Cycad Cone Mathematics

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indicated on the cone.



The large and compact cones or generative bodies of many species of Cycas are ideal materials to observe arcs and spiral patterns on them. These patterns are formed from the way the individual bracts (sporophylls or carpellary leaves) are arranged. In some slender cones, one may notice 3 spirals veering to the left and 5 spirals moving oppositely, or vice versa. In larger ones, 5 and 8 spirals can be noticed, and some still larger cones may display 8 and 13 spirals: each set moving opposite to the other. Giant cones may have as many as 21 spirals as the one seen in Fig. 1. These numbers (3, 5, 8, 13 & 21) form part of a numerical sequence known as Fibonacci Numbers which came to be known by the work of a 13th century Italian mathematician, Leonardo da Pisa. The sequence commences with these terms: 1, 1,2,3,5,8, 13,21,34, and so on. Except the first, any term in the sequence is obtained by adding the two preceding numbers. The sequence is thus non-ending.

The Fibonacci Numbers:

Leonardo da Pisa (1175-1250) wrote a book LiberAbaci in 1202 which remained obscure for centuries. In the 19th century, a great French Number Theorist, Edourd Lucas (1842-1891 ) who discovered it, was fascinated by a problem described in the first chapter of the book known as the famous 'breeding of rabbits problem'. When a female-male pair of adult rabbits is put in an enclosure to breed, and if the rabbits are assumed to produce a female-male pair of young ones every month, and when it is also assumed that the young ones start reproducing at the same frequency from the end of the second month of their birth, the number of pairs of rabbits will increase in the following manner: 1,2, 3, 5, 8, 13, 21, 34, 55, reaching 233 pairs at the end of one year. Any term in the sequence (excepting the first) is the sum of the two preceding terms. Lucas was very fascinated by the breeding of rabbits problem, as he could visualise some of their properties. He named the sequence as Fibonacci Numbers, or the son of Bonaccio numbers since Bonaccio was Leonardo's father(and filius or filio means son). Such a nickname was chosen for Leonardo's numbers because Lucas felt that he was not worthy of mentioning the real name of such a great mathematician who has given the gem of numbers. From the middle of the current century, the Fibonacci concept has taken great momentum thanks to the efforts of two great Californian mathematicians, Brother Alfred Brousseau and Prof. Verner E. Hoggatt. In 1962, a society, Fibonacci Association, was founded which started publishing a journal, "The Fibonacci Quarterly" from the following year. This periodical celebrated its silver jubilee year in 1987.

Some Properties of Fibonacci Numbers:

The elementary-looking numerical sequence (Fibonacci Numbers) has profound mathematical properties as evidenced from the innumerable articles that appear in The Fibonacci Quarterly contributed mostly by the over two thousand members of Fibonacci Association. However, my interest is in the applicaiton of some of Fibonacci properties to plants and animals. For example, the number of foliar spirals of any of the 2,70 0 species of palms included under the family Arecaceae, is always a Fibonacci number. Areca catechu has a single spiral, Arenga pinnata displays two spirals, Borassus flabellifer depicts on the stem three clear spirals, Cocos nucifera has five distinct spirals, and the leaf bases of Elaeis guineensis are arranged in eight spirals. Giant stems of Phoenix canariensis have numerous, prominent leaf scars from which thirteen spirals can be made out. It is also surprising that there is no palm showing 4, 6, 7, 9, 10, 11 or 12 spirals on its crown or stem. The flower heads of daisy, pansy, marygold, chrysanthemum, dahlias and sunflower, all belonging to the family Compositae, show spirals or arcs in the arrangement of individual flowers on the head. Their numbers always match with the Fibonacci numbers. So also the peduncles of most anthurium species, pineapple, pinecones and Cycas cones display spirals matching with Fibonacci numbers.

The reason for such an occurrence is because of a relationship between F. numbers and Golden Ratio (or Divine Proportion). The most remarkable property of Fibonacci series is that the ratio between two consecutive numbers is alternately greater or less than the Golden Ratio (0.618). As the series continues, the differences become less and less and the ratio approaches the Golden Ratio of 0.618033 9 ... (or its reciprocal 1.618033 9 ... which is an irrational number). Thus, when we work out the ratio between consecutive F. numbers, the values ultimately get reduced and reach very close to 0.61 8 as seen below. Hereafter, the value will continue to remain much close to the Golden Ratio. In other words, Golden Ratio is the value of Phi (0) whose expansion is (_/5-1)/2 = 0.618033 9 ... Such a Golden Ratio also appears when we consider the manner the various flowers in a sunflower head, pineapple or in the arrangement of bracts in cycas cones are arranged. If the points of origin of any two consecutive flowers are considered, the narrower angle between the two flowers or bracts will be approximately 137.5° and the wider angle (to complete a full revolution) will be 222.5°. The ratio between 137.5 and 222.5 makes the familiar Golden Ratio of 0.618. That is the reason why the cycas cone shows spiral patterns, and the number of spirals matches with a F. number.

The Cycas Cone and the Pineapple:

In Fig. 1 is shown the photograph of a pollen cone of Cycas circinalis collected from Waltair, India. From the compactly arranged bracts, spiral patterns become apparent. One of them is the 8 spirals running counter-clockwise. That is, by following this and the adioining 7 other spirals, the entire surface of the cone can be covered. Similarly, the 13 spirals which move opposite to the 8 spirals and at a steeper angle will cover the entire surface of the cone by moving clockwise. Again the 21 spirals which move much steeply, veer opposite to the 13 spirals. Thus, the rhythm of changing the direction of alternate spirals running to the left and right followed again by left, right etc. suggests one of the properties of Fibonacci Numbers. That is, the shifting of the Golden Ration between consecutive F. numbers to the plus and minus sides of 0.61 8 (or 1.618).

Sunshine Plantation at Nambour in Queensland, Australia.
veering spirals are marked.

In pineapples too, one can trace out 3, 5, 8, 13 or even 21 spirals according to their size, consecutive sets of spirals running opposite to each other. It is also true that spirals representing smaller numbers veer less steeply, but those that represent higher and higher F. numbers move steeper and steeper. This implies that no two oppositely moving sets of spirals will have the same steepness. But look at the giant pineapple dome (Fig. 2) constructed at the Sunshine Plantation at Nambour, Queensland. The marked spirals running opposite to each other move upward with the same steepness which is unrealistic. When I pointed out the difference between the inaccurate pattern on the giant dome and that on real pineapple (Fig. 3) to some workers at the Sunshine Plantation, they suggested me to write to their manager, Mr. Tony Jakeman as he was not on duty that day. When I wrote to Jakeman offering my assistance to design a realistic and scientifically accurate dome for the proposed new giant pineapple, he sent me on 11 February 1987 the following reply. "The article on the Pineapple in the Fibonacci Quarterly was fascinating, and I much appreciate your offer of assistance in redesigning the shape for the new one. Regretably the contract for the construction of the new skin for our "Big Pineapple" was let last November and the costs involved in changing the design now would be prohibitive. However, if at any time in the future we should require another Pineapple to be designed I would most certainly contact you to discuss the matter. " Mr. Jakeman could as well have a deeper look at the pine cone to design an efficient Fibonacci dome.

Contributed by:

T.A. Davis, JBS Haldane Research Centre, Nagercoil, Tamilnadu, India (from Palms & Cycads No 23. Oct-Dec 1989).

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