MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

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vi Preface Many of these plums may be absorbed through light skimming while others require considerable effort to digest. Caveat emptor: No attempt has been made either to distinguish between the two types or to segregate them. In Chapter 3, we take up the place of the equilateral triangle in Applied Mathematics. Some of the selected applications, such as antenna design and electrocardiography, are quite conventional while others, such as drilling a square hole and wrapping chocolates, are decidedly unconventional. I have based the selection of topics upon my desire to communicate the sheer breadth of such applications. Thus, the utilization of the equilateral triangle in detecting gravitational waves, the construction of superconducting gaskets, cartography, genetics, game theory, voting theory et cetera have all been included. The subject of Chapter 4 is the role of the equilateral triangle in Recreational Mathematics. Traditional fare such as dissection puzzles appear on the menu, but so do more exotic delicacies such as rep-tiles and spidrons. Devotees of the work of Martin Gardner in this area will instantly recognize my considerable indebtedness to his writings. Given his extensive contributions to Recreational Mathematics, this pleasant state of affairs is simply unavoidable. Chapter 5 contains a collection of olympiad-caliber problems on the equilateral triangle selected primarily from previous Mathematical Competitions. No solutions are included but readily available collections containing complete solutions are cited chapter and verse. Unless otherwise attributed, the source material for the biographical vignettes of Chapter 6 was drawn from Biographical Dictionary of Mathematicians [144], MacTutor History of Mathematics [230] and Wikipedia, The Free Encyclopedia [330]. Finally, we bid adieu to the equilateral triangle by taking a panoramic view of its many manifestations in the world about us. Thus, MOTET concludes with a Gallery of Equilateral Triangles that has been appended and which documents the multifarious and ubiquitous appearances of the equilateral triangle throughout the natural and man-made worlds. I owe a steep debt of gratitude to a succession of highly professional Interlibrary Loan Coordinators at Kettering University: Joyce Keys, Meg Wickman and Bruce Deitz. Quite frankly, without their tireless efforts in tracking down many times sketchy citations, whatever scholarly value may be attached to the present work would be substantially diminished. Also, I would like to warmly thank my Teachers: Harlon Phillips, Oved Shisha, Ghasi Verma and Antony Jameson. Each of them has played a significant role in my mathematical development and for that I am truly grateful. Once again, my loving wife Barbara A. (Rowe) McCartin has lent her Mathematical Artistry to the cover illustration thereby enhancing the appearance of this work. Brian J. McCartin Fellow of the Electromagnetics Academy Editorial Board, Applied Mathematical Sciences
Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1 History of the Equilateral Triangle 1 2 Mathematical Properties of the Equilateral Triangle 29 3 Applications of the Equilateral Triangle 78 4 Mathematical Recreations 103 5 Mathematical Competitions 129 6 Biographical Vignettes 148 A Gallery of Equilateral Triangles 188 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 vii
Page 1 and 2: MYSTERIES OF THE EQUILATERAL TRIANG
Page 3 and 4: Dedicated to our beloved Beta Katze
Page 5: Preface v PREFACE Welcome to Myster
Page 10 and 11: 2 History Lepenski Vir, located on
Page 12 and 13: 4 History counter the sister-states
Page 14 and 15: 6 History Figure 1.11: Chinese Wind
Page 16 and 17: 8 History Wasan which was usually s
Page 18 and 19: 10 History Figure 1.17: Pythagorean
Page 20 and 21: 12 History Figure 1.24: Five Platon
Page 22 and 23: 14 History Figure 1.26: Eight Conve
Page 24 and 25: 16 History (a) (b) (c) Figure 1.31:
Page 26 and 27: 18 History The equilateral triangle
Page 28 and 29: 20 History Figure 1.36: Gothic Maso
Page 30 and 31: 22 History Figure 1.40: Vesica Pisc
Page 32 and 33: 24 History Figure 1.43: Alchemical
Page 34 and 35: 26 History Modern sculpture has not
Page 36 and 37: 28 History (a) (b) Figure 1.49: Tri
Page 38 and 39: 30 Mathematical Properties The rela
Page 40 and 41: 32 Mathematical Properties Figure 2
Page 46 and 47: 38 Mathematical Properties 2.14(b)
Page 48 and 49: 40 Mathematical Properties - Combin
Page 52 and 53: 44 Mathematical Properties be the s
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48 Mathematical Properties Figure 2
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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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182 Biographical Vignettes meeting)
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184 Biographical Vignettes Figure 6
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186 Biographical Vignettes Figure 6
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Appendix A Gallery of Equilateral T
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190 Gallery Figure A.8: ET Wing Fig
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192 Gallery Figure A.18: ET Bowling
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194 Gallery Figure A.28: ET Escher
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196 Gallery Figure A.36: ET Street
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198 Gallery Figure A.44: ET Tragedy
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200 Bibliography [12] J. Aubrey, Br
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202 Bibliography [42] R. Calinger,
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204 Bibliography [74] J.-P. Delahay
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206 Bibliography [106] J. A. Flint,
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208 Bibliography [138] M. Gardner,
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210 Bibliography [169] H. Hellman,
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212 Bibliography [199] M. Kraitchik
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214 Bibliography [229] T. H. O’Be
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216 Bibliography [260] B. Russell,
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218 Bibliography [290] S. K. Stein,
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220 Bibliography [319] A. Weil, Num
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222 Index Barbier’s Theorem 56 ba
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224 Index Cruise, Tom 24 Crusades 2
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226 Index ture 157 Fermat’s Princ
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228 Index house (triangular) 197 Hu
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230 Index MacMahon, Percy Alexander
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232 Index oriented triangles 70 ori
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234 Index Riemann Surfaces 51, 167
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236 Index Tartaglian Measuring Puzz
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238 Index Weber, Wilhelm 164 Weiers
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