Voronoi Polygons in an Archimedean Spiral

It is well known that points placed along an Archimedean spiral in angular steps equal to the golden angle yield a pattern that closely matches that of a sunflower’s seeds. This Demonstration shows the patterns that emerge when you vary the angle. It also shows the Voronoi regions corresponding to each seed and computes the average of the set of ratios formed by comparing the area of the largest inscribed circle (centered at the seed) to the area of the Voronoi polygon containing the seed. The colors indicate, for each region, the value of this ratio (green means high ratio; red means low). This experiment illustrates that the greatest efficiency— is greatest—occurs for the golden angle. However, the result is unproved.

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Some researchers [1, 2, 4] have given mathematical explanations of why the golden angle is optimal in the seed pattern of sunflowers and other plants. The work of [2], which includes a proof of the properties claimed there, relates to maximizing the use of area, but is about leaves rising around a cylinder, as opposed to the planar configuration of seeds in a sunflower head.
Snapshot 1: this shows how the points lie on the spiral, with constant angular change 90° in this case
Snapshot 2: the step angle equals ; here is 150 and the average of the Voronoi region coverages by inscribed disks (the covering efficiency) is 61.9%
Snapshot 3: the step angle is 138°, near and ; the behavior is very different than for the golden angle
Snapshot 4: , which is near , a rational multiple of ; here and the covering efficiency is under 28%
Define to be the covering efficiency when there are seeds at step angle . It is not clear that , the limit of as , exists in all cases, but experiments indicate that it does. Note that will be a discontinuous function of ; for example, it is 0 at every rational multiple of . But when the seed count is fixed, then is continuous. The following figure shows a graph of for a one-degree interval around .
The figure illustrates the optimality of the (and some other numbers that, like , are highly irrational). The behavior is related to continued fractions. The continued fraction of is , and it appears that angles for which the continued fraction of ends in all 1s also have efficiency values at the maximum. Such numbers are considered to be the most irrational. In this regard, is far from being rational, but is close to being rational as it is well approximated by rationals. For more on irrationality measures of real numbers, see [3].
A nice introduction to this subject is the Numberphile video by Sparks [5]. This demonstration uses the Archimedean spiral, but Vogel [4] has made the argument that a Fermat spiral ) is more appropriate.
References
[1] P. Atela, "The Geometric and Dynamic Essence of Phyllotaxis," Mathematical Modelling of Natural Phenomena, 6(2), 2011 pp. 173–186. doi:10.1051/mmnp/20116207.
[2] F. Bergeron and C. Reutenauer, "Golden Ratio and Phyllotaxis, a Clear Mathematical Link," Journal of Mathematical Biology78(1), 2019 pp. 1–19. doi:10.1007/s00285-018-1265-3.
[3] E. W. Weisstein. "Irrationality Measure" from MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/IrrationalityMeasure.html.
[4] H. Vogel, "A Better Way to Construct the Sunflower Head," Mathematical Biosciences, 44, 3–4 (1979), 179–189.
[5] B. Sparks. The Golden Ratio (Why It Is So Irrational) [Video]. (May 21, 2020) www.youtube.com/watch?v=sj8Sg8qnjOg.
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