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Appendix S1
A meta algorithm for MGRS detection
The following algorithm decides if a configuration has evolved according to the MGRS. After numerically
simulating a configuration with N disks of decreasing size, starting with a (1,1)-front, we get a list of
(up, down) parastichy numbers of the successive fronts, from which we remove the elements equal to
(1,1). (We could do something similar if the algorithm started with an (n, n) front, for an arbitrary n.)
Denote by (up(k), down(k)) the kth element of this truncated list, with k going from 1 to some N . By
construction, (up(1), down(1)) = (1, 2) or (2,1). We select a “good” MGRS configuration as follows
(written in pseudo-code)
IF up(1) = 2, let high(n) = up(n) AND low(n) = down(n) for all n,
ELSE invert this prescription
ENDIF
next_high = 3;
Good = TRUE;
FOR k = 2 to N
IF low(k)<low(k-1) OR high(k) not= high(k-1) THEN
Good = FALSE, BREAK (the loop)
ENDIF
IF low(k) = next_high THEN
next_high = up(k)+down(k)
IF up(k)>down(k) THEN
let high(n) = up(n), low(n) = down(n) for all n,
ELSE invert this prescription;
ENDIF
ENDIF
ENDFOR
The output of the program is the Boolean variable Good, which is TRUE for a configuration following
the MGRS, FALSE otherwise.The first IF statement in the loop is the heart of the program, selecting
for monotone increase of the (initially) lowest parastichy number, and constancy of the other one. The
second IF statement checks for when the lowest parastichy number has achieved the sum of the original
two, and reverses the role of parastichy numbers as high and low. Note that this notation is paradoxical:
low(k) eventually becomes larger than high(k). It should be understood as “initially” low and high.
Figure S1 Number of iterations before front of length 3 in (1, 1) sweep
Figure S 1: Number of iterations before the first front of length 3. Using the same set of simulations as in
Figure 15, we recorded for each simulation the number of iterations necessary to reach the first front of length 3. The
gray-tone background of this figure is the contour plot of this function. We superimposed the Fibonacci MGRS points on
this plot to show how this number of iterations explains some of the self-similar structure of the MGRS set and of the
corresponding irregularity plot of Figure 16. The white, dashed vertical line corresponds to r = 1 , the minimal radius
√3
2
allowing quadrilateral transitions.
Appendix S2
Sweep over (2, 2) initial fronts
To simulate patterns with initial fronts of parastichy numbers (2, 2) and of given diameter b, we parameterized the space of such fronts by assuming each has a disk at (0, 0) - and thus its copy at (1, 0) in the
unrolled cylinder. Between them, we place 3 disks, starting with the middle disk, and letting the 2 other
disks be the children of the middle disk and the edge disk(s) at (0, 0) and (1, 0). The set of locations
where this construction is possible is limited: the center of the middle disk can’t be more than 2b away
from the edge disks, and thus must be confined to a lens-shaped region at the intersection of disks of
radius 2b centered at (0,0) and (1, 0). The middle disk can’t intersect with the edge disks, so we remove
from the lens region its intersections with the disks of radius b around (0,0) and (1, 0). Finally, some of
the configurations will yield intersections between the two middle disks, and this carves out two regions
in the middle of the lens region.1
The chains of disks obtained may have parastichy numbers (2, 2) (in blue in Figure S2A) but also (1,
2) and (3, 1) (in yellow). These are separated by the sets of configurations with horizontal vectors. It
is simple geometry to figure out that these sets are circles of radius b located at (b, 0) and (1 − b, 0)
respectively.
It turns out that, for the value of b chosen, each of these chains of disks is a front, that is, i the child disk
of each front is higher than the highest disk in the front.
We simulated the disk stacking process starting at each of the fronts represented in the parameter space
chosen. We used two speeds of decrease of b versus height: c = −.03 and c = −.06. The simulations
show a remarkable prevalence of Fibonacci phyllotaxis (in green). Only very symmetric (2, 2)-fronts yield
bijugate patterns, and only very asymmetric (2, 2)-fronts yield Lucas or trijugate patterns. As in the (1,
1
There are fronts of length 4 missing in this parametrization: the fronts corresponding to the upper blue region corresponds
to Λ shaped fronts, each of whose legs are concave down, as the middle point of the leg is the child of the other two. Another
copy of this space is not represented in our parametrization: the space of Λ-shaped (2, 2) fronts with concave up legs. And
the same holds for the bottom region, corresponding to V -shaped (2, 2)-fronts.
Space of initial fronts
Fibonacci Bijugate Trijugate Lucas
Monotone
Not Mon.
None
c = - 0.03
c = - 0.06
Figure S2: Sweep of (2, 2)-fronts and their final parastichy numbers.
A. Space of fronts of length 4, for diameter
b = .34. The colored areas correspond to centers of the middle disk of a front of length 4. One such front is represented as
bold black point. Points corresponding to fronts of parastichy numbers (2,2) are colored blue, those of parastichy numbers
(3,1) and (1,3) are colored yellow. The gray circles bound the region of fronts of length 4, and separate the (2, 2) fronts
from the others. B. Disk stacking simulation with rate of decrease of b vs. height c = −0.03. Each corresponding front
was iterated for 130 iterations. Points are colored according to the parastichy numbers of the final front, with the monotone
patterns in lighter colors. C. The same simulation but with c = −0.06. Note how, even though the speed of transition is
greater, more fronts yield monotone Fibonacci. Black points appear that do not belong to the previous categories. Many
correspond to quasi-symmetric patterns.
1)-fronts simulations, we detected MGRS patterns, starting at the first front of length 3 of the pattern
(either (1, 2) or (2, 1)).
Paradoxically, even though Fibonacci patterns are more prevalent overall at lower speeds, we
detected more MGRS at the higher speed (light green areas). But this can be explained by the fact
that, for flatter fronts, a triangle transition at lower speed will induce steeper vectors and greater
irregularity than at higher speed. Quasi-symmetric patterns appear at the higher speed as well – in the
guise of black points in the graph (71% of these have parastichy numbers less than 3 apart, 92%
less than 4).
N
N
O
P
Q
P
Figure S3: Front regularity and MGRS for (2, 2) fronts. In the center, in green, the Fibonacci
points from the previous simulation (Figure S2), with shade corresponding to monotonicity of the pattern.
MGRS points are in the lighter shade, the points with more than one pentagon are in the darker shade,
while the points with exactly one pentagon in
the intermediate shade. Note that, paradoxically, there are more MGRS points at the greater speed c =
−0.06. The gray
figures in the center right represent the contour plots of the irregularity function for the first (3, 2) or (2, 3)
front for the
respective speeds. Darker regions correspond to greater regularity, and they by and large correspond to the
regions of greater monotonicity. On either side, we plotted the parastichy number graphs and irregularity of
the successive fronts corresponding to the configurations stemming from the 4 points N, O, P, Q. The point N
is Fibonacci, and MGRS at both speeds of decrease.
O is bijugate in both cases, and monotonically so only for c = −0.03. P is one of the points that is MGRS
only at the greater speed c = −.06, but GRS at c = −.03. Q is Fibonacci GRS at lower speed, and
tending to a quasi-symmetric at greater speed. Points P and Q exhibit a stabilization due to sequences
of pentagon and triangle transitions for c = −0.03, seen
through the zigzagging of the parastichy numbers, and the decrease of irregularity outside of transitions.