A Golden Angle Conjecture

Update: this conjecture is broken, for n=7. See final note.

This conjecture is related to the process of phyllotaxis in plants, and understanding why nature would choose to iterate the Golden Angle for this process, after a sufficiently long period of evolution.

Essentially, I want to know if the Golden Angle is ideal for distributing plant structures around a 2-d radial point of origin at low iterations. For instance, if you want to distribute three seeds around a circle, iterating the Golden Angle three times will place a seed in each of the thirds of a circle starting from the angle origin. You want to add another seed, so you simply iterate another golden angle, placing the four seeds in each quarter of a circle, and so on.

My conjecture is

That for every integer $n > 0$, and for each integer $i$ in $[1...n]$ then $i$ maps to one unique integer $j$ from $[1...n]$ where

$\frac {i-1} n<(j\theta)\mod 1<\frac {i}n$

is true, when $\theta = \frac{3 - \sqrt 5}2$ (acute golden angle) or $\theta = \frac{\sqrt 5 - 1}2$ (obtuse golden angle)

In other words, if every value $j$ from $[1...n]$ is iterated, $j\theta $ will determine each of the $n$ regular sectors of the circle only once.

note: The angle being discussed here is in units of turns of a circle, and has nothing to do with $\pi$, other than considering this angle to be the distance traveled on the circumference a circle with $r=\frac{1}{2\pi}$.

note: Equality in some of the ordered comparisons could be used, but the assumption is that $\theta$ is some irrational number that can never be equal to a rational number, because this would obviously fail when the denominator of rational $\theta$ is a multiple of $n$.

Q1: Is there an easy proof of this conjecture, or derivation from another proof?

Q2: Is there another value of $\theta$ that this conjecture is also true?

Q3: Making a model where some specific number of seeds/structures is geometrically more likely than other adjacent numbers, could evolution chose a slightly different iterative angle that does a better job?

Notes: One simple verification of this conjecture for n=10 is to observe the mod 1 residues of 10 iterations of the obtuse Golden Angle on a calculator will cycle through all 10 digits: (and for the most arithmetically nimble of mind, you can also use this list to verify for n=[1..9] as well. )

     0.x cycles through all 10 digits:
     acute             obtuse
j=1  0.3819660113 i=4; 0.6180339887 i=7
j=2  0.7639320225 i=8; 0.2360679775 i=3
j=3  0.1458980338 i=2; 0.8541019662 i=8
j=4  0.5278640450 i=6; 0.4721359550 i=5
j=5  0.9098300563 i=9; 0.0901699437 i=1
j=6  0.2917960675 i=3; 0.7082039325 i=8
j=7  0.6737620788 i=7; 0.3262379212 i=4
j=8  0.0557280900 i=1; 0.9442719100 i=10
j=9  0.4376941013 i=5; 0.5623058987 i=6
j=10 0.8196601125 i=9; 0.1803398875 i=2

The Equi-distribution theorem is important to this conjecture, which states: that the sequence a, 2a, 3a, ... mod 1 is uniformly distributed on the circle R/Z, when a is an irrational number

The Three-Gap theorem, formerly the Steinhaus conjecture, proved in 1950, is possibly important to this, which states: if one places n points on a circle, at angles of θ, 2θ, 3θ ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two.

Intriguing is Hurwitz' Irrational Number theorem, which considers the golden ratio 'bad' in terms of tightening up Lagrange's Irrational number theorem for rational approximations of irrational numbers. I have not yet read the proof.

Final Note: This conjecture is quite broken, as I have discovered. I have since learned to use python code to test conjectures, so hopefully not embarrass myself in the future.

I ran used some python code to check how a distribution converges from the 1/n distribution around the circle for increasing values of n. It seems that an iteratively added golden angle will diverge from a sector distribution of $1/n$, and apparently will never converge. My conjecture had assumed that this was $x/n$ where $x<1$. It truth, it turns out that it will diverge, the maximum divergence at up to $n=1000$ is $\frac {2.6894...}n$ at $n=10000$, the maximum divergence is $\frac{3.894...}n$. I think this can be shown by applying the 3-gap theorem, and I think that this maximum divergence has something to do with an insufficient rate of reduction of the maximum gap angle, as n increases.

I made another test, calculating the sum $\sum _{k=1}^{n}e^{k\theta i}$ where $\theta$ is the acute golden angle in radians, that is, $\theta = \frac {2\pi (3-\sqrt 5)}2$. This would represent the center of gravity for $n$ iteratively added golden angle points around the unit circle in the complex plane. At $n=1$, the absolute value of this sum would obviously be $1$, but I thought it should converge to zero as $n$ increases, and it does, as the sum gets very close to (0,0) for many values of n, as the equi-distribution theorem would imply this. However, I was surprised to discover that the absolute value of the sum can exceed $1$ for some values of $n$, and for the values of $n$ up to 10000, I occasionally get a new maximums (keep in mind that the error in calculating the sum accumulates, and the last 5 significant digits can't be trusted near n=10000 and is further distorted by an 18 significant digit approximation of the angle):

n= 1  max= 1.0
n= 4  max= 1.0688158645695927
n= 17  max= 1.0726949676474582
n= 72  max= 1.0729112803221104
n= 305  max= 1.0729233354418457
n= 1292  max= 1.0729240072518482
n= 5473  max= 1.072924044690109
n= 23184  max= 1.072924046800302
n= 98209  max= 1.0729240468530004

In conclusion, I am a bit perplexed. Obviously, irrational angles are not easily tamed. The phyllotaxis process seems to be very tuned to using this angle, and I really want to know why it does so.

It follows from van Aardenne-Ehrenfest's discrepancy theorem (previously conjectured by van der Corput) that no sequence $(x_i)_i$ in $[0, 1)$ has the property that $x_1, \dots, x_r$ occupy different subintervals $[0, \frac{1}{r}), \dots, [1 - \frac{1}{r}, 1)$ for all $r$. In fact, Elwyn Berlekamp and Ron Graham showed that this cannot happen even for all $1 \leq r \leq 18$. So not only is there no single angle $\theta$ such that repeated rotations by $\theta$ yield your desired property; there is not even a sequence of angles $\theta_1, \theta_2, \dots$ that works (unless your plant produces 17 or fewer seeds).