MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

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56 Mathematical Properties (a) (b) Figure 2.37: Curves of Constant Width: (a) Sharp Reuleaux Triangle. (b) Rounded Reuleaux Triangle. Like the circle, it is a curve of constant breadth, the breadth (width) being equal to the length of the triangle side [291]. Whereas the circle encloses the largest area amongst constant-width curves with a fixed width, w, the Reuleaux triangle encloses the smallest area amongst such curves (Blaschke- Lebesgue Theorem) [39]. (The enclosed area is equal to (π− √ 3)w 2 /2 [125]; by Barbier’s theorem [39], the perimeter equals πw) Moreover, a constant-width curve cannot be more pointed than 120 ◦ , with the Reuleaux triangle being the only one with a corner of 120 ◦ [249]. Like all curves of constant-width, the Reuleaux triangle is a rotor for a square, i.e. it can be rotated so as to maintain contact with the sides of the square, but the center of rotation is not fixed [241]. In the case of the Reuleaux triangle, the square with rounded corners that is out swept out has area that is approximately equal to .9877 times the area of the square. (See Application 25.) The corners of the Reuleaux triangle can be smoothed by extending each side of the equilateral triangle a fixed distance at each end and then constructing six circular arcs centered at the vertices as shown in Figure 2.37(b). The constant-width of the resulting smoothed Reuleaux triangle is equal to the sum of the two radii so employed [125]. Property 57 (Least-Area Rotor). The least-area rotor for an equilateral triangle is formed from two 60 ◦ circular arcs with radius equal to the altitude of the triangle [291] (which coincides with the length of the rotor [322]) (Figure 2.38). (The incircle is the greatest-area rotor [39].) As it rotates, its corners trace the entire boundary of the triangle without rounding of corners [125], although it must slip as it rolls [291].
Mathematical Properties 57 Figure 2.38: Least-Area Rotor Figure 2.39: Triangular Rotor Property 58 (Equilateral Triangular Rotor). Obviously, the equilateral triangle is a rotor for a circle. Yet, it can also be a rotor for a noncircular cylinder. Figure 2.39 shows the curve given parametrically as x(t) = cost + .1 cos 3t; y(t) = sint + .1 sin 3t, within which rotates an equilateral triangular piston [291]. (See Application 26.) Figure 2.40: Kakeya Needle Problem [125]
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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
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
Page 66 and 67: 58 Mathematical Properties Property
Page 72 and 73: 64 Mathematical Properties (a) (b)
Page 78 and 79: 70 Mathematical Properties (a) (b)
Page 80 and 81: 72 Mathematical Properties The best
Page 82 and 83: 74 Mathematical Properties in colum
Page 86 and 87: Chapter 3 Applications of the Equil
Page 88 and 89: 80 Applications Application 2 (Sate
Page 90 and 91: 82 Applications The ei are the proj
Page 92 and 93: 84 Applications (a) Figure 3.9: (a)
Page 94 and 95: 86 Applications Figure 3.11: Warren
Page 96 and 97: 88 Applications Figure 3.14: Maxwel
Page 98 and 99: 90 Applications Figure 3.17: De Fin
Page 100 and 101: 92 Applications (a) (b) Figure 3.20
Page 102 and 103: 94 Applications (a) Figure 3.23: Lo
Page 104 and 105: 96 Applications Application 25 (Squ
Page 106 and 107: 98 Applications (a) Figure 3.28: Na
Page 108 and 109: 100 Applications (a) (b) (c) (d) Fi
Page 110 and 111: 102 Applications The eigenstructure
Page 112 and 113: 104 Mathematical Recreations Figure
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106 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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MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

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