Fibonacci numbers for palm foliar spirals

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 . ..) formulated by the Italian mathematician, Leonardo da Pisa has many interesting mathematical properties. One of these is the fairly constant proportion that works out between any two consecutive stages in the sequence excluding, of course, some lower stages. Such a proportion, spoken of as Golden Proportion, exists in many plant species having alternate phyllotaxis. The foliar arrangement in a few species of palms has been critically studied with a view to understanding the mathematics underlying it. Palms display one, two, three, five, eight, or thirteen foliar spirals, and the spirals in a palm may move clockwisely or counter-clockwisely, each species having both types of palms in almost equal proportions. Areca catechu, Arenga pinnata, Borassus flabellifer. Cocos nucifera. Elaeis guineensis. Phoenix canariensis respectively are examples for the above foliar spiral categories. All these numbers synchronise the Fibonacci Numbers, and no palm is known to have foliar spirals numbering 4,6, 7,9,10,11 or 12. Exceptional species, however, do not show any clear spirals, but their leaves are arranged vertically in two, three, five or eight rows, the numbers again synchronising the Fibonacci stages. A model has been suggested for the arrangement of leaves in any palm crown in which two consecutive leaves have been shown to subtend 137.5° between them, and so this angle makes a proportion of 0.618 with the remaining angle (222.5°) to complete one full revolution. It is this Golden Proportion that is responsible for palm leaves appearing in spirals. An explanation is offered for the apparent reversed situations in the positions of the flowerbunches in the coconuts (in left- and right-spiralled palms) and in the African oil palm. The drawing was prepared by Mr. S. K. De, the Senior Technical Assistant of the above Institute.