You are here

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 suggests that undergraduate mathematics libraries consider this book for acquisition.

[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The book was last revised in 1973, so it is occasionally out-of-date, although not frustratingly so.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
 
Tags: 

Dummy View - NOT TO BE DELETED