# Fibonacci Facts

INFORMATION SHEET ON FIBONACCI NUMBERS

The Fibonacci sequence first appeared as the solution to a problem in the Liber Abaci, a book written in 1202 by Leonardo Fibonacci of Pisa to introduce the Hindu-Arabic numerals used today to a Europe still using cumbersome Roman numerals. The original problem in the Liber Abaci asked how many pairs of rabbits can be generated from a single pair, if each month each mature pair brings forth a new pair, which, from the second month, becomes productive. The resulting Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ..., have been the subject of continuing research, especially by the Fibonacci Association, publisher of the Fibonacci Quarterly since 1963.

If Fn is the nth Fibonacci number, then successive terms are formed by addition of the previous two terms, as Fn+1 = Fn + Fn-1, F1 = 1, F2 =

1. The Fibonacci numbers are found to have many relationships to the Golden Ratio F = (1 + /5)/2, a constant of nature and a value which fascinated the ancient Greeks, appearing throughout Greek art and architecture. One can verify with a hand calculator that the ratio of Fn+1 to Fn is approximated by 1.6180339..., which is the decimal equivalent of the Golden Ratio.

Aside from an extraordinary wealth of mathematical relationships, there are numerous applications of the Fibonacci numbers in the most diverse fields. The Fibonacci Quarterly (FQ) has served as a focal point for widespread interest in the Fibonacci and related sequences, especially with respect to new results, challenging problems, and innovative proofs. The following references for Fibonacci number theory and applications in botany, biology, physics, music, and art provide an anthology of attractive mathematical ideas, selected from an abundance of material. Articles from all back issues of FQ are available from Bell & Howell Information and Learning, Article Clearinghouse Division, 300 North Zeeb Road, Ann Arbor MI 48106-1346.

1. BOTANY, BIOLOGY. The growth patterns of plants; the geneological tree of the male bee; the crossroads of mathematics and biology.

*S. L. Basin. "The Fibonacci Sequence Appears in Nature." FQ 1.1(1963): 53-56.

S. R. Braun. "Botany with a Twist." Science, May, l986, p. 63-64. Bro. A. Brousseau. "On the Trail of the California Pine." FQ 6.1(1968): 69-76.

H. S. M. Coxeter. Intro. to Geometry. New York: John Wiley, 1961. pp.169-172.

S. Douady and Y. Couder. "Phyllotaxis as a Physical Self-Organized Growth Process." Physical Review Letters 68.13 (1992): 2098-2101.

J. A. H. Hunter and J. S. Madachy. Mathematical Diversions. Princeton: Van Nostrand, 1963. Chapter 2.

Roger V. Jean. "Growth Matrices in Phyllotaxis." Mathematical Biosciences 32 (1976): 165-176.

Roger V. Jean. The Use of Continued Fractions in Botany: UMAP Module 571, Modules and Monographs in Undergraduate Mathematics and Its Applications Project, 1986.

Roger V. Jean. Mathematical Approach to Pattern and Form in Plant Growth, New York: John Wiley, 1984. Frank Land. The Language of Mathematics. New York: Doubleday, 1963.

*Sister Mary de Sales McNabb. "Phyllotaxis." FQ 1.4 (1963): 57-60. Ian Stewart. "Daisy, Daisy, Give me your answer, do." Scientific American 72.1 (1995): 96-99.

C. Sutton. "Sunflower Spirals Obey Laws of Mathematics." New Scientist , 18 April 1992, p. 16.

D'Arcy W. Thompson. On Growth and Form, New York: McMillan, 1944.

2. PHYSICAL SCIENCES. Atomic fission asymmetries; ladder and cascade electronic network analysis; reflection paths of light; musical properties; computer programming and search strategies; tributary patterns of streams and drainage patterns; fractal branching of diffusion aggregates.

J. Arkin. "Ladder Network Analysis Using Polynomials." FQ 3.2 (1965): 139-142.

A. Arneodo, F. Argoul, E. Bacry, J. F. Muzy, and M. Tabard. "Golden Mean Arithmetic in the Fractal Branching of Diffusion-Limited Aggregates." Physical Review Letters 68.23 (1992): 3456-3459.

R. E. Bellman and S. E. Dreyfus. Applied Dynamic Programming, Princeton: Princeton University Press, 1962. pp. 153-154.

*V. E. Hoggatt, Jr., and M. Bicknell. "Primer for Fibonacci Numbers: Part XIV, Morgan-Voyce Polynomials." FQ 12.2 (1974): 147-156.

V. E. Hoggatt, Jr,. and M. Bicknell-Johnson. "Reflections Across Two and Three Glass Plates." FQ 17.2 (1979): 118-142.

S. M. Johnson. "Best Exploration for Maximum is Fibonaccian." Report P-856. RAND Corporation, Santa Monica, 1956.

B. Junge and V. E. Hoggatt, Jr. "Polynomials Arising from Reflections Across Multiple Plates." FQ 11.3 (1973): 285-291.

P. Larson. "The Golden Section in the Earliest Notated Western Music." FQ 16.6 (1978): 513-515.

E. L. Lowman. "An Example of Fibonacci Numbers to Generate RhythmicValues in Modern Music." FQ 9.4 (1971): 423-426 and 436.

E. L. Lowman. "Some Striking Proportions in the Music of Bela Bartok." FQ 9.5 (1971): 527-528 and 536-537.

G. Markowsky. "Misconceptions about the Golden Ratio." College Mathematics Journal 23.1 (1992): 2-19. Extensive references.

H. Norden. "Proportions in Music." FQ 2:3 (1964): 219-222.

J. F. Putz. "The Golden Section and the Piano Sonatas of Mozart." Mathematics Magazine 68.4 (1995): 275-282.

W. E. Sharp. "Fibonacci Drainage Patterns." FQ 10.6 (1972): 643-655.

D. J. Wilde. Optimum Seeking Methods. Prentice-Hall, 1964, pp. 39-41.

J. Wlodarski. "The Golden Ratio and the Fibonacci Numbers in the World of Atoms." FQ 1.4 (1963): 61-63.

3. BUSINESS AND ECONOMICS, EDUCATION, POETRY. Stock market cycles; business cycles; teaching slow learners; analyzing poetry.

J. C. Curl. "Fibonacci Numbers and the Slow Learner." FQ 6.4 (1968): 266-274.

G. E. Duckworth. Structural Patterns and Proportions in Vergil's Aeneid, University of Michigan Press, 1962.

A. J. Faulconbridge. "Fibonacci Summation Economics: Parts I and II." FQ 2.4 (1964): 320-322, and FQ 3.4 (1965): 309-314.

A. J. Frost and R. R. Prechter, Jr. Elliott Wave Principle: Key to Stock Market Profits. Gainesville, Georgia: New Classics Library, 1985.

A. F. Horadam. "Further Appearance of the Fibonacci Sequence." FQ 1.4 (1963): 41-46.

B. Swancoat and E. Kasanjian. "Forecasting Market Turns Using Static and Dynamic Cycles." Technical Analysis of Stocks and Commodities, Sept., 1992, pp. 74-78.

4. THE GOLDEN RATIO, ARCHAEOLOGICAL STUDIES, ARCHITECTURE, FINE ARTS. The Great Pyramid of Gizeh; harmonic design in Minoan architecture; the Parthenon of the Acropolis in Athens; ancient Roman mosaics; the Golden Ratio in art and architecture.

R. S. Beard. "Fibonacci Drawing Board: Design of the Great Pyramid of Gizeh." FQ 6.1 (1968): 85-87.

D. Bergamini and Ed. Mathematics. New York: Time Inc., 1963. Ch. 4.

*M. Bicknell and V. E. Hoggatt, Jr. "Golden Triangles, Rectangles, and Cuboids." FQ 7.1 (1969): 73-91.

M. Bicknell-Johnson. "Fibonacci Chromotology or How To Paint Your Rabbit." FQ 16.5 (1978): 426-428.

R. Fischler. "How to find the "Golden Number" Without Really Trying." FQ 19.5 (1981): 406-410.

#Matila Ghyka. The Geometry of Art and Life. New York: Dover, 1977.

J. Hambidge. Dynamic Symmetry. Yale University Press, 1920.

Dan Harwell. Searching for Design with Fibonacci and Phi . Golden Spiral Publishing, 1842 Matador, Abilene, TX 79605.

H. Hedian. "The Golden Section and the Artist." FQ 14.4 (1976): 406-418.

#P. Hilton and J. Pedersen. Build Your Own Polyhedra. Addison-Wesley Publishing Company, 1988. Golden dodecahedron, pp. 107-110, 123.

#W. Hoffer. "A Magic Ratio Occurs Throughout Art and Nature." Smithsonian, December, 1975, pp. 110-120.

H. E. Huntley. The Divine Proportion. New York: Dover, 1970.

E. E. Kramer. The Main Stream of Mathematics. New York: Oxford University Press, 1955. Chapter 5.

G. Markowsky. "Misconceptions about the Golden Ratio." College Mathematics Journal 23.1 (1992): 2-19. Extensive references.

R. E. M. Moore. "Pattern Formation in Aggregations of Entities of Varied Sizes and Shapes as Seen in Mosaics." Nature 209.5019 (1966): 128-132.

R. E. M. Moore. "Mosaic Units: Pattern Sizes in Ancient Mosaics." FQ 8.3 (1970): 281-310.

R. E. M. Moore. "A Newly Observed Stratum in Roman Floor Mosaics." American Journal Archaeology 72.1 (1968): 57-68.

D. A. Preziosi. "Harmonic Design in Minoan Architecture." FQ 6.6 (1968): 370-384.

#G. E. Runion. The Golden Section. Dale Seymour (Box 5026, White Plains, NY 10602), 1990. Fibonacci numbers in art and nature; grades 7-12.

5. GENERAL REFERENCES.

M. Boles and R. Newman. Universal Patterns: The Golden Relationship: Art, Mathematics and Nature. Bradford Massachusetts 01835: Pythagorean Press, 1990. For liberal arts students.

B. A. Bondarenko (translated by R. C. Bollinger). Generalized Pascal Triangles and Pyramids: Their Fractals, Graphs, and Applications. Fibonacci Association, 1993. 253 pages; 532 references.

W. Boulger. "Pythagoras Meets Fibonacci." Mathematics Teacher 82.4 (1989): 277-282.

Brother A. Brousseau. An Introduction to Fibonacci Discovery. Fibonacci Association, 1965. For secondary students.

*M. Bicknell and V. E. Hoggatt, Jr. A Primer for the Fibonacci Numbers. Fibonacci Association,1973. For lower division students.

Richard A. Dunlap, The Golden Ratio and Fibonacci Numbers. World Scientific, New Jersey, 1997.

Martin Gardner. "Mathematical Games: The Multiple Fascinations of the Fibonacci Sequence." Scientific American, March, 1969, pp. 116-120.

#T. H. Garland. Fascinating Fibonaccis: Mystery and Magic in Numbers. Dale Seymour, 1987. Dale Seymour Publications, Box 5026, White Plains, NY 10602. Telephone (800) 872-1100. Grade 6 and up.

#T. H. Garland and C. V. Kahn. Math and Music. Dale Seymour, 1995. See above. #Joseph and Frances Gies. Leonard of Pisa and the New Mathematics of the Middle Ages, Reprinted by New Classics Library, Gainesville, GA, 1969.

#W. Hoffer. "A Magic Ratio Occurs Throughout Art and Nature." Smithsonian, December, 1975, pp. 110-120.

V. E. Hoggatt, Jr. Fibonacci and Lucas Numbers. Boston: Houghton-Mifflin, 1969. Now available from Fibonacci Association.

#V. E. Hoggatt, Jr. "Number Theory: The Fibonacci Sequence." 1977 Yearbook of Science and the Future, Encyclopaedia Britannica, pp. 178-191.

Ross Honsberger. "A Second Look at Fibonacci and Lucas Numbers." Chap. 8 in Mathematical Gems III: Dolciani Math Expositions No. 9. M. A. A., 1985.

A. F. Horadam. "Eight Hundred Years Young." Australian Mathemathics Teacher 31.4 (1975): 123-134. The life and mathematics of Fibonacci.

R. V. Jean, "The Fibonacci Sequence," The UMAP Journal V:1, 1984, pp. 23-47.

R. V. Jean and M. Johnson. "An Adventure into Applied Mathematics with Fibonacci Numbers." School Science and Mathematics 89.6(1989): 487-98. (Be sure you get the corrected version.)

E. A. Marchisotto. "Connections in Mathematics: An Introduction to Fibonacci via Pythagoras." FQ 31.1 (1993): 21-27. Extensive teaching references.

M. Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman,1991. Numerous references.

S. Vajda. Fibonacci & Lucas Numbers, and the Golden Section: Theory and Application . New York: Halsted Press, John Wiley & Sons, 1989.

J. Varnadore. "Pascal's Triangle and Fibonacci Numbers." Mathematics Teacher 84.4 (1991).

N. N. Vorobyvov. The Fibonacci Numbers. Boston: D. C. Heath, 1963.

#Mark Wahl. A Mathematical Mystery Tour: Higher-Thinking Math Tasks. Zephyr Press, Tucson, AZ, 1988. ISBN 0-913705-26-8. Patterns of nature using Fibonacci numbers and the Golden Section ratio. Middle school.

R. M. Young. Excursions in Calculus: An Interplay of the Continuous and the Discrete . Dolciani Mathematical Expositions Number 13, Mathematical Association of America, 1992. Chapter 3.

*References starred also appear in Primer anthology given above . The Primer for the Fibonacci Numbers (\$32.00) and Fibonacci and Lucas Numbers by Hoggatt (\$23.00) are available postpaid within the U.S. by sending a check to the Fibonacci Association, c/o Patty Solsaa, Subscription Manager, P. O. Box 320, Aurora, SD 57002-0320. Write for prices for international shipment.

#References marked # are particularly good for children.