Rhombic Tilings and Primordia Fronts of Phyllotaxis

In our numerical simulations, we consistently observed that, under iterations of $S$, all configurations converge to of “fat rhombic tilings”: lattice-like sets of points of the cylinder that are vertices of tilings with rhombic tiles that are not too thin (See Section 4.2). The tilings have, like the classical lattices of phyllotaxis, parastichies: strings of tangent primordia winding up and down the cylinder in somewhat irregular helices. And as with lattices, these parastichies come in two families winding in opposite directions (Fig 2).

Interestingly, we observed two distinct types of convergence to rhombic tilings: a finite time convergence and an asymptotic (infinite time) convergence. In our experiments, asymptotic convergence always involves at least one pair of pentagonal and triangular tiles, repeating along a parastichy (Fig. 2). One can see from the figure (see Proposition LABEL:prop:frontperiod for a proof) that when an orbit goes through segments of a given rhombic tiling of parastichy numbers $M,N$ its shape repeats periodically, with period $MN$. Hence, orbits that we have observed are either periodic (in the shape space), preperiodic or asymptotically periodic (with triangle and pentagon pairs). Moreover we observed large continua of tiling segments that are periodic orbits. In Theorem LABEL:thm:attractor, we prove the existence of such continua and of the preperiodicity of all orbits near steady state lattices. We will leave the analysis of the asymptotic convergence to a later work.

It is intuitively clear that, at a given time step. of the iteration, the top connected layer of primordia holds the key to the dynamics and the geometry of the orbits. We call such a layer a primordia front (Section 4.1). The number of “up” and “down” vectors forming the zigzagging curve as one travels from left to right on a front corresponds to parastichy numbers in the case of lattices and tilings (Proposition LABEL:prop:parastnum). We call them front parastichy numbers (Section 4.1). We contend that counting front parastichy numbers at each step of the iteration - which can be programmed in either simulations or data analysis - may be less misleading than the divergence angles commonly used in this type of experiment (See Figures 1 and 3 and the next section).

The litmus test for a model of phyllotaxis is its ability to reproduce aspects of the bifurcation diagram - or fixed point set - of Section LABEL:subsec:fp.po, and especially of its Fibonacci branch.

This diagram is formed by the generators (see Section 3) of the “good” lattices of phyllotaxis, that are steady states of the given model (in this case presented, of both $S$ and the Hofmeister map of [jns]). Each (dark) curve segment of the diagram corresponds to lattices with a given pair of parastichy numbers whose shapes are fixed under $S$. The $x$ coordinate of a generator corresponds to the so-called divergence angle between two consecutive points in the vertical ordering of the lattice. It also corresponds to the difference of $x$ coordinate of the new primordium and the next in an iteration process. The divergence angle, and its connection to parastichy numbers in lattices, has been widely used to explain and detect in models the Fibonacci phenomenon [douadycouder]. The Fibonacci branch of the bifurcation diagram is the largest in the diagram, and starts at lattices of parastichy numbers (1,1), corresponding to the beginning of the growth of most monocotyledonous plants. In our work on the Hofmeister map [jns], we showed that the steady state lattices are attractors, accounting for the fact that once near the Fibonacci branch, a configuration remains near it as the parameter (in that case the internodal distance) was decreased.

We were originally pessimistic about $S$ yielding Fibonacci transitions as the parameter varies. Indeed, we had observed numerically that a steady state lattice for $S$ is part of an attracting manifold of periodic orbits and the eigenvalues of the differential are either 0 or on the unit circle (A consequence of Theorem LABEL:thm:attractor). Hence the steady states for $S$ can at best be neutrally stable. However, our experiments (Fig. 3 (A & B)) show that, as we lower the diameter of the primordia while iterating $S$, the Fibonacci phenomenon, as measured by front parastichy numbers, is in fact much more robust in our $S$ model than the divergence angle measurements indicates: while the divergence angle can vary wildly even in an orbit close to a lattice, the parastichy numbers stay constant. Orbits do not have to stay too close to lattices to follow the Fibonacci route: It is sufficient that they stay in a neighborhood the substantially larger and attracting set of rhombic tilings. This flexibility allows for much faster transitions than previously thought, in a time scale observed in plants, as we will show in future work. Last but not least, the strong attraction of $S$ orbits to the set of rhombic tilings should make this set persist topologically in nearby systems.

In this paper, we concentrate on cylindrical phyllotaxis. We normalize the cylinder ${\mathcal{C}}$ to have circumference 1. Mathematically, ${\mathcal{C}}$ is the cartesian product ${\mathbb{S}^{1}}\times{\mathbb{R}}$ of the unit circle ${\mathbb{S}^{1}}={\mathbb{R}}/{\mathbb{Z}}$ with the reals. Note that fixing the circumference of the cylinder does not mean that we preclude lateral plant growth in our modeling. We make this convenient normalization choice without loss of generality since, in the patterns we study, the important parameters (such as the ratio $D$ of the size of primordia relative to the diameter of the meristem) are independent of scale. Both botanists and mathematicians often unroll cylindrical patterns on the plane ${\mathbb{R}}^{2}$, which can also be seen as the complex plane ${\mathbb{C}}$. This is the covering space of the cylinder (see Section 3.2). We will use the same notation for points and vectors in ${\mathbb{R}}^{2}$ and ${\mathcal{C}}$. By a configuration, we mean a finite set of points in ${\mathcal{C}}$ ordered by height. These points represent centers of primordia along the stem.

We often describe objects in the cylinder via their covers and lifts in the plane. The intuitive notion of cover of a set, in the case of the cylinder is simple: mark each point of the set with ink, and use the cylinder as a rolling press. As you roll the cylinder indefinitely on the plane, the points printed form the cover of the original set. Each piece of the cylindrical pattern is repeated at integer intervals along the $x$-direction. The cover of a helix, for example, is a collection of parallel lines. The lift of a helix at a point is the choice of one of these lines.

Here is a more rigorous description of these classical concepts [munkres] and notation that we will be using. The natural projection $\pi:{\mathbb{R}}^{2}\mapsto{\mathcal{C}}$ which maps a point $(x,y)$ to $(x\text{ mod }1,y)$ is a covering map and the plane ${\mathbb{R}}^{2}$ is a covering space of the cylinder ${\mathcal{C}}$. This means that $\pi$ is surjective, and that around any point $z$ of ${\mathcal{C}}$, there exists an open neighborhood $U$ such that $\pi^{-1}(U)$ (the inverse image of $U$) is a disjoint union $\cup U_{k}$ of open sets of the plane each homeomorphic to $U$. One says that $\pi$ is a local homeomorphism and that $U$ is evenly covered. In the case of the cylinder, $\pi$ is also a local isometry, for the metric induced by $\pi$ on the cylinder. A subset $X$ of ${\mathbb{R}}^{2}$ is a fundamental domain if $\pi:X\mapsto{\mathcal{C}}$ is a bijection. Any region of ${\mathbb{R}}^{2}$ of the form $\{(x,y)\in{\mathbb{R}}^{2}\mid a\leq x<a+1\}$ is a fundamental domain.

The cover of a subset $Y$ of ${\mathcal{C}}$ is the inverse image $\tilde{Y}=\pi^{-1}(Y)$ of $Y$. A set $\tilde{Y}$ of the plane is a cover of its projection $\pi(\tilde{Y})$ if and only if $\tilde{Y}+(1,0)=\tilde{Y}.$ The “tilde” notation as above is often used to denote covering spaces. In this paper, we also use the underline notation to denote the projection of a set in the plane to the cylinder: $\underline{X}=\pi(X)$.

As with all covering maps, $\pi$ has the lifting property: if $\gamma$ is a path in ${\mathcal{C}}$ and $c\in{\mathbb{R}}^{2}$ is a point “lying over” $\gamma(0)$ (i.e. $\pi(c)=\gamma(0)$), then there exists a unique path $\rho\in{\mathbb{R}}^{2}$ lying over $\gamma$ (i.e. $\pi\circ\rho=\gamma$) and with $\rho(0)=c$. The curve $\rho$ is called the lift of $\gamma$ at $c$.The lift of a path is only a connected part of its cover: for instance the lift at $(3,0)$ of the line of equation $\underline{x}=0$ of the cylinder is the line $x=3$ in the plane, whereas its cover is the union of all the lines $x=k,k\in{\mathbb{Z}}.$

A cylindrical lattice $L$ is a set of points in ${\mathcal{C}}$ whose cover $\tilde{L}$ is a lattice of ${\mathbb{R}}^{2}$:

$$\tilde{L}=\left\{m\vec{v}+n\vec{w}\in{\mathbb{R}}^{2}\mid m,n\in{\mathbb{Z}}\right\},$$

where $\vec{v},\vec{w}\in{\mathbb{R}}^{2}$ are independent generating vectors. Note that $\tilde{L}$ a discrete subgroup of ${\mathbb{R}}^{2}$ isomorphic to ${\mathbb{Z}}^{2}$. Since $\tilde{L}$ is a cover, it must be invariant under translation by $(1,0)$. Changing bases if necessary, one can assume that $\vec{w}=\left(\frac{1}{k},0\right)$ for some positive integer $k$, called the jugacy of the lattice.

If $k=1$, we say that $L$ is monojugate or that it is a helical lattice. In this case $\vec{w}=(1,0)=(0,0)\mod 1$ and $L$ has the unique generator $\vec{v}$. If $k>1$, $L$ is called a multijugate configuration or specifically a $k-$jugate configuration (or $k$-jugate lattice). A cylindrical lattice $L$ is a discrete subgroup of ${\mathcal{C}}={\mathbb{S}^{1}}\times{\mathbb{R}}$ isomorphic to ${\mathbb{Z}}\times{\mathbb{Z}}/{k{\mathbb{Z}}}$ (simply ${\mathbb{Z}}$ in the case of a helical lattice). In a $k$-jugate lattice, each point is part of a set of $k$ points, called a whorl, evenly spread around a horizontal circumference of ${\mathcal{C}}$.

Parastichies of a helical lattice $L$ are helixes joining each point of $L$ to its nearest neighbors. We now make this more precise. In general, there are two points of $\tilde{L}$ nearest to $0$ in the positive half plane. Say $z_{M}=M\vec{v}+(\Delta_{M},0)$ and $z_{N}=N\vec{v}+(\Delta_{N},0)$ nearest to 0, where $M,N,\Delta_{M},\Delta_{N}\in{\mathbb{Z}}$. Also assume that $\overset{\hbox{\rightarrowfill}}{0z_{M}}$ makes a larger angle with the horizontal than $\overset{\hbox{\rightarrowfill}}{0z_{N}}$, so in particular $\overset{\hbox{\rightarrowfill}}{0z_{M}}$ and $\overset{\hbox{\rightarrowfill}}{0z_{N}}$ are not colinear. The line through 0 and $z_{M}$ lifts a helix in ${\mathcal{C}}$ that contains all the points $\pi(kz_{M})=\pi(kM\vec{v}),k\in{\mathbb{Z}}$. The set of these points is called a parastichy, and the helix connecting them a connected parastichy. There are $M$ helixes, also called parastichies, parallel to this one. Each goes through a set of points $\left\{\pi(kz_{M}+jz_{N})\right\}_{k\in\mathbb{Z}}$, for a fixed $j\in\{0,\ldots,M-1\}$. To prove this, one shows that when $j=M$, one obtains another lift of the original parastichy, using

$$N\Delta_{M}-M\Delta_{N}=1$$

(1)

This is a consequence of $z_{M},z_{N}$ being closest to 0 ([jns], Proposition 4.2) and it implies that $M$ and $N$ are co-prime ($\gcd(M,N)=1$). Thus there are $M$ parastichies and they correspond to a cosets of the subgroup $M{\mathbb{Z}}$ of ${\mathbb{Z}}$. Likewise, there are $N$ parallel parastichies joining the second closest neighbors.

If $L$ is a $k$-jugate lattice, we can trace parastichies through nearest neighbors in a similar fashion. This time the parastichy numbers $M$ and $N$ must have the common divisor $k$. An intuitive way to see this is that, rescaling the cover $\tilde{L}$ of $L$ by $k$, one obtains the cover of a helical lattice, call it $L_{h}$. Build the parastichies through nearest neighbors for $L_{h}$ as before. Then rescale back by $1/k$ - the rescaled parastichies are parastichies of $L$. Since you need $k$ copies of the rescaled $L_{h}$ to go around the cylinder, you need $k$ rescaled copies of each parastichy of $L_{h}$, crossing the $x$-axis at intervals of $1/k$, to get all the parastichies of $L$.

Thus, all cylindrical lattices can be classified by their parastichy numbers $(M,N)=k(i,j)$ where $k$ is the number of primordia in a whorl. Helical lattices are the special case where $k=1$. Whorled configurations, where primordia in a new whorl are placed midway between those of the previous whorl, is another notable case, which corresponds to $k(1,1)$.

In Section 2.2, we presented a numerical simulation showing Fibonacci transitions along orbits of $S$ when the parameter $D$ is decreased. We argued briefly that the dynamical transitions observed mirrored the continuous geometric deformation of helical lattices along the main Fibonacci branch of the bifurcation diagram. The existence of the (connected) Fibonacci branch has been the basis of many explanations of the phenomenon since the $19^{th}$ century¹¹1This neat geometric fact has often been a source of confusion between the global deformation of a pattern and its transitions via a dynamical process with varying parameter. ([weisse], [vaniterson]). Unfortunately, this kind of argument, made rigorous for the Hofmeister model in [jns], cannot work for transitions involving a change of jugacy in the pattern. One of these transitions, from $2(1,1)$ (decussate) to $(2,3)$ (Fibonacci spiral) is the norm in the vast majority of dycotyledonous plants such as the sunflower, where after a few whorls of two leaves at $90^{o}$ angle, symmetry is broken and a spiral pattern emerges. This transition cannot be attributed to the proximity of iterated patterns to a continuous path within the set of lattices between lattices of parastichy numbers $(2,2)$ and $(2,3)$. Indeed, no such continuous path exists, since it would have to involve the continuous deformation of the vector $w=(1/2,0)$ into $(1,0)$ within the discrete set of rational vectors of the form $(1/k,0)$, which is clearly absurd.

Even in a Fibonacci transition, the global geometric deformation of lattices (orthostichies becoming parastichies when $D$ decreases) does not translate easily into a dynamical understanding of the transitional region. In short, we need more flexible and local geometrical tools to better describe dynamical transitions.

This subsection gives definitions regarding very general configurations of points of the cylinder. They can naturally be adapted to other geometries (cone or disk) as well (see [jpgr]). We consider general configurations of a number $K$ of disks of a given diameter $D$ in the cylinder. These configurations are given by their centers $(p_{1},\ldots,p_{K})$ and they form the set ${\mathcal{C}}^{K}$, Cartesian product of $K$ copies of the cylinder. Occasionally, we need to consider countably infinite configurations as well.

The ontogenetic order for a configuration in ${\mathcal{C}}^{K}$ is a choice of indices $\{1,\ldots,K\}$ for the points which corresponds to the following order of the points coordinates:

$$i>j\Leftrightarrow y(p_{i})<y(p_{j})\text{ or }\{y(p_{i})=y(p_{j})\text{ and }x(p_{i})>x(p_{j})\},$$

where we choose the fundamental domain $x\in\left(-\frac{1}{2},\frac{1}{2}\right]$.

Often, such as with lattices, we consider finite configurations that are pieces of infinite ones. A configuration ${{\bf p}}\in{\mathcal{C}}^{K}$ comprising all the points of an infinite configuration $X$ between some $p_{i}$ and $p_{i+K}$ in the ontogenetic order of $X$ is called a segment of $X$ of length $K$.

A primordium $p_{j}$ is a left (resp. right) parent of $p_{i}$ if it is tangent below and to the left (resp. right) of $p_{i}$. More precisely, we adopt the convention that, for $p_{j}$ to be left parent of $p_{i}$, the coordinates $x,y$ of the vector $\overset{\hbox{\rightarrowfill}}{p_{i}p_{j}}$ must satisfy $-1<x<0,y\leq 0$ and $x^{2}+y^{2}=D^{2}$, and $1>x\geq 0,y<0,x^{2}+y^{2}=D^{2}$ for $p_{j}$ to be right parent of $p_{i}$. In the obvious fashion, $p_{i}$ is a right (resp. left) child of $p_{j}$ if $p_{j}$ is a left (resp. right) parent of $p_{i}$.

The ontogenetic graph of a primordia configuration is the directed graph embedded in ${\mathcal{C}}$ whose vertices are the centers of the primordia and where oriented edges are drawn between primordia and their parents (if they have any).

Given an ontogenetically ordered configuration ${\bf p}$ of ${\mathcal{C}}^{K}$, we call parents data the information about which primordia are parents of which primordia. One way to represent this data is by a $K\times K$ parents data matrix, whose $(i,j)^{th}$ entry is $1$ if $p_{j}$ is left parent of $p_{i}$, $-1$ if $p_{j}$ is right parent of $p_{i}$ and 0 if $p_{i}$ is not a parent of $p_{j}$. Note that the absolute value of this matrix is just the adjacency matrix of the (directed) ontogenetic graph.

A primordia chain for a configuration is a subset $\{p_{i_{1}},\ldots p_{i_{q}}\}$ of distinct points in the configuration such that:

• •

  The chain is connected by tangencies: for all $k\in{1,\ldots,q}$, primordium $p_{i_{k+1}}$ is either a right parent or right child of $p_{i_{k}}$. A chain can thus be represented by a piecewise linear curve through the centers of its primordia, which can be lifted to ${\mathbb{R}}^{2}$.

• •

  The chain is closed and does not fold over itself: the point $p_{i_{q}}$ is either a left parent or left child of $p_{i_{1}}$ and any lift at a point $P$ with $\pi(P)=p_{i_{1}}$ of the chain is the graph of a piecewise linear function over the $x$ axis in ${\mathbb{R}}^{2}$ joining $P$ to its translate $P+(1,0)$.

The vector $\overset{\hbox{\rightarrowfill}}{p_{i_{k}}p_{i_{k+1}}}$ is an up vector of the chain if $p_{i_{k+1}}$ is a right child of $p_{i_{k}}$. The vector $\overset{\hbox{\rightarrowfill}}{p_{i_{k}}p_{i_{k+1}}}$ is a down vector if $p_{i_{k+1}}$ is a right parent of $p_{i_{k}}$. We call the number of up (resp.down) vectors in a chain its right (resp.left) parastichy number. If $p_{i_{k+1}}$ is always parent of $p_{i_{k}}$ for $k=1,\ldots m-1$ and then always a child for $i=m,\ldots,q$, we call the chain a necklace.

Given a configuration ordered ontogenetically, a front at $k$ is a chain with primordia of indices greater or equal to $k$, such that any primordium (not necessarily in the configuration) which is the child of a primordium in the chain, without overlapping any other primordium in the chain, is necessarily at a height greater or equal to that of $p_{k}$. The parastichy numbers of a front are called front parastichy numbers.

A Phyllotactic tiling is a set of points of ${\mathcal{C}}$ that can be obtained by summing to a base point cyclically ordered sums of “up” and “down” vectors. More precisely, a tiling ${T}$ is determined by a base point $(a,b)\in{\mathbb{R}}^{2}$, down vectors $\vec{d}_{1},\ldots,\vec{d}_{M}\in{\mathbb{R}}^{2}$ where each $\vec{d}_{k}$ has components $x\geq 0,y<0$, and up vectors $\vec{u}_{1},\ldots,\vec{u}_{N}\in{\mathbb{R}}^{2}$, each with components $x>0,y\geq 0$. Moreover, we ask that

$$\sum_{j=1}^{M}\vec{d}_{j}+\sum_{i=1}^{N}\vec{u}_{i}=(1,0)$$

(2)

We then define

$${T}=\{\underline{z}_{m,n}\in{\mathcal{C}}\mid m,n\in{\mathbb{Z}}\},$$

where $\underline{z}_{m,n}=\pi(z_{m,n})$ with:

$$z_{m,n}=(a,b)+sumdown(m)+sumup(n)$$

and where

$$sumdown(m)=\left\{\begin{array}[]{ccl}\sum_{j=1}^{m}\vec{d}_{j}&\text{if}&m>0\\ 0&\text{if}&m=0\\ \sum_{j=0}^{|m|-1}-\vec{d}_{M-j}&\text{if}&m<0\end{array}\right.$$

(3)

and we use the periodicity convention $\vec{d}_{j+M}=\vec{d}_{j},\forall j\in{\mathbb{Z}}$. The function $sumup(n)$ is a cyclical sum of up vectors defined similarly, with the convention that $\vec{u}_{i+N}=\vec{u}_{i},\forall i\in{\mathbb{Z}}$. The numbers $M,N$ of down and up vectors are called the parastichy numbers of the tiling, a terminology justified by Proposition LABEL:prop:paras.

The phyllotactic tiling ${T}$ is rhombic if all the up and down vectors have same length $D$, and we call $D$ the length of the tiling. A fat tiling is a phyllotactic tiling such that the angles between any two down and up vectors are in the interval $[\pi/3,2\pi/3]$. It is not hard to see that the ontogenetic graph of a rhombic tiling is the embedded graph in the cylinder whose vertices are the points of the tiling and the edges are the down and negative up vectors connecting them. For general phyllotactic tiling, we call this graph the graph of the tiling. We call the connected components of the complement of the graph its tiles.