Our goal is to derive an angle of rotation that in some sense is optimum: the resulting sunflower headconsists of well-spaced florets.Let us denote the rotation angle by 2πα.We first consider the possibility thatαis a rationalnumber, sayn/m, wherenandmare positive integers with no common factors, andn<m. Sinceaftermrotations florets will return to the radial line on which they started, the resulting sunflowerhead consists of florets lying alongmstraight lines. A simulation of such a sunflower head forα=1/7is shown in Fig.19.1a, where one observes seven straight lines. Evidently, rational values forαdo notresult in well-spaced florets.What about irrational values?Forαirrational, no number of rotations will return the florets totheir first radial line. Nevertheless, the resulting sunflower head may still not have well-spaced florets.For example, ifα=π-3, then the resulting sunflower head looks like Fig.19.1b.There, one cansee seven counterclockwise spirals. Recall that a good rational approximation toπis 22/7, which isslightly larger thanπ. On every seventh counterclockwise rotation, new florets fall just short of theradial line of florets created seven rotations ago.The irrational numbers that are most likely to construct a sunflower head with well-spaced floretsare those that can not be well-approximated by rational numbers. Here, we choose the golden angle,takingα=1-φ. The rational approximations to 1-φare given byFn/Fn+2, so that the number ofspirals observed will correspond to the Fibonacci numbers.Two simulations of the sunflower head withα=1-φare shown in Fig.19.2. These simulationsdiffer only by the choice of radial velocity,v0.In Fig.19.2a, one counts 13 clockwise spirals and21 counterclockwise spirals; in Fig.19.2b, one counts 21 counter clockwise spirals and 34 clockwisespirals, the same as the sunflower head shown in Fig.16.1.75

76LECTURE 19. THE GROWTH OF A SUNFLOWER(a)(b)Figure 19.1: Simulation of the sunflower model for (a)α=1/7; (b)α=π-3 and counterclockwiserotation.(a)(b)Figure 19.2: Simulation of the sunflower model forα=1-φand clockwise rotation. (a)v0=1/2; (b)v0=1/4.

Practice quiz: Fibonacci numbers innature1.The first two rational approximations toπfrom its continued fraction are 3 and 22/7. What is thenext rational approximation?

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