Table of Contents
Preface v
1 Geometry and geometric ideas 1
1.1 Geometric notions, models and geometric spaces 1
1.1.1 Geometric notions 3
1.2 Overview of Euclid's method and approaches to geometry 3
1.2.1 Incidence geometries - affine geometries, finite geometries, projective geometries 6
1.3 Euclidean geometry 7
1.3.1 Birkhoff's axioms for Euclidean geometry 9
1.4 Neutral or absolute geometry 9
1.5 Euclidean and hyperbolic geometry 12
1.5.1 Consistency of hyperbolic geometry 13
1.6 Elliptic geometry 14
1.7 Differential geometry 14
2 Isometries in Euclidean vector spaces and their classification in Rn 17
2.1 Isometries and Klein's Erlanger Programm 17
2.2 The isometries of the Euclidean plane R2 28
2.3 The isometries of the Euclidean space R3 36
2.4 The general case Rn with n ≥ 2 43
3 The conic sections in the Euclidean plane 51
3.1 The conic sections 51
3.2 Ellipse 59
3.3 Hyperbola 61
3.4 Parabola 62
3.5 The principal axis transformation 64
4 Special groups of planar isometries 67
4.1 Regular polygons 71
4.2 Regular tessellations of the plane 73
4.3 Groups of translations in the plane R2 78
4.4 Groups of isometries of the plane with trivial translation subgroup 79
4.5 Frieze groups 80
4.6 Planar crystallographic groups 84
5 Graph theory and platonic solids 101
5.1 Graph theory 101
5.2 Coloring of planar graphs 111
5.3 Euler line 112
5.4 Stereographic projection 116
5.5 Platonic solids 118
5.5.1 Cube(C) 120
5.5.2 Tetrahedron (T) 120
5.5.3 Octahedron (O) 121
5.5.4 Icosahedron (l) 122
5.5.5 Dodecahedron (D) 123
5.6 The spherical geometry of the sphere S2 124
5.7 Classification of the Platonic solids 127
6 Linear fractional transformation and planar hyperbolic geometry 135
6.1 Linear fractional transformations 135
6.2 A model for a planar hyperbolic geometry 142
6.3 The (planar) hyperbolic theorem of Pythagoras in H 146
6.4 The hyperbolic area of a hyperbolic polygon 148
7 Combinatorics and combinatorial problems 161
7.1 Combinatorics 161
7.2 Basic techniques and the multiplication principle 161
7.3 Sizes of finite sets and the sampling problem 165
7.3.1 The binomial coefficients 170
7.3.2 The occupancy problem 174
7.3.3 Some further comments 174
7.4 Multinomial coefficients 176
7.5 Sizes of finite sets and the inclusion-exclusion principle 178
7.6 Partitions and recurrence relations 185
7.7 Decompositions of naturals numbers, partition function 191
7.8 Catalan numbers 193
7.9 Generating functions 195
7.9.1 Ordinary generating functions 195
7.9.2 Exponential generating functions 205
8 Finite probability theory and Bayesian analysis 213
8.1 Probabilities and probability spaces 213
8.2 Some examples of finite probabilities 215
8.3 Random variables, distribution functions and expectation 217
8.4 The law of large numbers 220
8.5 Conditional probabilities 223
8.6 The goat or Monty Hall problem 230
8.7 Bayes nets 231
9 Boolean lattices, Boolean algebras and Stone's theorem 245
9.1 Boolean algebras and the algebra of sets 245
9.2 The algebra of sets and partial orders 245
9.3 Lattices 252
9.4 Distributive and modular lattices 257
9.5 Boolean lattices and Stone's theorem 262
9.6 Construction of Boolean lattices via 0-1 sequences 268
9.7 Boolean rings 272
9.8 The general theorem of Stone 275
Bibliography 279
Index 281
