/superscript

Chapters 3 through 5 describe the symmetries of polyhedra, first as permutation groups[3]Template:R/superscript and later, in the most innovative part of the book,[1]Template:R/superscript as the Coxeter groups, groups generated by reflections and described by the angles between their reflection planes. This part of the book also describes the regular tessellations of the Euclidean plane and the sphere, and the regular honeycombs of Euclidean space. Chapter 6 discusses the star polyhedra including the Kepler–Poinsot polyhedra.[3]Template:R/superscript

The remaining chapters cover higher-dimensional generalizations of these topics, including two chapters on the enumeration and construction of the regular polytopes, two chapters on higher-dimensional Euler characteristics and background on quadratic forms, two chapters on higher-dimensional Coxeter groups, a chapter on cross-sections and projections of polytopes, and a chapter on star polytopes and polytope compounds.[3]Template:R/superscript

Later editions

The second edition was published in paperback;[9]Template:R/superscript[11]Template:R/superscript it adds some more recent research of Robert Steinberg on Petrie polygons and the order of Coxeter groups,[9]Template:R/superscript[12]Template:R/superscript appends a new definition of polytopes at the end of the book, and makes minor corrections throughout.[9]Template:R/superscript The photographic plates were also enlarged for this printing,[10]Template:R/superscript[12]Template:R/superscript and some figures were redrawn.[12]Template:R/superscript The nomenclature of these editions was occasionally cumbersome,[2]Template:R/superscript and was modernized in the third edition. The third edition also included a new preface with added material on polyhedra in nature, found by the electron microscope.[13]Template:R/superscript[14]Template:R/superscript

Reception

The book only assumes a high-school understanding of algebra, geometry, and trigonometry,[2]Template:R/superscript[3]Template:R/superscript but it is primarily aimed at professionals in this area,[2]Template:R/superscript and some steps in the book's reasoning which a professional could take for granted might be too much for less-advanced readers.[3]Template:R/superscript Nevertheless, reviewer J. C. P. Miller recommends it to "anyone interested in the subject, whether from recreational, educational, or other aspects",[4]Template:R/superscript and (despite complaining about the omission of regular skew polyhedra) reviewer H. E. Wolfe suggests more strongly that every mathematician should own a copy.[7]Template:R/superscript Geologist A. J. Frueh Jr., describing the book as a textbook rather than a monograph, suggests that the parts of the book on the symmetries of space would likely be of great interest to crystallographers; however, Frueh complains of the lack of rigor in its proofs and the lack of clarity in its descriptions.[6]Template:R/superscript

Already in its first edition the book was described as "long awaited",[3]Template:R/superscript and "what is, and what will probably be for many years, the only organized treatment of the subject".[7]Template:R/superscript In a review of the second edition, Michael Goldberg (who also reviewed the first edition)[1]Template:R/superscript called it "the most extensive and authoritative summary" of its area of mathematics.[10]Template:R/superscript By the time of Tricia Muldoon Brown's 2016 review, she described it as "occasionally out-of-date, although not frustratingly so", for instance in its discussion of the four color theorem, proved after its last update. However, she still evaluated it as "well-written and comprehensive".[15]Template:R/superscript

See also

References

  1. ^ a b c d e f g h Page Module:Citation/CS1/styles.css has no content.Goldberg, M., "Review of Regular Polytopes", Mathematical Reviews, MR 0027148
  2. ^ a b c d e f Page Module:Citation/CS1/styles.css has no content.Allendoerfer, C.B. (1949), "Review of Regular Polytopes", Bulletin of the American Mathematical Society, 55 (7): 721–722, doi:10.1090/S0002-9904-1949-09258-3
  3. ^ a b c d e f g h i j Page Module:Citation/CS1/styles.css has no content.Cundy, H. Martyn (February 1949), "Review of Regular Polytopes", The Mathematical Gazette, 33 (303): 47–49, doi:10.2307/3608432, JSTOR 3608432
  4. ^ a b Page Module:Citation/CS1/styles.css has no content.Miller, J. C. P. (July 1949), "Review of Regular Polytopes", Science Progress, 37 (147): 563–564, JSTOR 43413146
  5. ^ Page Module:Citation/CS1/styles.css has no content.Walsh, J. L. (August 1949), "Review of Regular Polytopes", Scientific American, 181 (2): 58–59, JSTOR 24967260
  6. ^ a b Page Module:Citation/CS1/styles.css has no content.Frueh, Jr., A. J. (November 1950), "Review of Regular Polytopes", The Journal of Geology, 58 (6): 672, doi:10.1086/625793, JSTOR 30071213{{citation}}: CS1 maint: multiple names: authors list (link)
  7. ^ a b c Page Module:Citation/CS1/styles.css has no content.Wolfe, H. E. (February 1951), "Review of Regular Polytopes", American Mathematical Monthly, 58 (2): 119–120, doi:10.2307/2308393, JSTOR 2308393
  8. ^ Page Module:Citation/CS1/styles.css has no content.Tóth, L. Fejes, "Review of Regular Polytopes", zbMATH (in Deutsch), Zbl 0031.06502
  9. ^ a b c d Page Module:Citation/CS1/styles.css has no content.Robinson, G. de B., "Review of Regular Polytopes", Mathematical Reviews, MR 0151873
  10. ^ a b c Page Module:Citation/CS1/styles.css has no content.Goldberg, Michael (January 1964), "Review of Regular Polytopes", Mathematics of Computation, 18 (85): 166, doi:10.2307/2003446, JSTOR 2003446
  11. ^ a b Page Module:Citation/CS1/styles.css has no content.Primrose, E.J.F (October 1964), "Review of Regular Polytopes", The Mathematical Gazette, 48 (365): 344, doi:10.1017/s0025557200072995
  12. ^ a b c d Page Module:Citation/CS1/styles.css has no content.Yff, P. (February 1965), "Review of Regular Polytopes", Canadian Mathematical Bulletin, 8 (1): 124, doi:10.1017/s0008439500024413
  13. ^ a b Page Module:Citation/CS1/styles.css has no content.Peak, Philip (March 1975), "Review of Regular Polytopes", The Mathematics Teacher, 68 (3): 230, JSTOR 27960095
  14. ^ a b Page Module:Citation/CS1/styles.css has no content.Wenninger, Magnus J. (Winter 1976), "Review of Regular Polytopes", Leonardo, 9 (1): 83, doi:10.2307/1573335, JSTOR 1573335
  15. ^ a b c Page Module:Citation/CS1/styles.css has no content.Brown, Tricia Muldoon (October 2016), "Review of Regular Polytopes", MAA Reviews, Mathematical Association of America
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