MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

More documents

Recommendations

Info

98 Applications (a) Figure 3.28: Natural Equilateral Triangles: (a) High Altitude Snow Crystal. (b) South American Butterfly Species. [54] (a) Figure 3.29: Equilateral Triangular Maps: (a) Cahill’s Butterfly Map (Octahedron). (b) Fisher’s Likeaglobe Map (Icosahedron). [132] (b) (b)
Applications 99 also based upon the icosahedron. It differs from Fisher’s Likeaglobe in having the North and South poles on opposite faces at points slightly off center. Figure 3.30: 3-Frequency Geodesic Dome [190] Application 30 (Geodesic Dome). A geodesic dome is a (portion of a) spherical shell structure based upon a network of great circles (geodesics) lying on the surface of a sphere that intersect to form triangular elements [190]. Since the sphere encloses the greatest volume for a given surface area, it provides for economical design (largest amount of internal space and minimal heat loss due to decreased outer skin surface) and, because of its triangulated nature, it is structurally stable. In its simplest manifestation [190], the twenty triangular faces of an inscribed icosahedron are subdivided into equilateral triangles by partitioning each edge into n (the “frequency”) segments. The resulting vertices are then projected onto the surface of the circumscribing sphere. The projected triangles are no longer congruent but are of two varieties thereby producing vertices with a valence of either five or six. The resulting geodesic structure has 12 5-valent vertices, 10(n 2 − 1) 6-valent vertices, 30n 2 edges and 20n 2 faces [152]. See Figure 3.30 for the case n = 3. The geodesic sphere may now be truncated to produce a dome of the desired height. Such geodesic domes were popularized in the architecture of R. Buckminster Fuller [113]. About 1960, biochemists employed electron microscopy to discover that some viruses have recognizable icosahedral symmetry and look like tiny geodesic domes [59].
Page 1 and 2:
MYSTERIES OF THE EQUILATERAL TRIANG
Page 3 and 4:
Dedicated to our beloved Beta Katze
Page 5 and 6:
Preface v PREFACE Welcome to Myster
Page 7:
Contents Preface . . . . . . . . .
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 42 and 43:
Page 44 and 45:
Page 46 and 47:
38 Mathematical Properties 2.14(b)
Page 48 and 49:
40 Mathematical Properties - Combin
Page 50 and 51:
Page 52 and 53:
44 Mathematical Properties be the s
Page 54 and 55:
Page 56 and 57: 48 Mathematical Properties Figure 2
Page 64 and 65: 56 Mathematical Properties (a) (b)
Page 66 and 67: 58 Mathematical Properties Property
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 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
Page 116 and 117: 108 Mathematical Recreations (a) (b
Page 120 and 121: 112 Mathematical Recreations of pla
Page 128 and 129: 120 Mathematical Recreations and n
Page 132 and 133: 124 Mathematical Recreations (a) Fi
Page 136 and 137: 128 Mathematical Recreations Recrea
Page 138 and 139: 130 Mathematical Competitions Probl
Page 140 and 141: 132 Mathematical Competitions Figur
Page 156 and 157:
Chapter 6 Biographical Vignettes In
Page 158 and 159:
150 Biographical Vignettes Forms wh
Page 160 and 161:
152 Biographical Vignettes Apolloni
Page 162 and 163:
154 Biographical Vignettes He disco
Page 164 and 165:
156 Biographical Vignettes He fell
Page 166 and 167:
158 Biographical Vignettes give the
Page 168 and 169:
160 Biographical Vignettes Vignette
Page 170 and 171:
162 Biographical Vignettes Swiss 10
Page 172 and 173:
164 Biographical Vignettes sität B
Page 174 and 175:
166 Biographical Vignettes for Eucl
Page 176 and 177:
168 Biographical Vignettes ellipse
Page 178 and 179:
170 Biographical Vignettes Schwarz
Page 180 and 181:
172 Biographical Vignettes the numb
Page 182 and 183:
174 Biographical Vignettes a conseq
Page 184 and 185:
176 Biographical Vignettes topologi
Page 186 and 187:
178 Biographical Vignettes Vignette
Page 188 and 189:
180 Biographical Vignettes and Recr
Page 190 and 191:
182 Biographical Vignettes meeting)
Page 192 and 193:
184 Biographical Vignettes Figure 6
Page 194 and 195:
186 Biographical Vignettes Figure 6
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 240 and 241:
232 Index oriented triangles 70 ori
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
show all

MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Create successful ePaper yourself

Delete template?

Save as template?