Regular Polytopes

by H. S. M. Coxeter

Overview

Polytopes are geometrical figures bounded by portions of lines, planes, or hyperplanes. In plane (two dimensional) geometry, they are known as polygons and comprise such figures as triangles, squares, pentagons, etc. In solid (three dimensional) geometry they are known as polyhedra and include such figures as tetrahedra (a type of pyramid), cubes, icosahedra, and many more; the possibilities, in fact, are infinite! H. S. M. Coxeter's book is the foremost book available on regular polyhedra, incorporating not only the ancient Greek work on the subject, but also the vast amount of information that has been accumulated on them since, especially in the last hundred years. The author, professor of Mathematics, University of Toronto, has contributed much valuable work himself on polytopes and is a well-known authority on them.
Professor Coxeter begins with the fundamental concepts of plane and solid geometry and then moves on to multi-dimensionality. Among the many subjects covered are Euler's formula, rotation groups, star-polyhedra, truncation, forms, vectors, coordinates, kaleidoscopes, Petrie polygons, sections and projections, and star-polytopes. Each chapter ends with a historical summary showing when and how the information contained therein was discovered. Numerous figures and examples and the author's lucid explanations also help to make the text readily comprehensible.
Although the study of polytopes does have some practical applications to mineralogy, architecture, linear programming, and other areas, most people enjoy contemplating these figures simply because their symmetrical shapes have an aesthetic appeal. But whatever the reasons, anyone with an elementary knowledge of geometry and trigonometry will find this one of the best source books available on this fascinating study.

Product Details

ISBN-13:	9780486614809
Publisher:	Dover Publications
Publication date:	06/01/1973
Series:	Dover Books on Mathematics
Edition description:	3d ed
Pages:	368
Product dimensions:	5.50(w) x 8.50(h) x (d)

About the Author

H. S. M. Coxeter: Through the Looking Glass
Harold Scott MacDonald Coxeter (1907–2003) is one of the greatest geometers of the last century, or of any century, for that matter. Coxeter was associated with the University of Toronto for sixty years, the author of twelve books regarded as classics in their field, a student of Hermann Weyl in the 1930s, and a colleague of the intriguing Dutch artist and printmaker Maurits Escher in the 1950s.

In the Author's Own Words:
"I'm a Platonist — a follower of Plato — who believes that one didn't invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."

"In our times, geometers are still exploring those new Wonderlands, partly for the sake of their applications to cosmology and other branches of science, but much more for the sheer joy of passing through the looking glass into a land where the familiar lines, planes, triangles, circles, and spheres are seen to behave in strange but precisely determined ways."

"Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about the physical world. The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry."

"Let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused." — H. S. M. Coxeter

Table of Contents

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 αn,βn,γnαn​,βn​,γn"
13·4 New coordinates for αn and βn
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