Building a visualization of a sunflower spiral (i)

Sometimes you want to build a picture to illustrate something, but not necessarily do it by hand. The answer is to write an algorithm to do it, and code it in some visualization happy language… and there aren’t that many really. But using Processing works quite well. Imagine if we wanted to draw a sunflower spiral. What is a sunflower spiral you may ask? Okay, some background first.

Sunflower heads are interesting because of Fibonacci sequences. Fibonacci sequences appear quite commonly in nature, as two consecutive Fibonacci numbers, especially in plant life. Examples include the arrangement of leaves on a stem, the fruitlets of a pineapple, scales on a pine cone, artichoke flowers, fern fronds, or plant seed heads. This is an aspect of plant form known as phyllotaxis (the botanical study of the arrangement of phylla (leaves, petals, seeds, etc.) on plants)The seeds on a seed head are often distributed in the head in two distinct sets of spirals which radiate from the centre of the head to the outermost edge in clockwise and counterclockwise directions. The spirals are logarithmic in character. The number set exhibited by the double set of spirals is intimately bound up with Fibonacci numbers. It appears that the reason for this formation is to allow the seedheads to pack the maximum number of seed in the given area. The most perfect example of phyllotaxis are afforded by the common sunflower (Helianthus annuus,L.) [1]. The head of the sunflower which is approximately 3-5 inches across the disk will have exactly 34 long and 55 short curves. A larger head 5-6 inches in diameter will show 55 long curves crossing 89 shorter ones.

Fig 1: Sunflower head showing seeds [2]

Consider the image shown in Fig. 1, depicting a sunflower head. The shallower spiral of this sunflower repeats 34 times, whilst the steeper spiral repeats 55 times.

So to draw this we have to produce a spiral, which is sometimes called a Fibonacci spiral. A very simple model of the patterns in a sunflower seed head was developed by Helmut Vogel in 1979 [3]. He calculated the position of the seeds (k) in the head using polar coordinates (r, δ):

r(k) = c√k

θ(k) = kδ

where δ is the golden angle, 137.5°, and c is a constant scaling factor (some algorithms replace c√k with k). As the value of k increases, the position rotates through the golden angle, and the radius increases as the square root of k. This allows use to draw a sunflower head. This spiral is essentially identical to the orbit of a “nonrelativistic charged particle in a cyclotron” [3].

[1] Sutton, C, “Sunflower Spirals Obey Laws of Mathematics.” New Scientist , 18 April 1992, p. 16.

[2] Church, A.H., On the Relation of Phyllotaxis to Mechanical Laws, Williams and Norgate, London.Plate VII (1904)

[3] Vogel, H., “A better way to construct the sunflower head”, Mathematical Biosciences 44, pp.179–189 (1979).

 

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One thought on “Building a visualization of a sunflower spiral (i)

  1. Travis says:

    There’s a nice visual proof for fibonacci petal arrangements in this video, if I recall correctly.

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