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Posamentier A.S. Ч The Fabulous Fibonacci Numbers
Posamentier A.S. Ч The Fabulous Fibonacci Numbers

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Ќазвание: The Fabulous Fibonacci Numbers

јвтор: Posamentier A.S.


The most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout natureЧfrom the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world.

With admirable clarity, math educators Alfred Posamentier and Ingmar Lehmann take us on a fascinating tour of the many ramifications of the Fibonacci numbers. The authors begin with a brief history of their distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples). In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the "golden ratio." And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and PascalТs triangle, to name a few. Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2007

 оличество страниц: 385

ƒобавлена в каталог: 22.10.2010

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ѕредметный указатель
"Bigollo"      see "Fibonacci Leonardo"
"Book of Squares"      see "Liber quadratorum (Fibonacci)"
"Book on calculation"      see "Liber Abaci (Fibonacci)"
"Cartesian plane"      131
"divine proportion"      245
"Dow theory" of investing      178
"Equiangular spiral"      131
"factors of a multiplication"      23
"factors of a number"      23
"Nest of radicals"      173
"Pascaline" (calculator)      84
"Periodicity"      14 14n3 30 108n1 350
"Periodicity", periodic decimals      112n6
"queen of science," mathematics as      328
"Sign of the Devil, The" (Wang)      213
"Valuable Mirror of the Four Elements, The" (Khayyam)      84n2
"Vitruvian" man      257
"wave theory" of investing      178Ч193
$\Phi$      see "Golden ratio (section)"
$\pi$      119Ч120 165Ч166 166n5 237
0 (the number)      37 50 111 205Ч206
11 (the number), Fibonacci numbers divisible by      33 331 350Ч351
12 (the number), Fibonacci numbers divisible by      331
144 (the number)      208 217 328
2 (the number), Fibonacci numbers divisible by      47 210 330
2 (the number), power of      196 196n9
20 Steps around Globe (created by Niemeyer)      249Ч252 250n21
3 (the number), Fibonacci numbers divisible by      47 210 330Ч331
3 (the number), rule for divisibility by      47n24
39th Mersenne prime number      69 69n7
4 (the number), Fibonacci numbers divisible by      330Ч331
4 (the number), relationship to 89      213n19
5 (the number), Fibonacci numbers divisible by      47
6 (the number), Fibonacci numbers divisible by      331
666 (the number)      212Ч213
8 (the number), Fibonacci numbers divisible by      48 331
89 (the number)      213Ч217 213n19 365Ч366
abacus      12 19n3
Acropolis (Athens)      232 234
Adam and Eve (Duerer)      15 268
Adam and Eve (Raimondi)      268
Al-Khowarizmi      19
Algebra      12 19Ч20 19n2
Algebraic number      165n2
Algorithms      12 198 198n10 310
Algorithms and ancient Egyptians      195n8
Algorithms, multiplication algorithm and Fibonacci numbers      195Ч199
Alternating numbers      42 42n23 44Ч45 54 54n25 55 142
Altevogt, Rudolf      76
American Library Association      211n16
Angle and fractals      311 313 314
Angle and isosceles triangles      144
Angle and pentagons      151 157
Angle and right triangles      128 134 135
Angle, angle sum of a triangle      84
Angle, bisectors      123 145 146 362
Angle, divergence angle      64n3
Angle, golden angle      64 64n3 74 148Ч149 250 250n20
Angle, used in watch displays      217Ч220 220n21
Angle, vertex angle      144 147
Animali da 1 a 55 (Merz)      252
Antennas (in fractals)      322Ч325
Aphrodite of Melos (statue)      246Ч247
Apollo Belvedere (statue)      245Ч246
Arabic numerals      see "Hindu numerals"
Architecture, Fibonacci numbers in      232Ч244
Arte povera      252
Arts and Fibonacci numbers      231Ч269
Babbitt, Milton      288
Babylonian formulae      193
Bach, Johann Sebastian      286
Baptism of Christ, The (Piero della Francesca)      268
Bar over digits, meaning of      14 14n3 30 108n1
Barbara, Saint      260Ч261
Baroque period      277 286
Bartok, Bela      285Ч288
Base-sixty numbering system      21 21n6
Base-ten numbering system      11
Base-two numbering system      198
Bathers at Asneres (Seurat)      263
Bear markets      179 182
Bees, male      13 59Ч61
Beethoven, Ludwig van      280Ч282
Beothy, Etienne      247Ч248
Bernoulli family      296n3
Bernoulli, Daniel      296
Bernoulli, Jacob      131Ч132
Bernoulli, Nicolaus, I      296
Bijugate spirals      69n8
Binary forms of music      274Ч275 277
Binary numbering system      198
Binet formula      207
Binet formula for finding a particular Fibonacci number      293Ч301 328 366Ч368
Binet formula for finding a particular Lucas number      301Ч302
Binet, Jacques-Philippe-Marie      28 293Ч294 296
Binomial coefficient      88n3
Binomial expansion      87Ч88 88n3
Binomial theorem      356
Birth of Venus, The (Botticelli)      259
Bisectors of an angle      123 145 146 362
Bizet, George      11
Bonacci, Guilielmo (William)      17Ч18
Bonacci, son of      see "Fibonacci Leonardo"
Boncompagni, Baldassarre      19n4
Borgliese, Pietro      See "Piero della Francesca"
Botticelli, Sandro      259
Boulez, Pierre      288
Bracts, counting of      13 27 63Ч64 65Ч66
Braun, A.      27
Brouseau, Alfred      65
Bruch, Hellmut      254Ч255
Brunelleschi, Filippo      239
Bulbs (in fractals)      322Ч325
Bull markets      179
Bury, Claus      255Ч256
Business applications of Fibonacci numbers      177Ч183
Cage, John      290
Calculator or computer used to find a Fibonacci number      303Ч304 368Ч369
Cantor, Georg      308
Cardiods (in fractals)      322Ч325 322n7
Carmen (Bizet)      11
Cassai, Tommaso      see "Masaccio"
Cathedral of Chartres (France)      237Ч238
Center for International Light Art (Germany)      253Ч254
Checkerboard, covering a      188Ч191
Cheops (Khufu) Pyramid (Giza)      see "Great Pyramid (Giza)"
Chinese remainder theorem      24
Chopin, Frederic      272Ч274
Chords of a circle      124 124n14
Chromatic scale      286
Chu Shih-Chieh      84n2
Circles and constructing a pentagon      156Ч158
Circles, chords of      124 124n14
Circles, concentric circles      133
Circles, congruent circles      139Ч140
Circles, great circle      250 250n19
Circles, sequence of      80Ч81 84 87
Circles, used to construct golden ratio      121Ч122 124 360Ч361
Circus Parade (Seurat)      263
Claus, M.      see "Lucas Francois-Edouard-Anatole
Coda in music      281 281n4
Codetta in music      281
Coins and vending machines      183Ч184
Collins, Charles      178 179
Cologne (Germany)      255Ч256
Columbia University, School of Library, Economy      211n16
Common differences      64 78 82 330 331 364
Common factors      33Ч34 33n20 94 192 192n5 351Ч352 see
Common fractions      20n5 24n12 30
complex numbers      120 320
Complex plane      320 320n5
Composite numbers      35 35n21 53Ч54 352Ч353
Composition with Colored Areas and Gray Lines I (Mondrian)      269n36
Composition with Gray and Light Brown (Mondrian)      269n36
Composition with Red Yellow Blue (Mondrian)      269n36
Compound interest      24
Computer or calculator used to find a Fibonacci number      303Ч304 368Ч370
Computers and music      289
Concentric circles      133
Congruent circles      139Ч140
Consecutive numbers      43Ч44 55 56
Consecutive numbers, four consecutive Fibonacci numbers      211
Consecutive numbers, odd numbers      295
Consecutive numbers, ratios of      109Ч110 210
Construction, constructing a pentagon      155Ч158
Construction, constructing fractals      310Ч317
Construction, constructing the golden ratio      120Ч124 362Ч365
Continued fractions and Fibonacci numbers      161Ч175 162n1
Continued fractions and Fibonacci numbers, finite continued fractions      163
Continued fractions and Fibonacci numbers, golden ratio as a continued fraction      166Ч172
Continued fractions and Fibonacci numbers, infinite continued fractions      163Ч164
Cook, Theodore Andrea      245
Corrective waves      178 179
Cossali, Pietro      17n1
Couder, Yves      74
Credit cards, measurements of      182
Crucifixion (Raphael)      268
Crystallography      342
Cubits (as a measurement)      236
CURVES      72 131 132Ч133 247 308 322
Curves of Life, The (Cook)      245
Curves, Gaussian curve      246 246n13
Dali, Salvador      268
Davis, T. Antony      76
De divina proportione (Pacioli)      245 257 260
De Moivre, Abraham      296
Decimal expansion      108n2
Denominator      23
Deposition from the Cross (Weyden)      268
Der goldene Schnitt (sculpture by Ulrichs)      248Ч249
Der goldene Schnitt (Zeising)      115 115n8
Der goldene Schnitt. Ein Harmoniege-setz und seine Anwendung (Hagenmaier)      248
Descartes, Rene      131
Devaney, Robert      319 319n4
Development in music      276 280 281 286
Devil, sign of      212 213
Dewey Decimal classification system      211 211n16 271n1
Dewey, Melvil      211n16 271n1
di Bondone, Giotto      see "Giotto (di Bondone)"
di Gherardo, Giovanni (da Prato)      239
Di minor guisa (Fibonacci)      20
Diagonal of the golden rectangle      138Ч139
Diatonic scales      272
Differences and sum of successive powers      297Ч298
Differences in Lucas numbers      301
Differences, common differences      64 78 82 330 331 364
Differences, Fibonacci differences      78Ч80 82 83 95n9
Differences, pattern in differences of squares      43Ч44 45Ч46 54 55 357 364
Differences, sequences of progressive differences      102
Differences, sum and difference      296Ч298
DIGITS      see "Integers" "Numbers"
Digits, curiosity of      208
Digits, first-digit patterns      207Ч208
Digits, last-digit patterns      206Ч207
Dionysius' Procession (relief at Villa Albani)      267
Distances, conversion of measures of      200Ч203
Divergence angle      64n3
Division, common divisor      337
Division, denominator      23
Division, divisibility of Fibonacci numbers      47Ч48 55Ч56 330 341 350 358
Division, divisors of composite numbers      353
Division, numerator      23
Division, sequences of remainders      31Ч32
dodecahedron      231
Dodgson, Charles Lutwidge      140
Dominant in music      285 285n5
Dominoes, covering a checkerboard      188Ч191
Dominoes, knocking down as example of mathematical induction      349Ч350
Door, height of      241
Douady, Stephane      74
Drones      13 59Ч61
Duchamp, Gaston      247 247n14
Dudley, Underwood      119
Duerer, Albrecht      15 155 158 258Ч259 259n26 268
Dyad in music      285 285n7
Dynamic symmetry      245 245n10
e      see "Euler's number"
Egyptian pyramids      180 234Ч237
Elements (Euclid)      20 171
Elements of Dynamic Symmetry, The (Hambidge)      245
Elliott, Ralph Nelson      177 178Ч183
Ellipse, area of      133 133n18
Empire State Building, climbing stairs of      185
Equal binary form of music      274Ч275 277
Equations, integer polynomial equation      165n2
Equations, linear equations      23
Equations, polynomial equation      165n2
Essor II (sculpture)      248
Euclid      12 19 20 84 171 307
Euler's number      120 165 165n2
Euler, Leonhard      120 165n2 296
Even-positioned Fibonacci numbers      47 210 330
Even-positioned Fibonacci numbers, sum of      37Ч38 54 353
Event seating and Fibonacci numbers      221Ч222
Evolution: Progression and Symmetry III (Mields)      266Ч267
Evolution: Progression and Symmetry IV (Mields)      266Ч267
Expansion, binomial      87Ч88 88n3
Exposition in music      280
Factors, common      33Ч34 33n20 192 192n5 351Ч352
Fatou, Pierre      308
Fechner, Gustav      115Ч117
Feet (as a measurement)      241 241n4
Feininger, Lyonel      268
Fibonacci Applications and Strategies for Traders (Fischer)      182
Fibonacci Association      17 28 329
Fibonacci multiplication algorithm      198Ч199
Fibonacci Napoli (Merz)      252
Fibonacci nim      225Ч226
Fibonacci numbers, mathematical aspects      see "Golden angle" "Golden "Golden "Golden
Fibonacci numbers, mathematical aspects and Lucas numbers      15 97Ч98 104Ч105 227 228Ч230 297Ч298
Fibonacci numbers, mathematical aspects and Pythagorean triples      192Ч194
Fibonacci numbers, mathematical aspects and the Pascal triangle      90Ч98 99 102 103 104Ч105
Fibonacci numbers, mathematical aspects, definition of      53
Fibonacci numbers, mathematical aspects, introduction to      26Ч32
Fibonacci numbers, mathematical aspects, list of first 500      343Ч348
Fibonacci numbers, mathematical aspects, proofs of Fibonacci relationships      349Ч358 360Ч369
Fibonacci numbers, mathematical aspects, properties of      33Ч56 337Ч341
Fibonacci numbers, mathematical aspects, testing to determine if it is      304Ч305 305n6
Fibonacci numbers, use of, and fractals      307Ч325
Fibonacci numbers, use of, and painting a house      186
Fibonacci numbers, use of, and physics      203Ч206
Fibonacci numbers, use of, and watch displays      217Ч220
Fibonacci numbers, use of, business applications of      177Ч183
Fibonacci numbers, use of, climbing a staircase      184Ч185
Fibonacci numbers, use of, converting miles and kilometers      200Ч203
Fibonacci numbers, use of, determining path of fish in a hatchery      222Ч224
Fibonacci numbers, use of, determining seating at events      221Ч222
Fibonacci numbers, use of, in art and architecture      231Ч269
Fibonacci numbers, use of, in geometry      136Ч143
Fibonacci numbers, use of, in music      271Ч291
Fibonacci numbers, use of, in nature      12 13 25Ч27 57 59Ч76 132 133
Fibonacci numbers, use of, in optics      203Ч206
Fibonacci phyllotaxis      64n3 70Ч74
Fibonacci Ratios      94 107Ч110 180 182 see
Fibonacci spirals      see "Golden spirals"
Fibonacci's Temple (Bury)      255Ч256
Fibonacci, Leonardo      11 17Ч22 23nn8Ч23nn9 56Ч57 266 327
Fifth Symphony (Beethoven)      280Ч282
Finding a particular Fibonacci number in a fractal      319 319n4
Finding a particular Fibonacci number, using a calculator or a computer      303Ч304 368Ч369
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