(3) If, on a shallow spiral, one puts a pointĮvery f = 137.5. Or branches in plants such as broccoli or a rose vine, each petal or branch (2) In the spiral of petals in an artichoke, (1) Plants such as sunflowers, pineapples, etc., have two families of spirals the number of spirals in each family is a Fibonacci number (the two numbers being consecutive in the Fibonacci sequence). The brief account, above, contains the following assertions. 6 of Stewart's book provides an informative discussion of spiral structures in plants. Barabé, eds., Symmetry in Plants, Word Scientific,įor an excellent introduction to Fibonacci sequences and their applications, see Knott's website. Garcia-Ruiz et al., Plenum Press, New York, 1993.ĥ. Couder, Phyllotaxis as a Self-Organized Growth Process, In: Growth Patterns in Physical Sciences and Biology, ed, by J. Garland, Fascinating Fibonaccis, Dale Seymore Publications, 1987 (1-80).Ĥ. Ian Stewart, Life's Other Secret, Wiley, 1998.ģ. Theįormation and nature of spiral structures in plants is an ongoing topic of researchĢ. The underlying biological mechanisms are not well understood. Quite well-both qualitively and quantitatively-to the observed phenomena.Īlthough the model of Douady and Couder is a good contribution, much remains In 1993,ĭouady and Couder developed a biologically plausible model whose results correspond That attempts to provide an answer date back to 1868 (Hofmeister). Of this question is beyond the scope of the present project. Thus the question reduces to: Why are the primordia laidĭown at successive increments of f? A detailed investigation Of cells) laid down at regular time intervals at the boundary of the apex (tip But what is the biologicalĮxplanation of this? The botanical elements evolve from primordia (little clumps That the botanical elements are laid down at successive increments of f This mathematical model is purely descriptive: it expresses the fact Thus the observed phenomena can be described by an interesting mathematical Whole, accessible to high school students. The mathematics of these spirals is, on the This is easily illustrated by using mathematical software Spirals is a purely geometric consequence of the fact that these objects lie (The continuous curve that one uses to join the points is, of course, a mathematicalįiction.) The observational phenomenon that the botanical elements lie on certain Is not just a mathematical fiction but a biological reality it can be seenĭirectly in the case of of artichokes, broccoli, and certain trees and shrubs. The generative spiral (as a discrete structure) In other words, the botanical elements (florets, scales or bracts) are laidĭown at successive increments of f = 137.5. is the Fibonacci sequence 0, 1, 1, 3, 5, 8, 13. The florets, scales or bracts lie on a spiral at angular increments of f Why is the number of spirals with a given pitch a Fibonacci number? This isĮxplained by the fact that there is an underlying generative spiral. The two families have consecutive Fibonacci numbers, the shallower spirals having the lower number. Typically, two families of spirals will be visible-one clockwise and one counterclockwise. Simply by counting the spirals, we see that the number of spirals with a given pitch (slope) is a Fibonacci number. Spiral patterns occur in certain plants such as sunflower heads, pineapples and artichokes. Hence our treatment is addressed primarily to high school mathematics teachers.īrief Account of the Biological Phenomena and Mathematical Ideas to be Investigated Our main emphasis is on a mathematical discussion of the spirals. This is a good subject matter for projects by high school students in other words, projects in which the students investigate the spiral structures in plants The purpose of these web pages is to provide material which will help high school teachers design such projects for their students. are are easily observed in our daily environment and they are of considerable scientific interest-from the viewpoint of both mathematics and biology. The spiral structures found in sunflowers, pineapples, pine cones, etc.
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