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Simulations of Sunflower Spirals and Fibonacci Numbers

Computer simulations to produce spiral patterns are made by an algorithm to put successive points on the 2D plane with a fixed angle increase around a center and a fixed distance increase form the center. After putting each point, it is connected to the nearest point among the points already put. When the angle increase is equal to that by golden section of 2π, the curves made of the point connections become spirals growing from the center, and their number is one of the Fibonacci numbers. These spirals make branching as growing, and several Fibonacci numbers coexist. This situation is seen in real sunflowers. Next, a new algorithm is proposed to produce the Fibonacci sequence. In contrary to the conventional one, i.e. to add a new term a n (=a n-2 + a n-1 ) successively at the right end, we add a new term a 1 (=1) at the left as the initial term and shift the other terms to the right. Under certain conditions neighboring terms are allowed to merge to a new term with summation of the two terms. Then, at discrete steps we have Fibonacci sequences growing successively. Relations between the productions of the sunflower spirals and the Fibonacci sequences are discussed.