Simulations of Sunflower Spirals and Fibonacci Numbers

Ryuji Takaki¹*, Yuri Ogiso², Mamoru Hayashi³ and Akijiro Katsu⁴

¹Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan
²School of Bioscience and Biotechnology, Tokyo Institute of Technology, Yokohama, Kanagawa 226-8501, Japan
³School of Engineering, Tokyo Institute of Technology, Meguro, Tokyo 152-8552, Japan
⁴Science Group, Sumisho-electronics, Chiyoda, Tokyo 101-8453, Japan
*E-mail address: takaki@cc.tuat.ac.jp

(Received November 14, 2003; Accepted January 28, 2004)

Keywords: Sunflower Spiral, Golden Ratio, Fibonacci Number, New Algorithm, Computer Simulation

Abstract. 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 2p, 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 a₁ (=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.