Regular Polytopes

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