The (Fabulous) Fibonacci Numbers. Alfred S. Posamentier & Ingmar Lehmann. Afterword by Herbert Hauptman, Nobel Laureate

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1 The (abulous) ibonacci Numbers Alfred S. Posamentier & Ingmar Lehmann Afterword by Herbert Hauptman, Nobel Laureate Amherst (New York), Prometheus Books, 007, 385 p. ISBN Translation by Nikkei Business Publications, Inc. arranged through Tuttle-Mori Agency, Inc., Tokyo, 00 ISBN Publisher s notes (On Book Jacket) The most ubiquitous, and perhaps most intriguing, number pattern in all of mathematics is the ibonacci sequence. In this strikingly simple pattern, beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (,,, 3, 5, 8, 3,, ad infinitum). ar from being just a mathematical curiosity, however, this sequence recurs throughout nature from the regeneration patterns of bees and rabbits to the arrangement of spirals on pinecones and pineapples. All of which is astounding evidence for the deep mathematical basis of the natural world. With admirable insight and clarity, math educators Alfred Posamentier and Ingmar Lehmann take us on an utterly fascinating tour of the many ramifications of the ibonacci numbers. The authors begin with a brief history of their distinguished thirteenth-century Italian mathematician Leonardo of Pisa (more commonly known as ibonacci), whose other accomplishments including popularizing the use of Arabic numerals in the West. Turning to the field of botany, the authors demonstrate through illustrative diagrams, the many amazing connections between the ibonacci numbers and natural forms (including pinecones, pineapples, sunflowers, and daisies). In art, architecture, the stock market, and others areas of society and culture, Posamentier and Lehmann find an almost endless array of instances where the ibonacci sequence, as well as its derivative, the golden ratio, makes an appearance. And, of course, as the authors amply demonstrate, there are almost boundless applications in probability, algebra, and Pascal s triangle, to name but a few. Thoroughly accessible and appealing to even the math-phobic individual, this fun-filled and enlightening book allows the reader to appreciate the true elegance of mathematics and its amazing applications in the world around us.

2 Contents Acknowledgments 9 Introduction Chapter : A History and Introduction to the ibonacci Numbers 7 Chapter : The ibonacci Numbers in Nature 59 Chapter 3: The ibonacci Numbers and the Pascal Triangle 77 Chapter 4: The ibonacci Numbers and the Golden Ratio 07 Chapter 5: The ibonacci Numbers and Continued ractions 6 Chapter 6: A Potpourri of ibonacci Number Applications 77 Chapter 7: The ibonacci Numbers ound in Art and Architecture 3 Chapter 8: The ibonacci Numbers and Musical orm 7 Chapter 9: The amous Binet ormula for inding a Particular ibonacci Number 93 Chapter 0: The ibonacci Numbers and ractals 307 Epilogue 37 Afterword by Herbert A. Hauptman 39 Appendix A: List of the irst 500 ibonacci Numbers, with the irst 00 ibonacci Numbers actored 343 Appendix B: Proofs of ibonacci acts 349 References 37 Index 375

3 The abulous ibonacci Numbers By Alfred S. Posamentier and Ingmar Lehmann Prometheus Books, 008 Errata irst Edition: Third Printing 3

4 The abulous ibonacci Numbers By Alfred S. Posamentier and Ingmar Lehmann Prometheus Books, 007 Errata irst Edition: Second Printing Page, ootnote : Change the word Hindu to Hindu-Arabic. Page, Line 9: Change the last digit of to an 8, so as to read: Page, Line 8 from the bottom: change 4 4 to read Page, ootnote 7 change the word integer to number. Page 3, ootnote 8: Change the word Hindu to Hindu-Arabic. Page 5, Line 6: Make a plural: notes included): and Line 4 from bottom remove quotation marks at end. Page 9, last Line: Delete either. Page 3, igure -6: The entries for each of the following must be corrected: = 6,765 = = 0,946 = = 37,8 = 45, = 83, 040 = 8, Page 34, igure -9: Some exponents are not written as exponents: e.g. n = 6: 3 instead of 3 n n actors unit unit 3 prime 4 3 prime 5 5 prime prime prime prime

5 n n actors prime prime prime Page 4, Line 6: To read with the insertion: merely the product of the last number that is squared and Page 44, Line 6: Change sum to difference : to read: the difference of the squares Page 45, Line from bottom: Change differ to differs. Page 47, Second Line from the bottom: Change 765 to 6,765, 75,05, and Page 47, ootnote 4: Delete 75,05, Page 48, igures -4 and -5: Both squares contain an odd number of rectangles; n = 7 and n = 9. Page 48, next-to-last Line: To read: to n. The square then (Drop the +) Page 49, Line 7: In igure -5, where n = 9, the sum of the rectangles is = = 56 = 34 = 9 5

6 Page 5, Line 6/7: Insert twice the in the last two expressions, to read as follows: L + L + L 3 + L 4 + L 5 + L 6 = = 50 = 5 = 8 9 = L 6 L 7. Page 54, ootnote 5: Insert position and should read: ibonacci number with the position number n Page 55, Line : Delete ± before the colon. Page 56, item 3 should read: The sum of the products of consecutive ibonacci numbers is either the square of a ibonacci number or one less than the square of a ibonacci number. Page 60, igure -: In the column emale Bees the last two numbers should be 8 and 3 (not 3 and ). Page 6, Line 5 from bottom: Change p = to p n = n +. n n Page 68, Line 6: The last ratio is to be: 89:44. Page 79, igure 3-3: Unbold 5 and 3. Page 80: ootnote should read `fourth difference', not `third difference'. Page 00: Last five lines (ig. 3-6) have some incomplete plus signs that look something like this: Page 0, last Line: Insert ca. in (ca BCE) Page 0, ootnote 5: Phidias or Pheidias; in Greek: Φειδίας. Page 4, Line 8: Change we havebegun to we have begun Page 6, igure 4-4: The second entry in the middle column: Change.0 to 0.. Page 7, Line 7 from the bottom: Change the for an l in the equation. To read: ww ( + l) = l. Page 7: Move footnote 0 to page 3, line 7 after the word formula. Page, Line 4 after the parentheses ) add the following:, where BC = AB." Page 33, ootnote 8, Line 3: Change formula to formulas. Page 33, ootnote 8, Line 4: Interchange the a and b. a b Page 33, ootnote 8, Line 7: Change to b a. 6

7 Page 38, Line 5: point S divides the diagonal in the golden ratio should read: point S divides the diagonal in the square of the golden ratio. Page 4, Line : Change from 5 3 to 5 3 Page 50, Line 4: Delete therefore. Page 5, Line 7 from bottom: Change last triangle from CEA to CEB. Page 57, Line 3: Change it s to its. Page 63, Line 4: ourth convergent make the c in convergent as a lower case. Page 70: The third continued fraction should show n times the end part. Page 7, Line 6: Change Φ < μ < to μ < < Φ. Page 94, third Line from bottom: Delete last two terms + k+ k+ Page 09, Line 5: Change number to numbers. Page, 0, Line 3: Delete can enhance this by staying and replace it with recall. Page 6, Line 8: Should read 0 3 = , where an additional 0 is added on. Page 8, Line 4: hour markers or minute markers Page 3, Line 6 from bottom: Change ΦΙΔΙΑΣ to Φειδίας. Page 35, first line: Change dates to ( ca. 484 ca. 44 BCE). Page 6, Line 5: Change ration to ratio. Page 95: Last line period missing at end. n Page 98: Replace the last three lines with the following: φ ( ) n n n n = φ. Page 307, Line 6 from bottom: Remove the period after the word Their. Page 3, ootnote : Replace Waclaw with Wacław. Page 340, Line 3: Change usthe to us the Page 35, Line 3 from bottom: Replace induction with proof by contradiction ; last line: delete induction. φ φ 7

8 Page 35, Line 8: Replace induction with proof by contradiction. Page 35, ootnote : Insert that to read: A subsidiary proposition that is used to prove Page 353, Line 4: Delete or equal to to read for n greater than, Page 356, Lines 3 and 4 from bottom: Delete the binomial theorem. Page 357, Last line: the exponent for ( ) should be k, not k + to read = ( ) k Page 364: Delete the first sentence at the top of the page. 8

9 The abulous ibonacci Numbers By Alfred S. Posamentier and Ingmar Lehmann Prometheus Books, 007 Errata irst Edition: irst Printing Title Page: Title should read Nobel Laureate Page, ootnote : Change the word Hindu to Hindu-Arabic. Page, Line 9: Change the last digit of to an 8, so as to read: Page, Line 8 from the bottom: change 4 4 to read Page, ootnote 7 change the word integer to number. Page 3, ootnote 8: Change the word Hindu to Hindu-Arabic. Page 5, Line 6: Make a plural: notes included): and Line 4 from bottom remove quotation marks at end. Page 9, last Line: Delete either. Page 3, igure -6: The entries for each of the following must be corrected: = 6,765 = = 0,946 = = 37,8 = 45, = 83, 040 = 8, Page 34, igure -9: Some exponents are not written as exponents: e.g. n = 6: 3 instead of 3 n n actors unit unit 3 prime 4 3 prime 5 5 prime prime prime prime

10 n n actors prime prime prime Page 4, Line 6: To read with the insertion: merely the product of the last number that is squared and Page 44, Line 6: Change sum to difference : to read: the difference of the squares Page 45, Line from bottom: Change differ to differs. Page 47, Second Line from the bottom: Change 765 to 6,765, 75,05, and Page 47, ootnote 4: Delete 75,05, Page 48, igures -4 and -5: Both squares contain an odd number of rectangles; n = 7 and n = 9. Page 48, next-to-last Line: To read: to n. The square then (Drop the +) Page 49, Line 7: In igure -5, where n = 9, the sum of the rectangles is =

11 = 56 = 34 = 9 Page 5, Line 6/7: Insert twice the in the last two expressions, to read as follows: L + L + L 3 + L 4 + L 5 + L 6 = = 50 = 5 = 8 9 = L 6 L 7. Page 54, ootnote 5: Insert position and should read: ibonacci number with the position number n Page 55, Line : Delete ± before the colon. Page 56, item 3 should read: The sum of the products of consecutive ibonacci numbers is either the square of a ibonacci number or one less than the square of a ibonacci number. Page 60, igure -: In the column emale Bees the last two numbers should be 8 and 3 (not 3 and ). Page 6, Line 5 from bottom: Change p = to p n = n +. n n Page 68, Line 6: The last ratio is to be: 89:44. Page 79, igure 3-3: Unbold 5 and 3. Page 80: ootnote should read `fourth difference', not `third difference'. Page 00: Last five lines (ig. 3-6) have some incomplete plus signs that look something like this: Page 0, last Line: Insert ca. in (ca BCE) Page 0, ootnote 5: Phidias or Pheidias; in Greek: Φειδίας. Page 4, Line 8: Change we havebegun to we have begun Page 6, igure 4-4: The second entry in the middle column: Change.0 to 0.. Page 7, Line 7 from the bottom: Change the for an l in the equation. To read: ww ( + l) = l. Page 7: Move footnote 0 to page 3, line 7 after the word formula. Page, Line 4: After the parentheses ) add the following:, where BC = AB." Page, Lines and : To do the construction begin with a right triangle ABC where AB = a and AC = a/. Then draw a circle with center C and radius length CB. Page 33, ootnote 8, Line 3: Change formula to formulas.

12 Page 33, ootnote 8, Line 4: Interchange the a and b. Page 33, ootnote 8, Line 7: Change a b to b a. Page 38, Line 5: point S divides the diagonal in the golden ratio should read: point S divides the diagonal in the square of the golden ratio. Page 4, Line : Change from 5 3 to 5 3 Page 50, Line 4: Delete therefore. Page 5, Line 7 from bottom: Change last triangle from CEA to CEB. Page 57, Line 3: Change it s to its. Page 63, Line 4: ourth convergent make the c in convergent as a lower case. Page 70: The third continued fraction should show n times the end part. Page 7, Line 6: Change Φ < μ < to μ < < Φ. Page 85: The next to last line should end with 76, the last line is to begin with: Page 88, Line 8, right column: Replace 0 = + 6 = + 8 with 0 = = + 8. Page 94, third Line from bottom: delete last two terms + k+ k+ Page 04: igure 6-4 is missing, instead figure 6-5 appears twice. igure 6-4 should be: igure 6-4 Page 09, Line 5: Change number to numbers. Page, 0, Line 3: Delete can enhance this by staying and replace it with recall. Page 6, Line 8: Should read 0 3 = , where an additional 0 is added on. Page 8, Line : Replace the 55 with 5. Page 8, Line 4: hour markers or minute markers

13 Page 3, Line 6 from bottom: Change ΦΙΔΙΑΣ to Φειδίας. Page 35, first line: Change dates to ( ca. 484 ca. 44 BCE). Page 6, Line 5: Change ration to ratio Page 93, last Line of first paragraph should be offer not after. Page 95: Last line period missing at end. n Page 98: Replace the last three lines with the following: φ ( ) Page 307, Line 6 from bottom: Remove the period after the word Their. n n n n = φ. Page 98, last Line should say = φ n ( /φ ) n instead of = φ n (/φ ) n. Page 3, ootnote : Replace Waclaw with Wacław. Page 333, Line 9: The 4 should be changed to 5. Page 340, Line 3: Change usthe to us the Page 35, Line 3 from bottom: Replace induction with proof by contradiction ; last line: delete induction. Page 35, Line 8: Replace induction with proof by contradiction. Page 35, ootnote : Insert that to read: A subsidiary proposition that is used to prove Page 353, Line 4: Delete or equal to to read for n greater than, Page 356, Lines 3 and 4 from bottom: Delete the binomial theorem. Page 357: Last line: the exponent for ( ) should be k, not k + to read = ( ) k Page 358: Line : Delete the complete line: = ( ) k Page 359: Lines 3, 7, and 7 need a + sign before the L k term. Page 364: Delete the first sentence at the top of the page. φ φ 3

14 We appreciate any comments about the book as well as any typographical errors that have not yet been detected so that they can be incorporated in future printings of the book. Alfred S. Posamentier: The City College of the City University of New York Ingmar Lehmann: Humboldt University of Berlin 4