# Regular Polytopes

###### H. S. M. Coxeter
Publisher:
Dover Publications
Publication Date:
1973
Number of Pages:
321
Format:
Paperback
Edition:
3
Price:
16.95
ISBN:
9780486614809
Category:
Monograph
BLL Rating:

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Tricia Muldoon Brown
, on
10/30/2016
]

Regular Polytopes is densely packed, with definitions coming rapid-fire and results following quickly, much like Stanley’s Enumerative Combinatorics. Years of results are elegantly summarized with just enough details for clarity, but not so many as to increase the length to a burdensome amount. Most of the chapters are definition-heavy, but still very readable. The key vocabulary is italicized with definitions given more casually within the narrative rather than set apart in a formal style. Similarly, theorems, propositions, and proof also occur naturally in the text, with section headings giving reference to the theorem or proof being addressed.

The readability is further enhanced by the consistent use of concrete examples with each new topic. For instance, reflection groups are illustrated with mirrors (Chapter 5) and illustrations or diagrams of the polytopes are given throughout the text when possible. These illustrations and figures are not flashy, but are good, clear, and effective.

The book was last revised in 1973, so it is occasionally out-of-date, although not frustratingly so. For example, in Chapter 5, Coxeter discusses the “novelty” of the use of Dynkin diagrams. In Chapter 1, we are reminded that the four-color problem is “unanswered.” Vocabulary has also experienced some changes over time, i.e. Coxeter’s use of “reciprocal” polytopes, which in more recent times are usually referred to as “dual” polytopes. Rather than detracting from the text, I found that these occasional differences give insight into the mathematical progress that has been made in the last half-century.

One of my favorite parts of this book are the historical remarks found at the end of each chapter. Coxeter carefully associates the results from the chapter with the major contributing mathematicians, but also adds a few interesting details setting the context for the mathematics. Personal details are also included, both about the mathematicians and occasionally about Coxeter himself; I particularly enjoyed reading about his acquaintance with Alicia Boole Stott in Chapter 13.

Overall, like the illustrations and diagrams, the book provides a well-written and comprehensive coverage of regular polytopes that is clear and effective without being elaborate or excessively detailed. Further, the historical perspective, found at the end of each chapter as well as in the treatment of the topics, gives this book a distinctly more entertaining style than a standard mathematical textbook.

Tricia Muldoon Brown (patricia.brown@armstrong.edu) is an Associate Professor at Armstrong State University with an interest in commutative algebra, combinatorics, and recreational mathematics.

 I. POLYGONS AND POLYHEDRA 1·1 Regular polygons 1·2 Polyhedra 1·3 The five Platonic Solids 1·4 Graphs and maps 1·5 "A voyage round the world" 1·6 Euler's Formula 1·7 Regular maps 1·8 Configurations 1·9 Historical remarks II. REGULAR AND QUASI-REGULAR SOLIDS 2·1 Regular polyhedra 2·2 Reciprocation 2·3 Quasi-regular polyhedra 2·4 Radii and angles 2·5 Descartes' Formula 2·6 Petrie polygons 2·7 The rhombic dodecahedron and triacontahedron 2·8 Zonohedra 2·9 Historical remarks III. ROTATION GROUPS 3·1 Congruent transformations 3·2 Transformations in general 3·3 Groups 3·4 Symmetry opperations 3·5 The polyhedral groups 3·6 The five regular compounds 3·7 Coordinates for the vertices of the regular and quasi-regular solids 3·8 The complete enumeration of finite rotation groups 3·9 Historical remarks IV. TESSELLATIONS AND HONEYCOMBS 4·1 The three regular tessellations 4·2 The quasi-regular and rhombic tessellations 4·3 Rotation groups in two dimensions 4·4 Coordinates for the vertices 4·5 Lines of symmetry 4·6 Space filled with cubes 4·7 Other honeycombs 4·8 Proportional numbers of elements 4·9 Historical remarks V. THE KALEIDOSCOPE 5·1 "Reflections in one or two planes, or lines, or points" 5·2 Reflections in three or four lines 5·3 The fundamental region and generating relations 5·4 Reflections in three concurrent planes 5·5 "Reflections in four, five, or six planes" 5·6 Representation by graphs 5·7 Wythoff's construction 5·8 Pappus's observation concerning reciprocal regular polyhedra 5·9 The Petrie polygon and central symmetry 5·x Historical remarks VI. STAR-POLYHEDRA 6·1 Star-polygons 6·2 Stellating the Platonic solids 6·3 Faceting the Platonic solids 6·4 The general regular polyhedron 6·5 A digression on Riemann surfaces 6·6 Ismorphism 6·7 Are there only nine regular polyhedra? 6·8 Scwarz's triangles 6·9 Historical remarks VII. ORDINARY POLYTOPES IN HIGHER SPACE 7·1 Dimensional analogy 7·2 "Pyramids, dipyramids, and prisms" 7·3 The general sphere 7·4 Polytopes and honeycombs 7·5 Regularity 7·6 The symmetry group of the general regular polytope 7·7 Schäfli's criterion 7·8 The enumeration of possible regular figures 7·9 The characteristic simplex 7·10 Historical remarks VIII. TRUNCATION 8·1 The simple truncations of the genral regular polytope 8·2 "Cesàro's construction for 3, 4, 3" 8·3 Coherent indexing 8·4 "The snub 3, 4, 3" 8·5 "Gosset's construction for 3, 3, 5" 8·6 "Partial truncation, or alternation" 8·7 Cartesian coordinates 8·8 Metrical properties 8·9 Historical remarks IX. POINCARÉ'S PROOF OF EULER'S FORMULA 9·1 Euler's Formula as generalized by Schläfli 9·2 Incidence matrices 9·3 The algebra of k-chains 9·4 Linear dependence and rank 9·5 The k-circuits 9·6 The bounding k-circuits 9·7 The condition for simple-connectivity 9·8 The analogous formula for a honeycomb 9·9 Polytopes which do not satisfy Euler's Formula X. "FORMS, VECTORS, AND COORDINATES" 10·1 Real quadratic forms 10·2 Forms with non-positive product terms 10·3 A criterion for semidefiniteness 10·4 Covariant and contravariant bases for a vector space 10·5 Affine coordinates and reciprocal lattices 10·6 The general reflection 10·7 Normal coordinates 10·8 The simplex determined by n + 1 dependent vectors 10·9 Historical remarks XI. THE GENERALIZED KALEIDOSCOPE 11·1 Discrete groups generated by reflectins 11·2 Proof that the fundamental region is a simplex 11·3 Representation by graphs 11·4 "Semidefinite forms, Euclidean simplexes, and infinite groups" 11·5 "Definite forms, spherical simplexes, and finite groups" 11·6 Wythoff's construction 11·7 Regular figures and their truncations 11·8 "Gosset's figures in six, seven, and eight dimensions" 11·9 Weyl's formula for the order of the largest finite subgroup of an infinite discrete group generated by reflections 11·x Historical remarks XII. THE GENERALIZED PETRIE POLYGON 12·1 Orthogonal transformations 12·2 Congruent transformations 12·3 The product of n reflections 12·4 "The Petrie polygon of p, q, . . . , w" 12·5 The central inversion 12·6 The number of reflections 12·7 A necklace of tetrahedral beads 12·8 A rational expression for h/g in four dimensions 12·9 Historical remarks XIII. SECTIONS AND PROJECTIONS 13·1 The principal sections of the regular polytopes 13·2 Orthogonal projection onto a hyperplane 13·3 "Plane projections an,ßn,?n" 13·4 New coordinates for an and ßn 13·5 "The dodecagonal projection of 3, 4, 3" 13·6 "The triacontagonal projection of 3, 3, 5" 13·7 Eutactic stars 13·8 Shadows of measure polytopes 13·9 Historical remarks XIV. STAR-POLYTOPES 14·1 The notion of a star-polytope 14·2 "Stellating 5, 3, 3" 14·3 Systematic faceting 14·4 The general regular polytope in four dimensions 14·5 A trigonometrical lemma 14·6 Van Oss's criterion 14·7 The Petrie polygon criterion 14·8 Computation of density 14·9 Complete enumeration of regular star-polytopes and honeycombs 14·x Historical remarks Epilogue Definitions of symbols Table I: Regular polytopes Table II: Regular honeycombs Table III: Schwarz's triangles Table IV: Fundamental regions for irreducible groups generated by reflections Table V: The distribution of vertices of four-dimensional polytopes in parallel solid sections Table VI: The derivation of four-dimensional star-polytopes and compounds by faceting the convex regular polytopes Table VII: Regular compunds in four dimensions Table VIII: The number of regular polytopes and honeycombs Bibliography Index
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