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.

[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 Biology*,

**78**(1), 2019 pp. 1–19.

doi:10.1007/s00285-018-1265-3.

[4] H. Vogel, "A Better Way to Construct the Sunflower Head,"

*Mathematical Biosciences,* **44**, 3–4 (1979), 179–189.