Íàøëè îïå÷àòêó? Âûäåëèòå åå ìûøêîé è íàæìèòå Ctrl+Enter
Íàçâàíèå: The Fabulous Fibonacci Numbers
Àâòîð: Posamentier A.S.
Àííîòàöèÿ:
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 nature—from 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 Pascal’s 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.
Complex plane320320n5 Composite numbers3535n2153—54352—353 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 interest24 Computer or calculator used to find a Fibonacci number303—304368—370 Computers and music289 Concentric circles133 Congruent circles139—140 Consecutive numbers43—445556 Consecutive numbers, four consecutive Fibonacci numbers211 Consecutive numbers, odd numbers295 Consecutive numbers, ratios of109—110210 Construction, constructing a pentagon155—158 Construction, constructing fractals310—317 Construction, constructing the golden ratio120—124362—365 Continued fractions and Fibonacci numbers161—175162n1 Continued fractions and Fibonacci numbers, finite continued fractions163 Continued fractions and Fibonacci numbers, golden ratio as a continued fraction166—172 Continued fractions and Fibonacci numbers, infinite continued fractions163—164 Cook, Theodore Andrea245 Corrective waves178179 Cossali, Pietro17n1 Couder, Yves74 Credit cards, measurements of182 Crucifixion (Raphael)268 Crystallography342 Cubits (as a measurement)236 CURVES72131132—133247308322 Curves of Life, The (Cook)245 Curves, Gaussian curve246246n13 Dali, Salvador268 Davis, T. Antony76 De divina proportione (Pacioli)245257260 De Moivre, Abraham296 Decimal expansion108n2 Denominator23 Deposition from the Cross (Weyden)268 Der goldene Schnitt (sculpture by Ulrichs)248—249 Der goldene Schnitt (Zeising)115115n8 Der goldene Schnitt. Ein Harmoniege-setz und seine Anwendung (Hagenmaier)248 Descartes, Rene131 Devaney, Robert319319n4 Development in music276280281286 Devil, sign of212213 Dewey Decimal classification system211211n16271n1 Dewey, Melvil211n16271n1 di Bondone, Giottosee "Giotto (di Bondone)" di Gherardo, Giovanni (da Prato)239 Di minor guisa (Fibonacci)20 Diagonal of the golden rectangle138—139 Diatonic scales272 Differences and sum of successive powers297—298 Differences in Lucas numbers301 Differences, common differences647882330331364 Differences, Fibonacci differences78—80828395n9 Differences, pattern in differences of squares43—4445—465455357364 Differences, sequences of progressive differences102 Differences, sum and difference296—298 DIGITSsee "Integers""Numbers" Digits, curiosity of208 Digits, first-digit patterns207—208 Digits, last-digit patterns206—207 Dionysius' Procession (relief at Villa Albani)267 Distances, conversion of measures of200—203 Divergence angle64n3 Division, common divisor337 Division, denominator23 Division, divisibility of Fibonacci numbers47—4855—56330341350358 Division, divisors of composite numbers353 Division, numerator23 Division, sequences of remainders31—32 dodecahedron231 Dodgson, Charles Lutwidge140 Dominant in music285285n5 Dominoes, covering a checkerboard188—191 Dominoes, knocking down as example of mathematical induction349—350 Door, height of241 Douady, Stephane74 Drones1359—61 Duchamp, Gaston247247n14 Dudley, Underwood119 Duerer, Albrecht15155158258—259259n26268 Dyad in music285285n7 Dynamic symmetry245245n10 esee "Euler's number" Egyptian pyramids180234—237 Elements (Euclid)20171 Elements of Dynamic Symmetry, The (Hambidge)245 Elliott, Ralph Nelson177178—183 Ellipse, area of133133n18 Empire State Building, climbing stairs of185 Equal binary form of music274—275277 Equations, integer polynomial equation165n2 Equations, linear equations23 Equations, polynomial equation165n2 Essor II (sculpture)248 Euclid12192084171307 Euler's number120165165n2 Euler, Leonhard120165n2296 Even-positioned Fibonacci numbers47210330 Even-positioned Fibonacci numbers, sum of37—3854353 Event seating and Fibonacci numbers221—222 Evolution: Progression and Symmetry III (Mields)266—267 Evolution: Progression and Symmetry IV (Mields)266—267 Expansion, binomial87—8888n3 Exposition in music280 Factors, common33—3433n20192192n5351—352 Fatou, Pierre308 Fechner, Gustav115—117 Feet (as a measurement)241241n4 Feininger, Lyonel268 Fibonacci Applications and Strategies for Traders (Fischer)182 Fibonacci Association1728329 Fibonacci multiplication algorithm198—199 Fibonacci Napoli (Merz)252 Fibonacci nim225—226 Fibonacci numbers, mathematical aspectssee "Golden angle""Golden"Golden"Golden Fibonacci numbers, mathematical aspects and Lucas numbers1597—98104—105227228—230297—298 Fibonacci numbers, mathematical aspects and Pythagorean triples192—194 Fibonacci numbers, mathematical aspects and the Pascal triangle90—9899102103104—105 Fibonacci numbers, mathematical aspects, definition of53 Fibonacci numbers, mathematical aspects, introduction to26—32 Fibonacci numbers, mathematical aspects, list of first 500343—348 Fibonacci numbers, mathematical aspects, proofs of Fibonacci relationships349—358360—369 Fibonacci numbers, mathematical aspects, properties of33—56337—341 Fibonacci numbers, mathematical aspects, testing to determine if it is304—305305n6 Fibonacci numbers, use of, and fractals307—325 Fibonacci numbers, use of, and painting a house186 Fibonacci numbers, use of, and physics203—206 Fibonacci numbers, use of, and watch displays217—220 Fibonacci numbers, use of, business applications of177—183 Fibonacci numbers, use of, climbing a staircase184—185 Fibonacci numbers, use of, converting miles and kilometers200—203 Fibonacci numbers, use of, determining path of fish in a hatchery222—224 Fibonacci numbers, use of, determining seating at events221—222 Fibonacci numbers, use of, in art and architecture231—269 Fibonacci numbers, use of, in geometry136—143 Fibonacci numbers, use of, in music271—291 Fibonacci numbers, use of, in nature121325—275759—76132133 Fibonacci numbers, use of, in optics203—206 Fibonacci phyllotaxis64n370—74 Fibonacci Ratios94107—110180182see Fibonacci spiralssee "Golden spirals" Fibonacci's Temple (Bury)255—256 Fibonacci, Leonardo1117—2223nn8—23nn956—57266327 Fifth Symphony (Beethoven)280—282 Finding a particular Fibonacci number in a fractal319319n4 Finding a particular Fibonacci number, using a calculator or a computer303—304368—369