,
(The growth patterns of plants; the genealogical tree of the male bee; the crossroads of mathematics and biology)

  1. Basin, S.L. "The Fibonacci Sequence as it appears in Nature", FQ 1:1, Feb., 1963, pp. 53-56.
  2. Stephen R. B., "Botany with a Twist", Science, May, 1986, pp. 63-64.
  3. Brother Alfred Brousseau, "On the Trail of the California Pine", FQ 6:1, Feb., 1968, pp. 69-76.
  4. Coxeter, H.S.M., Introduction to Geometry, John Wiley and Sons, New York, 1961.
  5. Douady, S., and Couder, Y., "Phyllotaxis as a Physical Self-Organized Growth Process", Physical Review Letters 68:13, 30 mar. 1992, pp. 2098-2101.
  6. Hunter, J.A.H. and Madachy, J.S., Mathematical Diversions, Van Nostrand, Princeton, 1963, Chapter 2.
  7. Roger, V. J., "Growth Matrices in Phyllotaxis", Mathematical Biosciences, 32, 1976, pp. 165-176.
  8. Roger, V.J., The Use of Continued Fractions in Botany: UMAP Module 571, Modules and Monographs in Undergraduate Mathematics and its Applications Project, 1986.
  9. Roger, V. J., Mathematical Approach to Pattern and Form in Plant Growth, John Wiley, New York, 1984.
  10. Land, F., The Language of Mathematics, Double day, New York, 1963.
  11. Sister Mary de sales McNabb, "Phyllotaxis", FQ 1:4, Dec., 1963, pp. 57-60.
  12. Stewart, I., "Daisy, Daisy, Give me your answer, do", Scientific American, 272:1, Jan., 1995, pp. 96-99.
  13. Sutton, C., "Sunflower Spirals Obey Laws of Mathematics", New Scientist, 18 April, 1992, p.16.
  14. Thompson, D'Arcy W., On Growth and Form, McMillan, New York, 1944.


(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)

  1. Arneodo, A., Argoul, F., Bacry, E., Muzy, J.F., and Tabard, M., "Golden Mean Arithmetic in the Fractal Branching of Diffusion-Limited Aggregates", Physical Review Letters, 68:23, June, 8, 1992, pp. 3456-3459.
  2. Belliman, R.E., and Dreyfus, S.E., Applied Dynamic Programming, Pronceton University Press, Princeton, 1962, pp. 153-154.
  3. Hoggat, V.E., Jr., and Bicknell, M., "Primer for Fibonacci Numbers": Part XIV, Morgan-Voyce Polynomials", FQ 12:2, April, 1974, pp.147-156.
  4. Hoggat, V.E., Jr., and Bicknell-Johnson, M., "Reflections Across Two and Three Glass Plates", FQ 17:2, April, 1979, pp.118-142.
  5. Johnson, M., "The Best Exploration for Maximum is Fibonaccian", Report P-856, RAND Corporation, Santa Monica, 1956.
  6. Bjarne Junge, and Hoggat, V.E., Jr., "Polynomials Arising from Reflection Across Multiple Plates", FQ 11:3,October, 1973, pp. 285-291.
  7. Larson, P., "The Golden Section in the Earliest Notated Western Music", FQ 16:6, Dec., 1978, pp. 513-515.
  8. Lowman, E.L., "An Example of Fibonacci Numbers to Generate Rhythmic Values in Modern Music", FQ 9:4, Oct., pp. 423-426, 436.
  9. Lowman, E.L., "Some Striking Proportions in the Music of Bela Bartok", FQ 9:5, Dec., 1971, pp. 527-528 and 536-537.
  10. Markowsky, G., "Micro-conceptions about the Golden Ratio", College Mathematics Journal 23:1, Jan., 1992, pp. 2-19.
  11. Norden, H., "Proportions in Music", FQ 2:3, Oct., 1964, pp. 219-222.
  12. Putz, J.F., "The Golden Section and the Piano Sonatas of Mozart", Mathematics Magazine 68:4, Oct., 1995, pp. 275-282.
  13. Sharp, W.E., "Fibonacci Drainage Patterns", FQ 10:6, Dec., 1972, pp. 643-655.
  14. Wilde, D.J., Optimum Seeking Methods, Prentice-Hall, 1964, pp.39-41.
  15. Wlodarsky, J., "The Golden Ratio and the Fibonacci Numbers in the World of Atoms", FQ 1:4, Dec., 1963, pp. 61-63.

, ,
(Stock market cycles; business cycles; teaching slow learners; analyzing poetry)

  1. Curl, J.C., "Fibonacci Numbers and the Slow Learner", FQ 6:4, Oct., 1968, pp. 266-274.
  2. Duckworth, G.E., Structural Patterns and Proportions in Vergil's Aeneid, University of Michigan Press, 1962.
  3. Faulconbridge, A.J., "Fibonacci Summation Economics: Parts I and II", FQ 2:4, Dec., 1964, pp. 320-322, and FQ 3:4, pp. 309-314.
  4. Frost, A.J., and Prechter, R.R., Jr., Elliot Wave Principle: Key to Stock Market Profits, New Classics Library, Gainseville, Georgia, 1985.
  5. Fisher, Robert, Fibonacci Applications and Strategies for Traders, New York, John Wiley & Sons, Inc., 1993.
  6. Horadam, A.F., "Further Appearance of the Fibonacci Sequence", FG 1:4, Dec., 1963, pp. 41-46.
  7. Prechter, Robert R., The Wave Principle of Human Social Behavior and the New Science of Socionomics, New Classics Library, Gainseville, Georgia, 1999.
  8. Swancoat, Brad, and Kasanjian, Ed., "Forecasting Market Turns Using Static and Dynamic Cycles", Technical Analysis of Stocks and Commodities, Sept., 1992, pp. 74-78.

, , ,
(The Great Pyramid of Gizes; harmonic design in Minoan architecture; the Parthenon of the Acropolis in Athens; ancient Roman mosaics; the Golden Ratio in art)

  1. Beard, R.S., "Fibonacci Drawing Board: Design of the Great Pyramid of Gizes", FQ 6:1, Feb., 1968, pp. 85-87.
  2. Bicknell, M., and Hoggat, V., Jr., "Golden Triangles, Rectangles, and Cuboids", FQ 7:1, Feb., 1969, pp. 73-91.
  3. Bicknell-Johnson, M., "Fibonacci Chromotology or How to paint Your Rabbit", FQ 16:5, Oct., 1978, pp. 426-428.
  4. Fishler, R., "How to find the "Golden Number" without Really Trying", FQ 19:5, Dec., 1981, pp. 406-410.
  5. Ghika, Matila, The Geometry of Art and Life, Dover, New York, 1977.
  6. Hambidge, J., Dynamic Symmetry, Yale University Press, 1920.
  7. Hedian, H., "The Golden Section and the Artist', FQ 14:4, Dec., 1976, pp. 406-418.
  8. Hilton, P., and Pedersen, J., Build Your Own Polyhedra, Addison-Wesley Publishing Company, 1988. Golden Dodecahedron, pp. 107-110, 123.
  9. Hoffer, W., "A Magic Ratio Occurs throughout Art and Nature", Smithsonian, Dec., 1975, pp. 110-120.
  10. Huntley, H.E., The Divine Proportion, Dover, New York, 1970.
  11. Kramer, E.E., The main Stream of Mathematics, Oxford University Press, New York, 1955, Chapter 5.
  12. Moore, R.E.M., "Pattern Formation in Aggregations of Entities of Varied Sizes and Shapes as Seen in Mosaics", Nature 209:5019, January, 8, 1996, pp. 128-132.
  13. Moore, R.E.M., "Mosaic Units: Pattern Sizes in Ancient Mosaics", FQ 8:3, April, 1970, pp. 281-310.
  14. Moore, R.E.M., "A Newly Observed Stratum in Roman Floor Mosaics", American Journal Archeolgy 72:1, January, 1968, pp. 57-68.
  15. Presiosi, D.A., "Harmonic Design in Minoan Architecture", FQ 6:6, Dec., 1968, pp. 370-384.
  16. Runion, G.E., The Golden Section, Dale Seymour, 1990.

  1. Roles, M., and Newmann, R., Universal Patterns: The Golden Relationship: Art, Mathematics, and Nature, Pythagorean Press, Bradford Massachusetts, 1990.
  2. Bondarenko, Boris A., Generalized Pascal Tringles and Pyramids: Their Fractals, Graphs, and Applications, Fibonacci Association, 1993.
  3. Boulanger, W., "Pythagoras Meets Fibonacci", Mathematics Teacher, April, 1989.
  4. Brother Alfred Brousseau, An Introduction to Fibonacci Discovery, Fibonacci Association, San Jose, California, 1965.
  5. Bicknell, M., and Hoggat, V., Jr., A Primer for the Fibonacci Numbers, Fibonacci Association, San Jose, California, 1973.
  6. Gardner, M., "Mathematical Games: The Multiple Fascinations of the Fibonacci Sequence", Scientific American, March, 1969, pp. 116-120.
  7. Garland, T.H., Fascinating Fibonaccis: Mystery and Magic in Numbers, Dale Seymour, 1987.
  8. Garland, T.H., and Kahn, C.V., Math and Music, Dale Seymour, 1995.
  9. Hoggat, V., Jr., Fibonacci and Lucas Numbers, Houghton-Mifflin, Palo Alto, California, 1969.
  10. Hoggat, V., Jr.,"Number Theory: The Fibonacci Sequence", Yearbook of Science and the Future, Encyclopaedia Britannica, pp. 178-191.
  11. Honserberger, R., Mathematical Games: Dolciany Mathematical Expositions No 9, Mathematical Association of America, 1985, Chapter 8.
  12. Roger, J.,V., "The Fibonacci Sequence', The UMAP Journal V:1, 1984, pp. 23-47.
  13. Roger, J.V., and Johnson, M., "An Adventure into Applied Mathematics with Fibonacci Numbers", School Science and Mathematics, Oct., 1989, pp. 487-498.
  14. Schroeder, M., Fractals, Chaos, Power Laws: Minutes from Infinite Paradise, Numerous references, W.H. Freeman, New York, 1991.
  15. Varnadore, J., "Pascal's Triangle and Fibonacci Numbers", Mathematics Teacher, April,1991.
  16. Wahl, M., A Mathematical Mystery Tour: Higher-Thinking Math Tasks, Zephyr Press, Tuscon, 1988.

"" Q-

  1. Frame, J.S. "Continued fractions and matrices", Amer. Math. Monthly, 56 (1949), 98-103.
  2. Brenner, J.L. "Lucas' matrix", Amer. Math. Monthly, 58 (1951), 220-221.
  3. Brenner, J.L. "Linear recurrence relations", Amer. Math. Monthly, 61 (1954), 171-173.
  4. Rosenbaum, R.A. "An application of matrices to linear recursion relations", Amer. Math. Monthly, 66 (1959), 792-793.
  5. Miles, E.P., Jr. "Generalized Fibonacci numbers and associated matrices", Amer. Math. Monthly, 67 (1960), 745-752.
  6. Robinson, D.W. "The Fibonacci matrix modulo m", FQ, 1 (1963), No.2, 29-36.
  7. Basin, S.L. & Hoggatt, V.E., Jr. "A primer on the Fibonacci sequence, Part II", FQ, 1 (1963), No.2, 61-68.
  8. Hoggatt, V.E., Jr. & Ruggles, I.D. "A primer on the Fibonacci sequence, Part III", FQ, 1 (1963), No.3, 61-65.
  9. Hoggatt, V.E., Jr. & Ruggles, I.D. "A primer on the Fibonacci sequence", Part IV, FQ, 1 (1963), No.4, 65-71.
  10. Brennan, Terrence A. "Fibonacci powers and Pascal's triangle in a matrix, Part I", FQ, 2 (1964), No.2, 93-103.
  11. Brennan,Terrence A. "Fibonacci powers and Pascal's triangle in a matrix", Part II, FQ, 2 (1964), No.3, 177-184.
  12. Bicknell Marjory. "Fibonacci fantasy: The square root of the Q-matrix", FQ, 3 (1965), No.1, 67-71.
  13. Lind, Douglas A. "The Q matrix as a counterexample in group theory", FQ, 5 (1967), No.1, 44, 80.
  14. Waddill, Marcelus E. & Sacks, Lois. "Another generalized Fibonacci sequence", FQ, 5 (1967), No.3, 209-222.
  15. Gale, Gene B. "Factorization of 2 ´ 2 integral matrices with determinant ±1", FQ, 6 (1968), 6 (1968), No.1, 3-21.
  16. Hoggatt, V.E., Jr. "Related acknowledgment", FQ, 6 (1968), No.3, 85.
  17. Hoggatt, V.E., Jr. & Lind, D.A. "Symbolic substitutions into Fibonacci polynomials", FQ, 6 (1968), No.5, 55-74.
  18. Bicknell, Marjory. "A primer for the Fibonacci numbers, Part VII, An introduction to Fibonacci polynomials and their divisibility properties", FQ, 8 (1970), No.4, 407-420.
  19. Ivie, John. "A general Q-matrix", FQ, 10 (1972), No.3, 255-261, 264.
  20. Bicknell, Marjory & Hoggatt, V.E., Jr. A primer for the Fibonacci Numbers, Fibonacci Association, San Jose, California, 1972.
  21. Hoggat, V.E., Jr. & Bicknell, Marjory. "Roots of Fibonacci polynomials", FQ, 11 (1973), 271 -274.
  22. Hoggat, V.E., Jr. & Bicknell, Marjory. "Generalized Fibonacci polynomials", FQ, 11 (1973), No.5, 457-465.
  23. Hoggat, V.E., Jr. & Bicknell, Marjory. "A primer for the Fibonacci numbers, Part XIV", FQ, 12 (1974), No.2, 147-156.
  24. Small, Donald B. "A matrix sequence associated with continued fraction expansion of a number", FQ, 15 (1977), No.2, 123-130.
  25. Waddill, M.E. "Some properties of a generalized Fibonacci sequence modulo m", FQ, 16 (1978), 344-353.
  26. Hoggat, V.E., Jr. & Bicknell-Johnson, Marjory. "Divisibility properties of polynomials in Pascal's triangle", FQ, 16 (1978), No.6, 501-513.
  27. Ercolano, Joseph. "Matrix generators of Pell sequences", FQ, 17 (1979), No.1, 71-77.
  28. Azevedo, J. C. de Almedo. "Fibonacci numbers", FQ, 17 (1979), No.2, 162-165.
  29. Pollin, Jack M. & Schoenberg, I. J. "On the matrix approach to Fibonacci numbers and the Fibonacci pseudo-primes", FQ, 18 (1980), No.3, 261-268.
  30. Gould, H.W. "A history of the Fibonacci Q-matrix and a higher-dimensional problem", FQ, 19 (1981), No.3, 250-257