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Regular Polytopes
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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 Schafli'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 "Cesaro'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. POINCARE'S PROOF OF EULER'S FORMULA 9*1 Euler's Formula as generalized by Schlafli 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

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

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