TABLE OF CONTENTS: Preface xi
INTRODUCTION Gnomons 3
Of Gnomons and Sundials 6
On Geometric Similarity 9
Geometry and Number 10
Of Gnomons and Obelisks 13
CHAPTER I Figurate and m-adic Numbers 15
Figurate Numbers 15
Property of Triangular Numbers 17
Property of Square Numbers 20
m-adic Numbers 21
Powers of Dyadic Numbers 22
The Dyadic Hamiltonian Path 25
Powers of Triadic Numbers 29
CHAPTER II Continued Fractions 31
Euclid's Algorithm 31
Continued Fractions 33
Simple Continued Fractions 34
Convergents 35
Terminating Regular Continued Fractions 37
Periodic Regular Continued Fractions 38
Spectra of Surds 40
Nonperiodic Nonterminating Regular Continued Fractions 42
Retrovergents 43
Appendix 44
Summary of Formulae 45
CHAPTER III Fibonacci Sequences 49
Recursive Definition 50
The Seed and Gnomonic Numbers so
Explicit Formulation of Fm,n 52
Alternative Explicit Formulation 56
The Monognomonic Simple Periodic Fraction 58
The Dignomonic Simple Periodic Fraction 61
Arbitrarily Terminated Simple Periodic Fractions 63
m Is Very Small: From Fibonacci to Hyperbolic and Trigonometric Functions 66
Appendix: The Polygnomonic SPF 67
Summary of Formulae 69
CHAPTER IV Ladders: From Fibonacci to Wave Propagation 74
The Transducer Ladder 74
The Electrical Ladder 76
Resistance Ladders 77
Iterative Ladders 79
Imaginary Components 83
The Transmission Line 85
The Mismatched Transmission Line 86
Wave Propagation Along a Transmission Line 88
Pulley Ladder Networks 91
Marginalia 95
A Topological Similarity 95
CHAPTER V Whorled Figures 96
Whorled Rectangles 96
Euclid's Algorithm 96
Monognomonic Whorled Rectangles 99
Dignomonic Whorled Rectangles 102
Self-Similarity 108
Improperly Seeded Whorled Rectangles 109
Two Whorled Triangles III I
Marginalia 113
Transmission Lines Revisited 1 13
CHAPTER VI The Golden Number 114
From Number to Geometry 117
The Whorled Golden Rectangle 118
The Fibonacci Whorl 120
The Whorled Golden Triangle 121
The Whorled Pentagon 121
The Golden Section: From Antiquity to the Renaissance 123
Marginalia 132
The Sneezewort 132
A Golden Trick 134
The Golden Knot 134
CHAPTER VII The Silver Number 135
From Number to Geometry 137
The Silver Pentagon 138
The Silver Spiral 139
The Winkle 142
Marginalia 143
Golomb's Rep-Tiles 143
A Commedia dell'Arte 146
Repeated Radicals 148
CHAPTER VIII Spirals 151
The Rotation Matrix 151
The Monognomonic Spiral 153
Self-similarity 158
Equiangularity 159
Perimeter of the Spiral 161
The Rectangular Dignomonic Spiral 165
The Archimedean Spiral 168
Damped Oscillations 171
The Simple Pendulum 174
The RLC Circuit 177
The Resistor 178
The Capacitor 179
The Inductor 180
The Series RLC Circuit 180
Appendix: Finite Difference Equations 183
CHAPTER IX Positional Number Systems 187
Division 187
Mixed Base Positional Systems 191
Finding the Digits of an integer 195
CHAPTER X Fractals 198
The Kronecker Product Revisited 198
Associativity of the Kronecker Product 201
Matrix Order 205
Commutativity of the Kronecker Product 206
Vectors 208
Fractal Lattices 209
Pascal's Triangle and Lucas's Theorem 211
The Sierpinky Gasket and Carpet 215
The Cantor Dust 219
The Thue-Morse Sequence and Tiling 223
Higher-Dimensional Lattices 225
Commutativity and Higher Dimensions 227
The Three Dimensional Sierpinsky Pyramid and Menger Sponge 227
The Kronecker Product with Respect to Other
Operations 231
Fractal Linkages 233
The Koch Curve 234
The Peano Space-Filling Curve 237
A Collection of Regular Fractal Linkages 238
Mixed Regular Linkages and Corresponding Tesselations 244
An Irregular Fractal Linkage: The pentagonal "Eiffel Tower" 246
Appendix: Simplifying Symbols 248
Index 253
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