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1 Burnside Mathematics- Differential Geometry

Regular Polytopes 3RD Edition

by

Regular Polytopes 3RD Edition Cover

 

Synopses & Reviews

Publisher Comments:

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.

Synopsis:

Foremost book available on polytopes, incorporating ancient Greek and most modern work. Discusses polygons, polyhedrons, and multi-dimensional polytopes. Definitions of symbols. Includes 8 tables plus many diagrams and examples. 1963 edition.

Synopsis:

Foremost book available on polytopes, incorporating ancient Greek and most modern work done on them. Beginning with polygons and polyhedrons, the book moves on to multi-dimensional polytopes in a way that anyone with a basic knowledge of geometry and trigonometry can easily understand. Definitions of symbols. Eight tables plus many diagrams and examples. 1963 edition.

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 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

Product Details

ISBN:
9780486614809
Author:
Coxeter, H. S. M.
Publisher:
Dover Publications
Author:
Mathematics
Author:
Coxeter, H. S. M.
Author:
Coxeter
Location:
New York
Subject:
Mathematics
Subject:
Geometry - General
Subject:
Geometry
Subject:
Algebra - General
Subject:
Topology
Subject:
Polytopes
Subject:
General Mathematics
Subject:
Mathematics - Algebra
Copyright:
Edition Number:
3
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Series Volume:
no. 12
Publication Date:
19730631
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
350
Dimensions:
8.5 x 5.38 in 1 lb

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Regular Polytopes 3RD Edition Used Trade Paper
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Product details 350 pages Dover Publications - English 9780486614809 Reviews:
"Synopsis" by ,
Foremost book available on polytopes, incorporating ancient Greek and most modern work. Discusses polygons, polyhedrons, and multi-dimensional polytopes. Definitions of symbols. Includes 8 tables plus many diagrams and examples. 1963 edition.

"Synopsis" by ,
Foremost book available on polytopes, incorporating ancient Greek and most modern work done on them. Beginning with polygons and polyhedrons, the book moves on to multi-dimensional polytopes in a way that anyone with a basic knowledge of geometry and trigonometry can easily understand. Definitions of symbols. Eight tables plus many diagrams and examples. 1963 edition.

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