MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

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232 Index oriented triangles 70 origami 127 orthic triangle 54 orthocenter 39, 46 Ostrowski, Alexander 173 Oughtred, William 148 Our Lord 18 Oxford University 150 Pach, J. 55 Pacioli, Luca 11, 153 packings v, 44 Padovan sequence 70 Padovan spirals 70 Padovan whorl 70 pagoda 6-7 paper folding 127 palimpsest 151 Pappus’ Centroid Theorem 152 Pappus’ Hexagon Theorem 152 Pappus of Alexandria 13, 151-152, 183 Pappus’ Problem 152 parallelogram properties 37 Parthenon 9 partition 72-73 partridge number 72 tiling v, 72 Pascal, Blaise 58, 157, 159, 184 Pascal’s Law of Pressure 159 Pascal’s Mystic Hexagram Theorem 159 Pascal’s Triangle 58-59, 74, 159 Pascal’s Wager 159 Pauca sed matura 164 pedal mapping 54 pedal triangle 38-39, 54, 118, 127 Pedersen, Jean J. 117 Pedoe, Dan 31 peg solitaire 169 Pell, John 148 Pell’s equation 157 Peloponnesian War 149 pengala 5 pentagon 107 pentagonal dipyramid 15 Perelman, Grigory 170 perfect coloring 67 tiling 70 triangulation 70 periodic orbit 118-120 Perplexities 181 Pestalozzi, J. H. 165 Peterhouse, Cambridge 168 Pfaff, J. F. 164 Philosophia Sacra 9 Picard, Émile 166 Pinchasi, R. 55 ping-pong 178 plaited polyhedra 117 planar soap bubble clusters 52 plastic number 70 pentagon 70 Platonic solid 11-13, 117, 150 Platonic triangles 11 Platonism 150 Plato of Athens 11, 149, 183 Plato’s Academy 149-150 playscape (triangular) 195 Poincaré-Bendixson Theorem 170 Poincaré Conjecture 170 Poincaré disk 75 Poincaré group 170 Poincaré, Henri 166, 170, 185 Poincaré inequality 170 Poincaré-Linstedt method 170 Poincaré map 170 Poincaré metric 170 Poincaré, Raymond 170 Polydeuces 83 polygonal dissections 106-107
Index 233 Polygonal Number Theorem 50, 163 polyhedral formula 11, 156, 161 polyhedron 11, 13, 15 Polyominoes 178 Pompeiu’s Theorem 49-50 Poncelet-Steiner Theorem 165 Pontormo, Jacopo 17-18 pool balls (triangular) 192 pool-ball triangle 117-118 positively / negatively equilateral triangle 39-40 Post, K. 43 Post’s Theorem 43 Potter, Harry 28 Precious Mirror of the Four Elements 59 Pressman & Co., J. 16 primary color 86 Prime Number Theorem 173, 177 Princeps mathematicorum 163 Princeton University 109, 176-177, 180 principal frequency 100 Principia Mathematica (Newton) 1, 160-161 Principia Mathematica (Whitehead & Russell) 173 Principle of Least Action 162 Principle of the Equilateral Triangle (electrocardiography) 81 Principle of Virtual Work 163 Problem of Apollonius 151-152 Problems in the Distribution of Prime Numbers 178 projective plane 91-92 Propeller Theorem 64 Ptolemy 152 pumpkin (triangular) 193 Puzzles and Curious Problems 181 pyramid 2-3 Pythagoras of Samos 149, 183 Pythagoreans 9, 22, 149 Pythagorean Theorem 149, 153 QFL diagram 88-89 Qianxun Pagoda 6 Qing Dynasty 6 quadratix of Hippias 152 Quan Fengyou, King 6 quantum mechanics 102 Queen Anne 160 Queen Christina 156 Queen of the Sciences 164 quilt (triangular) 194 Rabbinic Judaism 16 random point 50 ranking region 91 rational triangle 48 Ravensburger Spieleverlag GmbH 16 Recreational Mathematics vi, 103, 178, 182 regular heptadecagon 164 polytope 73-74 rhombus 61 simplex 73 tessellations (planar) 61 tilings 15, 156 Renaissance 18, 22 representation triangle 91 rep-tiles vi, 112-113, 178 restricted three-body problem 83 Resurrection 18 Resurrection Stone 28 Reuleaux Triangle 55-56, 96 Ricatti, Vincenzo 162 Richmond, H. W. 66 Riemann, Bernhard 164, 166, 185 Riemann Hypothesis 167, 173 Riemannian metric 167 Riemannian space 167 Riemann integrability 167 Riemann-Lebesgue Lemma 174
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MYSTERIES OF THE EQUILATERAL TRIANG
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Dedicated to our beloved Beta Katze
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Preface v PREFACE Welcome to Myster
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Contents Preface . . . . . . . . .
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2 History Lepenski Vir, located on
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4 History counter the sister-states
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6 History Figure 1.11: Chinese Wind
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8 History Wasan which was usually s
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10 History Figure 1.17: Pythagorean
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12 History Figure 1.24: Five Platon
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14 History Figure 1.26: Eight Conve
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16 History (a) (b) (c) Figure 1.31:
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18 History The equilateral triangle
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20 History Figure 1.36: Gothic Maso
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22 History Figure 1.40: Vesica Pisc
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24 History Figure 1.43: Alchemical
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26 History Modern sculpture has not
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28 History (a) (b) Figure 1.49: Tri
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30 Mathematical Properties The rela
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32 Mathematical Properties Figure 2
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38 Mathematical Properties 2.14(b)
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40 Mathematical Properties - Combin
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44 Mathematical Properties be the s
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56 Mathematical Properties (a) (b)
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58 Mathematical Properties Property
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72 Mathematical Properties The best
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74 Mathematical Properties in colum
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Chapter 3 Applications of the Equil
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80 Applications Application 2 (Sate
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82 Applications The ei are the proj
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84 Applications (a) Figure 3.9: (a)
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86 Applications Figure 3.11: Warren
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88 Applications Figure 3.14: Maxwel
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90 Applications Figure 3.17: De Fin
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92 Applications (a) (b) Figure 3.20
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94 Applications (a) Figure 3.23: Lo
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96 Applications Application 25 (Squ
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98 Applications (a) Figure 3.28: Na
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100 Applications (a) (b) (c) (d) Fi
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102 Applications The eigenstructure
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104 Mathematical Recreations Figure
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108 Mathematical Recreations (a) (b
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112 Mathematical Recreations of pla
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120 Mathematical Recreations and n
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124 Mathematical Recreations (a) Fi
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128 Mathematical Recreations Recrea
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130 Mathematical Competitions Probl
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132 Mathematical Competitions Figur
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Chapter 6 Biographical Vignettes In
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150 Biographical Vignettes Forms wh
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152 Biographical Vignettes Apolloni
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154 Biographical Vignettes He disco
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156 Biographical Vignettes He fell
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158 Biographical Vignettes give the
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160 Biographical Vignettes Vignette
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162 Biographical Vignettes Swiss 10
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164 Biographical Vignettes sität B
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166 Biographical Vignettes for Eucl
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168 Biographical Vignettes ellipse
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170 Biographical Vignettes Schwarz
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172 Biographical Vignettes the numb
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174 Biographical Vignettes a conseq
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176 Biographical Vignettes topologi
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178 Biographical Vignettes Vignette
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180 Biographical Vignettes and Recr
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Page 192 and 193: 184 Biographical Vignettes Figure 6
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Page 196 and 197: Appendix A Gallery of Equilateral T
Page 198 and 199: 190 Gallery Figure A.8: ET Wing Fig
Page 200 and 201: 192 Gallery Figure A.18: ET Bowling
Page 202 and 203: 194 Gallery Figure A.28: ET Escher
Page 204 and 205: 196 Gallery Figure A.36: ET Street
Page 206 and 207: 198 Gallery Figure A.44: ET Tragedy
Page 208 and 209: 200 Bibliography [12] J. Aubrey, Br
Page 210 and 211: 202 Bibliography [42] R. Calinger,
Page 212 and 213: 204 Bibliography [74] J.-P. Delahay
Page 214 and 215: 206 Bibliography [106] J. A. Flint,
Page 216 and 217: 208 Bibliography [138] M. Gardner,
Page 218 and 219: 210 Bibliography [169] H. Hellman,
Page 220 and 221: 212 Bibliography [199] M. Kraitchik
Page 222 and 223: 214 Bibliography [229] T. H. O’Be
Page 224 and 225: 216 Bibliography [260] B. Russell,
Page 226 and 227: 218 Bibliography [290] S. K. Stein,
Page 228 and 229: 220 Bibliography [319] A. Weil, Num
Page 230 and 231: 222 Index Barbier’s Theorem 56 ba
Page 232 and 233: 224 Index Cruise, Tom 24 Crusades 2
Page 234 and 235: 226 Index ture 157 Fermat’s Princ
Page 236 and 237: 228 Index house (triangular) 197 Hu
Page 238 and 239: 230 Index MacMahon, Percy Alexander
Page 242 and 243: 234 Index Riemann Surfaces 51, 167
Page 244 and 245: 236 Index Tartaglian Measuring Puzz
Page 246: 238 Index Weber, Wilhelm 164 Weiers
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