ARTICLE - Finding the Center of a Phyllotactic Pattern - Scott Hotton (Journal Article)

8/22/2019 ARTICLE - Finding the Center of a Phyllotactic Pattern - Scott Hotton (Journal Article)

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Finding the Center of a Phyllotactic Pattern

Scott Hotton

Department of Mathematics and Statistics, Miami University, Oxford, OH 45056

Abstract: The calculation of divergence angles

between primordia in a plant apex depends on the

point used as the center of the apex. In mathemat-

ically ideal phyllotactic patterns the center is well

defined but there has not been a precise definition

for the center of naturally occurring phyllotactic pat-

terns. A few techniques have been proposed for esti-

mating the location of the center but without a pre-

cise definition for the center the accuracy of thesemethods cannot be known. This paper provides a

precise definition that can be used as the center of a

phyllotactic pattern and a numerical method which

can accurately find it. These tools will make it easier

to compare theory against experiment in phyllotaxis.

1. Introduction

It is well known that plant organs tend to formregular patterns and that the study of these pat-

terns is called phyllotaxis. Most plant organsdevelop from meristematic tissue in plant apicescalled the shoot apical meristem (SAM). Exten-sive cell divisions along the boundary of the meris-tems produce primordia which develop into var-ious plant organs. The pattern exhibited by thecollection of primordia is preserved as they de-velop.

Phyllotactic patterns do not form all at oncebut rather in a sequential order. To understandhow a particular pattern arises we need to un-derstand its ontogeny. This calls for a dynamical

model of meristematic development. To comparea dynamical model with a developing apex wewant to observe each primordium form. Primor-dia are tiny and can only be precisely observedunder a microscope. The instrument of choice

Supplementary appendices are available atdoi:00000000000000000000000000000.

is the scanning electron microscope (SEM). Op-tical microscopes lack the depth of field neededto get well focused images. Unfortunately theact of observing an SAM under an SEM disturbsits growth. To get around this difficulty tech-niques based on making molds of a shoot apexas it grows have been developed (see (Williams& Green 1988), (Hernandez et al., 1991), (Hill,

2001), (Dumais & Kwiatkowska, 2002)).Spiral phyllotaxis is characterized by the pres-

ence of two sets of distinctive spiral rows of botan-ical units. The spirals are called parastichies.One set of parastichies coils in the opposite di-rection from the other. The pair of numbers thatenumerate how many parastichies are in each setare called the parastichy numbers. The paras-tichy numbers are often successive Fibonacci num-bers.

Another important quantity characterizing a

phyllotactic pattern is the divergence angle, i.e.the angle between primordia that form consecu-tively. This angle often varies only slightly aboutsome particular value as the plant develops. Indistichous phyllotaxis the divergence angle is about180o while in spiral phyllotaxis it is usually about137.5o. The Bravais brothers (Bravais & Bra-vais, 1837) showed that the parastichy numbersof a spiral lattice with a divergence angle closeto 137.5o are consecutive Fibonacci numbers.

Hofmeister (Hofmeister, 1868) performed one

of the first microscopic studies of shoot apices.From his observations he proposed a set of hy-pothesis for SAM ontogeny. Dynamical models(Atela et al., 2002), (Douady & Couder, 1996)based on Hofmeisters hypothesis lead to disti-chous patterns and to spiral lattices with a di-vergence angle of about 137.5o. So there is nicecongruence between theory and fact.


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It is often convenient to assume the apex iscircularly symmetric. While this is a useful ap-proximation real plant apices often deviate no-ticeably from perfect circular symmetry. Thereare an endless variety of shapes that plant apicescan assume e.g. ellipsoidal, polygonal, or hour-glass shaped. Furthermore the parastichies arenot always strictly spiral shaped. Sometimesthey can undulate. Also real phyllotactic pat-terns exhibit dislocations and the parastichy num-bers can sometimes be off by one or two from aFibonacci number especially when the parastichynumbers are large. So a more detailed analysisof plant patterns is necessary to connect theorywith fact.

This paper is about finding a good point onan apical surface to calculate divergence anglesfrom. Not surprisingly it turns out that the prob-

lem of finding this point in a naturally occurringphyllotactic pattern depends on how we defineit. This is discussed in section 2. To deal withreal phyllotatic patterns we need some way ofdefining the center that is general enough to dealwith the diversity of phyllotatic patterns. In sec-tion 3 we introduce the concept of the minimalvariation center. It provides an objective andreasonable way to deal with this diversity. Sec-tions 4 and 5 provide an algorithm for finding theminimal variation center and illustrate its usage

with some examples.Plants generate primordia in a precise fashion

producing the crystalline like phyllotactic pat-terns we see in many specimens. Even when thepattern we see in a particular plant is not asperfect as we would like we should not assumethat primordia formation is not as precisely de-termined by the plant as it is in more ideal spec-imens. To learn how plants use positional in-formation in more general situations we need tomeasure the positions of primordia as precisely

as possible.To obtain positional data on morphologicalforms certain features are usually selected to marktheir location. For example in members of theAsteraceae family disk floret primordia often dis-play a five-pointed star shaped invagination whenthe petals begin to form. The center of thisstar can be used as a marker for the position of

the floret primordia. More often young primor-dia tend to have fairly smooth surfaces and lackdistinctive features that can be use as markers.This has made it difficult to measure primordiaposition precisely. We can choose a point on thesurface of a young primordium which appearsto be the furthest above the surrounding meris-tem surface to be a marker for the primordiumsposition. In many cases young primordia areapproximately hemispherical. In these cases wecan choose a point which is about equidistant tothe perceived boundary of the primordium to bethe marker. Since primordia are not all exactlythe same shape the various methods for choos-ing a marker are not entirely consistent withone another. Currently we cannot be confidentthat measurements of primordia location havethe same degree of precision as the process which

generated them. In section 6 we show how theminimal variation center can also be used to re-duce the impact uncertainties in primordia posi-tion have on the computation of divergence an-gles. Section 7 will illustrate this with four ex-amples.

It is easier to obtain precise measurementsof primordia location when the apices are fairlyflat like those from the Asteraceae family. Theflowering shoots in this family are often referredto as capitula. In these capitula an observer can

work from a single micrograph which one tries totake as parallel to the surface as possible. Mostapices, though, are dome shaped and the pri-mordia form a three dimensional configuration.There are fairly accurate methods for obtainingthree dimensional data from plant apices (see(Williams, 1975) and (Dumais & Kwiatkowska,2002)) but they are not widely used. This paperwill focus mainly on how to analyze apices thatare approximately flat.

While the center of a two dimensional pat-

tern is a point the center of a three dimensionalpattern is a curve. In a circularly symmetricapex this curve is a straight line and even with-out circular symmetry the center can still be astraight line. Section 8 will discuss how to findthis line. There are instances, however, where aplant shoot does not grow in a straight manner(for examples see (Sattler, 1973)). These cases

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are much more complicated and will have to bedealt with in a further paper.

2. Previous Methods for Finding the Cen-

ter

Lets consider some methods that have been em-

ployed in the past to find the center of phyllotac-tic patterns. This is not a review of all possiblemethods that have been used. Rather we areonly choosing some of the simpler methods tohighlight the limitation of ad hoc approaches.This should help clarify the need for the moresystematic approach presented in the followingsections. We shall apply the methods of this sec-tion to two configurations. The first configura-tion will be a mathematically ideal pattern whilethe second configuration will be a randomly per-

turbed version of the first one.Well begin by describing the form of an equi-

angular spiral lattice. This is a discrete set ofpoints sitting on an equiangular spiral (see figure1). Mathematically the center is easy to definefor this pattern. We simply follow the spiral in-ward and (in the limit) it will take us there. Thedivergence angle is the angle between consecu-tive lattice points along the spiral. This angleis computed from the center of the spiral andit is the same between any two consecutive lat-

tice points. We will denote the divergence angleby . The plastochron ratio is the ratio of thedistances between the center and two consecu-tive lattice points. This is also the same for anypair of consecutive lattice points in an equiangu-lar spiral lattice. We will denote the plastochronratio by a.

Given the position of three consecutive pointsin an equiangular spiral lattice we can derivethe plastochron ratio, divergence angle, and thecenter of the spiral lattice using the propertiesof similar triangles (see supplementary appendixA). Let the coordinates of the three consecu-tive lattice points be (X1, Y1), (X2, Y2), (X3, Y3)where (X1, Y1) is the outermost and (X3, Y3) isthe innermost. And let (xc, yc) be the coordi-nates for the center of the spiral lattice. The

Figure 1: An equiangular spiral and two config-urations. The spiral lattice, Pattern C, is indi-cated by the solid dots. Pattern C is indicatedby the open boxes.

plastochron ratio is given by:

a =

(X2 X1)2 + (Y2 Y1)2

(X3 X2)2 + (Y3 Y2)2

The divergence angle is

= atan2(Y3 Y2, X3 X2)

atan2(Y2 Y1, X2 X1)

The atan2 function is explained in supplemen-tary appendix B. Define the four quantities:

x =(a cos() 1)(X3 X2) a sin()(Y3 Y2)

1 + a2 2a cos()

y =(a cos() 1)(Y3 Y2) a sin()(X3 X2)

1 + a2 2a cos()

and the pair of points:

((x+) + X3, (y) + Y3)

((x) + X3, (y+) + Y3)

When = 0o or = 180o these two points areactually the same and are equal to (xc, yc) (thecase = 0o doesnt resemble a phyllotactic pat-tern but the case = 180o is quite common).When = 0o and = 180o these are two dis-tinct points and only one of them is (xc, yc). The

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Mathematically Ideal Pattern Realistic PatternOuter Triplet Inner Triplet Outer Triplet Inner Triplet

P1 (40.496,0.0) (8.9840,11.708) (42.239,-0.49173) (8.6569,11.379)P2 (-21.326,19.542) (-10.381,-1.8305) (-20.873,18.595) (-10.450,-1.9776)P3 (1.8007,-20.582) (6.3503,-4.0456) (2.2537,-20.230) (6.2918,-5.3393)a 1.4000 1.4000 1.4590 1.3652 137.50o 137.50o 137.61o 133.69o

(xc, yc) (0.00000,0.00000) (0.00000,0.00000) (0.38883,-0.65946) (0.14202,-0.48471)

Table 1: Results from the triplets of lattice points shown in figure 1. The calculations were doneto ten digits and have been rounded to five digits in the table.

correct choice can be made by inspection of thefigure. A formal procedure to make the correctchoice is in supplementary appendix A.

We shall call this the three-point method. Ma-ksymowych & Erickson (Maksymowych & Erick-son, 1977) derived the same formula for a and a

formula like the one here for . Meicenheimerexpanded on their work to derive a formula sim-ilar to the one here for the coordinates of thecenter which he applied to Linum shoot apices(Meicenheimer, 1986).

In this mathematically ideal situation it doesnot matter which three consecutive points of theequiangular spiral lattice we use - we will get thesame value for the center. For example considerthe equiangular spiral lattice in figure 1. Thisspiral lattice resembles a commonly seen phyl-lotactic pattern. Its plastochron ratio is 1.4, its

divergence angle is 137.5o and the center is lo-cated at (0, 0). The positions of the lattice pointsare

xj = 1.4(12j) cos(137.5o(j 1))

yj = 1.4(12j) sin(137.5o(j 1))

where j = 1, . . . , 6. Well shall call this configu-ration pattern C.

In this demonstration we select two distincttriplets of consecutive lattice points from this

spiral lattice. In particular we choose the outermost triplet

(X1, Y1) = (x1, y1)

(X2, Y2) = (x2, y2)

(X3, Y3) = (x3, y3)

and the inner most triplet

(X1, Y1) = (x4, y4)

(X2, Y2) = (x5, y5)

(X3, Y3) = (x6, y6).

We apply the three-point method to these two

triplets to obtain two sets of values for the plas-tochron ratio, divergence angle, and coordinatesof the center for the spiral lattice. Table 1 showsthe results. We see in the middle two columnsthat when we have very precise values for thecoordinates of the lattice points we can get vir-tually the same values for the coordinates of Cusing different triplets of lattice points.

We now want to consider what happens whenwe add noise to pattern C. Well perturb eachpoint in pattern Cby adding Gaussian noise with

a standard deviation of 1. Well call the resultingconfiguration pattern C (see figure 1). Thereare dual interpretations for the meaning of theadded noise. First we can think of the addednoise as the deviation an individual plant makesfrom a mathematically ideal pattern. Second wecan think of the added noise as experimental er-rors made in measuring the location of primor-dia. We shall consider both viewpoints, the firstviewpoint will be taken up presently and the sec-ond viewpoint will be taken up in section 7.

When we think of pattern C

as the devia-tion a plant makes from a mathematical idealwe cannot assume the center of the pattern is(0, 0). The perturbations are in random direc-tions which roughly tend to cancel each otherout but not p erfectly. The effect of the randomperturbations is to both distort the ideal patternand to move it somewhat. So we dont have an

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exact position for the center of pattern C as wedid for pattern C.

We can apply the three-point method to theinner and outer triplet of points in pattern C totry and find the center. The results are shown inthe last two columns of table 1. We see that thethree-point method only yields approximately thesame results with the two triplets. The resultsare close to (0, 0) as we expect since we hardlyperturbed pattern C to get pattern C, but westill dont have an exact position for the cen-ter. Without a precise definition for the centerof pattern C we cant even estimate the errorof approximation for the three-point method inthese examples.

There is another commonly used techniquefor finding the center that does give us an unam-biguous result (Matkowski et al., 1998). This is

a center of mass calculation. It is very much likecomputing the mean of a data set. One treats in-dividual points as having a unit mass and addstogether the position vectors of all the points ina data set. The vector sum is then divided by thenumber of points used in the sum. Let J denotethe number of points in a phyllotactic patternand let the coordinates of the points in the pat-tern be (xj , yj) where j = 1, . . . , J . The centerof mass calculation is:

1J

Jj=1

(xj , yj)

With the three-point method we could onlyuse three points at a time and we got differentresults using different triplets of points. The cen-ter of mass calculation overcomes this problemby making use of the whole data set. It alsohas the advantage of being much easier to use.However it has a serious drawback. Even with amathematically perfect spiral lattice the center

of mass method will not give accurate results.For example using the center of mass method onpattern C gives (4.32058, 0.79858) which is noteven close to (0, 0).

Only in the limit as the number of pointsused goes to infinity will the center of mass cal-culation yield the correct position for a perfectspiral lattice. With only a finite number of lat-

tice points there must be some error. The size ofthe error depends on the number of lattice pointsused and the divergence angle of the spiral lat-tice (Matkowski et al., 1998). Since the center ofmass method isnt very accurate in mathemati-cally ideal cases it cant be expected to gives usgood results for more realistic patterns.

We can combine the three-point method andthe center of mass method into a single method.We can use the three-point method on every con-secutive triple of points in the pattern to pro-duce a set of points each of which approximatesthe center of the pattern. We can then findthe center of mass for this set of points. Thishas the advantage of using all of the points andproducing an unambiguous result. Furthermorewhen we apply this combination method to amathematically ideal case like pattern C it accu-

rately gives us the center. So this combinationmethod has overcome some of the difficulties weran into using the three-point method or the cen-ter of mass method individually. However themost serious difficulty remains. When we applythe combination method to pattern C we get(0.15215,0.39930). This appears to be a rea-sonable approximation for the center of patternC but without an exact definition for the centerwe are unable to estimate the error of approx-imation. We cannot even claim that the com-

bination method is any more accurate than thethree-point method.

Nor is pattern C a pathological case. Farmore distorted phyllotactic patterns are not hardto find. Often in naturally occurring phyllotac-tic patterns the plastochron ratio is not constant.When the plastochron ratio varies the resultingspiral lattice is no longer an equiangular spirallattice. A square root spiral lattice is a com-mon occurrence (Williams, 1975). In principala three point type method can be developed for

this type of spiral lattice but its extremely cum-bersome. As with the equiangular spiral latticesany deviation in a sample pattern from a perfectsquare root spiral lattice will just lead to uncer-tainty about the location of the center.

Not only can the plastochron ratio vary asa plant develops but shoot apices can be ellip-soidal or have even stranger shapes. What is

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needed is a definition that does not assume thatphyllotactic patterns conform to any mathemat-ical ideal. There is no limit to the number of adhoc methods that can be devised for finding thecenter of patterns that we believe resemble somemathematically ideal form but without a generaldefinition for the center we cant say which of themany methods is the best.

3. The Minimal Variation Center of a

Phyllotactic Pattern

To obtain a general definition for the center of aphyllotactic pattern lets consider how the cen-ter is used to study these patterns. The centeris the point from which the divergence anglesare computed. In well formed phyllotactic pat-terns the divergence angles are nearly constant.

In more distorted specimens there appears to bea greater variability in the divergence angles. Itis an open question as to how the variation in thedivergence angles arises as we observe increas-ingly distorted apices. Even though phyllotacticpatterns can be strikingly well formed no plantis perfect. To a greater or lesser extent the vari-ation in the divergence angles always plays somerole in the developmental process and wed liketo have a better understanding of that role. Forexample how does the shape of the apex affect

the shape of the parastichies?If we have only a vague idea of where the cen-ter is we can only have a vague idea of what thedivergence angles are. We could only say thatdivergence angles tend to become more variablein distorted apices and we would be unable toeven address a question like: what affect doesthe curvature along an ellipsoidal apex have ondivergence angles? Finding the center of a phyl-lotactic pattern will not by itself give us com-plete answers to questions like this but it is animportant step in that direction.

However even in a mathematically ideal pat-tern we can end up observing variation in the di-vergence angles if we dont work from the correctpoint. In a pattern that has been constructed sothat the divergence angles are constant an ob-server can ascertain that the divergence anglesare all the same only if they can determine the

right point to use to calculate the divergence an-gles from. If the observer uses a different pointthey will see some variation in the divergence an-gles. There are several ways to assess variationin a data set. We shall use the standard devi-ation to assess the amount of variation in thedivergence angles. But regardless of which mea-sure of variation that is used it tends to be thecase that the further the point used to calculatethe divergence angles is from the point used toconstruct the pattern the greater the variationthat will be observed in the divergence angles.

With a real phyllotactic pattern it will almostnever be possible to find a point so that the di-vergence angles computed relative to that pointwill be exactly the same between consecutive pri-mordia. As with the mathematically ideal casesthe amount of variation that will be observed

depends on the point used to calculate the di-vergence angles. There will be some point insidethe pattern that will give the smallest amountof variation. The further the point we use tocalculate the divergence angles is from the pointthat minimizes the variation the larger the varia-tion tends to be. This raises the question of howmuch of the observed variation is intrinsic to thepattern itself and how much is due to our choicefor the center. The best way to provide an ob-jective analysis of this variation is to choose the

point in each pattern which minimizes this vari-ation. In mathematically ideal cases the pointthat does this is of course the point which wealready consider to be the center of the spirallattice (the divergence angles are all the samerelative to this point). In real phyllotactic pat-terns the best we can hope for is to minimize thevariation in the divergence angle. Formally:

We define the minimal variation center of atwo dimensional phyllotactic pattern to be thepoint inside the pattern that gives the small-

est value for the standard deviation of the di-vergence angles as computed from this point.

The minimal variation center for pattern C

is (0.161339,0.504142). Table 2 shows thestandard deviation of the divergence angles inpattern C using the points obtained from the

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Dist. Std. dev.

outer triplet 0.572 3.33o

inner triplet 0.304 2.43o

combination 0.331 2.58o

min variation 0 2.00o

Table 2: The rows refer to four points used asthe center of pattern C. The second columnshows the distance of the point from the minimalvariation center. The third column shows thestandard deviation of the divergence angles

three-point and combination methods to calcu-late the divergence angles from. It also showsthe standard deviation we get using the min-imal variation center. It is interesting to seehow much the standard deviation depends on the

point used as the center of pattern C

. No pointof pattern C is any closer to any other point ofthe pattern than a distance of 15. Yet using apoint only a distance 0.572 away from the mini-mal variation center to calculate the divergenceangles from has increased the standard deviationfrom 2.00 to 3.33. This could give an observera somewhat exaggerated view of the extent towhich the divergence angles in pattern C devi-ate from one another.

We have explicitly assumed in the definitionof the minimal variation center that the apex

is two dimensional. There is also an implicitassumption which is that the divergence anglestend to vary only slightly about their mean valueso that it is worthwhile to look for the minimalvariation center. We have made no assumptionsabout the shape of a shoot apex or the type ofspiral the parastichies form. Thus the minimalvariation center is applicable to a wide variety ofphyllotactic patterns.

4. An Algorithm for Finding the Min-

imal Variation Center

An advantage of having precisely defined our goalis that we can construct algorithms specificallyto reach that goal. In this section we will con-struct one fairly simple algorithm for finding theminimal variation center. This algorithm will beaccurate in the sense that in the absence of ex-

perimental errors in the data set it will find theminimal variation center of the pattern up to theprecision of the computer used to implement thealgorithm.

We let (xmvc, ymvc) be the coordinates forthe minimal variation center. The algorithm willtake an estimate for (x

mvc, y

mvc) and produce a

new point that will be even closer to (xmvc, ymvc).We let (x0, y0) be the coordinates for an esti-mate of the minimal variation centers location.By iteration the value of (x0, y0) will converge to(xmvc, ymvc).

Let there be J primordia in a sample phyl-lotactic pattern. Then there will be J 1 diver-gence angles to work with. Let j be the angularposition of (xj , yj) relative to (x0, y0). Let the di-vergence angle between (xj+1, yj+1) and (xj, yj)be j = j+1 j. The sequence of divergence

angles is 1, 2, . . . , J1.We seek to minimize the standard deviation,

s, of the divergence angles. The value of s iscomputed from the statistical quantity known asthe sum of squares, S, according to the formulas =

S/(J 2). Consequently minimizing the

value of s is equivalent to minimizing the valueof S. It is sufficient, therefore, for the algorithmto just minimize the sum of squares. The sum ofsquares is defined as

S =J1j=1

(j )2

where is the mean of the data set {1, . . . , J1}.Since the value ofSdepends on 1, . . . , J1 whichin turn depend on (x0, y0) we can think of thesum of squares as a function which associatesto each point in the plane, (x0, y0), the non-negative number S. To indicate the dependenceof S on (x0, y0) we write S(x0, y0). The goal ofthe algorithm is to find the value for (x0, y0) that

minimizes S(x0, y0).Computing the gradient of S(x0, y0) gener-ally gives us a vector that points in the direc-tion of greatest increase for S(x0, y0) while thenegative of the gradient points in the directionof greatest decrease for S(x0, y0). By following apath that always points in the direction of great-est decrease we generally end up reaching a min-

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imum. This is known as the method of gradientdescent.

There are several ways to implement gradientdescent but one of the simplest will prove quitesufficient for minimizing S. We scale the gradi-ent vector by a small number, , and then choosea new position for the estimate of the minimalvariation center by subtracting the scaled downvector from the old position. Let (X0, Y0) denotethe new estimate. Our method can be writtenout as

(X0, Y0) = (x0, y0) S(x0, y0)

We then replace the value of (x0, y0) with thevalue of (X0, Y0) and do the calculation again.By repeating this calculation over and over thepoint (x0, y0) tends to converge to (xmvc, ymvc).

We stop when the magnitude of the gradient vec-tor becomes too small to produce any significantchange in the position of (x0, y0). Even for acomplicated pattern the algorithm can finishedin a matter of a few minutes on a modern com-puter. More information on this algorithm isavailable in supplementary appendix B,1

5. The Algorithm in Operation

In this section we illustrate how the algorithmworks by applying it to three configurations.

The first illustration uses pattern C

and itis displayed in figure 2. A sparse configurationlike pattern C shows that it is important thatthe initial estimate be reasonable. It should becloser to the minimal variation center than anyof the points in the pattern. Otherwise gradi-ent descent can lead outside of the configura-tion and move away from it. From the viewof a point far away from the pattern the diver-gence angles are all close to zero. Consequentlythe standard deviation of the divergence angles

is also close to zero. Once we are too far fromthe minimal variation center moving even fur-ther away can lower the values of the divergenceangles and their standard deviation. Thereforethe algorithm can lead away from the pattern.

1 A sample C code that implements this algorithmalong with instructions for its use is at http://math.smith.edu/phyllo/Research/findcenter/findcenter.html

Figure 2: The boxes show the points of patternC. Six different initial estimates for the mini-mal variation center have b een used. The circleshows the location of the center of mass which isone of the initial estimates. The six curves showthe paths followed by the algorithm with theseinitial estimates. Three of the initial estimatesincluding the center of mass are close enough tothe minimal variation center that the paths goto it. The other three estimates are too far awayand the paths go off to infinity.

It is not difficult to choose an initial estimate

by inspection but this is not necessary. A usefulstarting estimate can be obtained from the cen-ter of mass method (explained in section 2) sothat a computer program only needs to be pro-vided with the coordinates for the position of thepoints in the configuration. The center of massfor pattern C is one of the initial estimates forthe minimal variation center shown in figure 2.

We now define two mathematically ideal pat-terns for illustrating this algorithm. We willcall these patterns A and B. These are squareroot lattices and they are very similar, respec-

tively, to the patterns shown in figures 4A and4B by Matkowski et al. to illustrate their algo-rithm. As mentioned earlier square root latticesare common in nature (Williams, 1975). We in-dex the points differently here so that the firstone corresponds to the oldest primordium of thepattern and the last point corresponds to thenewest primordium. The divergence angle is the

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A B

Figure 3: Results of the algorithm with the pattern A and pattern B. The circles denote thepositions of the p oints forming the pattern. The curves show the paths followed by gradientdescent to the center from three different initial estimates. The initial estimates lie on the verticesof an equilateral triangle. In pattern A the initial estimates are at a distance of 75 from the centerand in pattern B they are at a distance of 160 from the center.

constant 137.5

o

= 360

o

222.5

o

. Pattern A is:xj = 39.23725

27j cos (222.5o(26j) + 90o)

yj = 39.23725

27j sin(222.5o(26j) + 90o)

where j = 1, . . . , 26. Pattern B is:

xj = 40.57925

62j cos (222.5o(61j) + 90o)

yj = 40.57925

62j sin(222.5o(61j) + 90o)

where j = 1, . . . , 40. Figure 3 shows patternsA and B and the paths that the algorithm fol-lows from different initial conditions. There aremany more points in these two patterns and the

starting estimates for the minimal variation cen-ter can be much further away from the minimalvariation center than in pattern C. This is par-ticular clear from the illustration for pattern A.For these two mathematically ideal patterns gra-dient descent went to (0, 0) (up to the precisionof the computer). We know that this is the min-imal variation center because we have designedthe pattern to have constant divergence angleswhen calculated from (0, 0). This gives zero forthe standard deviation of the divergence angles

which is the smallest value the standard devia-tion can have.It is worth pointing out that the methods em-

ployed by Matkowski et al. to find the center ofthese patterns gave approximations that are no-ticeably different from (0, 0). And the divergenceangles obtained with their estimates of the cen-ters fluctuated by several degrees (Matkowski et

al., 1998) even though the patterns themselveswere mathematically perfect.To simulate more realistic patterns we have

added Gaussian noise with a standard deviationof 5 to A to obtain pattern A and Gaussian noisewith a standard deviation of 10 to pattern B toget pattern B. Figure 4 shows the perturbedpatterns. With this much Gaussian noise theparastichies of the patterns are barely recogniz-able. They are not unlike what we sometimessee in naturally occurring patterns. The gradi-ent descent algorithm was given the same start-

ing estimates with the perturbed patterns as itwas with the unperturbed patterns and it be-haved pretty much as it did before. The minimalvariation center of pattern A is (2.03,1.15).The minimal variation center of pattern B is(2.28,0.77). It is worth noting that for pat-terns A, B, and C the addition of noise resultedin the minimal variation center moving by lessthan half the standard deviation of the addednoise.

6. The Minimal Variation Center andUncertainties in the Positions of Primor-

dia

We have defined the minimal variation center ofa phyllotactic pattern and we have constructedan algorithm which can accurately find it whenthere is no uncertainty about the position of the

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A B

Figure 4: Results of the algorithm with pattern A and pattern B. The circles denote the positionsof the points forming the pattern. The curves show the paths followed by gradient descent to thecenter from the same starting points as figure 3

primordia in a phyllotactic pattern. But as wasexplained in the introduction we can not mea-

sure primordia positions precisely. This leadsto some uncertainty in the values of the diver-gence angles. A sequence of divergence anglesdoes not form a set of statistically independentevents though. In this section we will see howthat fact can work to our advantage. Computingthe minimal variation center from less than per-fect information on primordia position reducesthe effect the uncertainties have on the compu-tation of the divergence angles.

While naturally occurring phyllotactic pat-terns do resemble spiral lattices we can not as-sume that if we had precise data on primordia lo-cation it would exactly conform to a mathemat-ically ideal configuration or that the divergenceangles would all have the same value. We do ex-pect in many cases that the values we computefor the divergence angles will vary only slightlyabout their mean but we still want to be able todetect the variation in the original sequence asbest as we can.

We let (x1, y1), . . . , (xJ, yJ) stand for the lo-cation of the primordia in an apex and 1, . . . ,

J1 be the divergence angles of the pattern com-puted using the minimal variation center for thereasons explained in section 3 i.e. we dont wantthe amount of variation in the divergence anglesto be an artifact of the point used as the center.We let (x1, y

1), . . . , (x

J, y

J) stand for imprecisemeasurements of the primordias positions. The

sequence of divergence angles computed from theimprecise data will be denoted by 1, . . . ,

J1.

We will not assume that this sequence of diver-gence angles was necessarily computed from theminimal variation center of (x1, y

1), . . . , (x

J, y

J).We want the sequence 1, . . . ,

J1 to be asclose as possible to the original sequence 1, . . . ,J1. One way to measure the distance betweentwo sequences is to form the sum of squares ofthe differences between corresponding members.To keep the distance the same scale across se-quences of different length we divide the sum byJ1, the length of the sequence. Finally we takethe square root to make the distance the samescale as the differences between correspondingmembers. This can be written out asJ1

j=1 (

j j)2

J 1(1)

Our goal is to make this distance as small as wecan without knowledge of the divergence angles1, . . . , J1 or the positions (x1, y1), . . . , (xJ, yJ).Without this knowledge we cannot hope to findthe exact minimum of (1) but we can at least

approximate it.To approximately minimize (1) we will makeuse of the fact that the means of the two se-quences 1, . . . , J1 and

1, . . . ,

J1 are closeto each other. The reason for this is as follows.Starting from the first point in the configuration,(x1, y1), and going around from one point to thenext until we reach the last point (xJ, yJ) we

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turn about the minimal variation center by a to-tal amount of

J1j=1 j. If instead we start with

(x1, y

1) and go around from one point to the nextuntil we reach (xJ, y

J) the total amount that we

turn will be the sumJ1

j=1

j . In either casethe total amount that we turn depends only on

the angular position of the first and last point- not on the angular position of the intermedi-ate points. The more points there are in thepattern the less it matters that the angular po-sition of (x1, y

1) differs slightly from angular po-sition of (x1, yj) and that the angular position of(xJ, y

J) differs slightly from the angular positionof (xJ, yJ). These slight differences will be smallin proportion to the size of the sums

J1j=1 j ,J1

j=1

j. The mean divergence angles are thesesums divided by J1 so the means will be closeto each other2

Since

we getJ1j=1 (

j )2

J 1

J1j=1 (

j

)2

J 1(2)

Now since we use the minimal variation cen-ter to prevent undue variation in the sequence1, . . . , J1 we keep the values of j close to .This gives us the approximationJ1

j=1 (

j j)2

J 1

J1j=1 (

j )2

J 1(3)

If the divergence angles 1, . . . , J1 are all thesame then (3) becomes a strict equality. As thestandard deviation of1, . . . , J1 rises above zerothe left and right hand sides of (3) can moveapart. So long as the standard deviation is smallenough the right hand side is a good approxima-tion of the left hand side.

We can combine expressions (2) and (3) toget J1

j=1 (

j j)2

J 1

J1j=1 (

j

)2

J 1(4)

2The mean divergence angle is closely related to theconcept of the rotation number of an orientation preserv-ing diffeomorphism of a circle to itself. The argumentpresented above for the close proximity of the mean di-vergence angles is a variation of a step in the proof of thetheorem that the rotation numbers ofC0-close orientationpreserving diffeomorphism of a circle have approximatelythe same value. For a concise presentation of a proof ofthis theorem see (Devaney, 1989).

The left hand side is the expression under theradical in (1) which we want to minimize. Theright hand side can be computed entirely from(x1, y

1), . . . , (x

J, y

J). (An estimate for the er-ror of approximation in expression (4) is in sup-plementary appendix C. As was explained insection 4 using the minimal variation center tocompute the divergence angles 1, . . . , J1 min-imizes the numerator of the right hand side of(4). Thus finding the minimal variation center ofthe points (x1, y

1), . . . , (x

J, y

J) provides us witha way of approximately minimizing (1). We canuse it under those circumstances where both (2)and (3) are good approximations i.e. when thereare a large number of divergence angles and thedivergence angles vary only slightly about theirmean.

7. Finding Divergence Angles: Four

Examples

We now provide four demonstrations of the use-fulness of the minimal variation center for re-covering the underlying sequence of divergenceangles of a phyllotactic pattern from imprecisedata. In this section we will now interpret noiseadded to a pattern as experimental errors madein the measurement of primordia positions. Wewill compute approximate values for the diver-

gence angles of the original patterns using onlythe noisy data. The divergence angles will b ecomputed using both the minimal variation cen-ter and the center of mass. We will see thatthe minimal variation center does a better job offinding the divergence angles.

We begin with patterns A and B which haveover two dozen points each and whose divergenceangles are a constant 137.5o. These two patternsare too p erfect to be natural. They representideal cases where we expect to get good resultsfor the divergence angles because approximation(3) in section 6 is an equality for these cases. Weadd Gaussian noise with a standard deviation of2 to pattern A to get pattern A and we addGaussian noise with a standard deviation of 4 topattern B to get pattern B .

The divergence angles are shown in figures 5and 6. The original sequences are constant and

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125

130

135

140

145

150

5 10 15 20 25

Figure 5: The thick line marks 137.5o - the value for every divergence angle in pattern A. Thethin solid line passing through the open circles shows the sequence of divergence angles in patternA using the minimal variation center to compute the angles. The dotted line passing through thecrosses shows the sequence of divergence angles in pattern A using the center of mass to computethe angles.

125

130

135

140

145

150

5 10 15 20 25 30 35

Figure 6: The thick line marks 137.5o - the value for every divergence angle in pattern B. Thethin solid line passing through the open circles shows the sequence of divergence angles in patternB using the minimal variation center to compute the angles. The dotted line passing through thecrosses shows the sequence of divergence angles in pattern B using the center of mass to computethe angles.

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125

130

135

140

145

150

5 10 15 20 25

Figure 7: The thick line indicates the sequence of divergence angles in pattern A

using the minimalvariation center to compute the angles. The thin solid line passing through the open circles showsthe sequence of divergence angles in pattern A using the minimal variation center to computethe angles. The dotted line passing through the crosses shows the sequence of divergence angles inpattern A using the center of mass to compute the angles.

125

130

135

140

145

150

5 10 15 20 25 30 35

Figure 8: The thick line indicates the sequence of divergence angles in pattern B using the minimalvariation center to compute the angles. The thin solid line passing through the open circles showsthe sequence of divergence angles in pattern B using the minimal variation center to computethe angles. The dotted line passing through the crosses shows the sequence of divergence angles inpattern B using the center of mass to compute the angles.

13


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the sequences obtained from the perturbed pat-terns using the minimal variation center is closerto being level than the sequences obtained us-ing the center of mass. We can not expect tocompletely overcome the effects of the noise thathas been added to patterns A and B. We seein figures 5 and 6 that some of the divergenceangles deviate from 137.5o by a few degrees re-gardless of whether we use the minimal varia-tion center or the center of mass. For the vastmajority of divergence angles, though, the valueobtained using the minimal variation center iscloser to original value than the value obtainedusing the center of mass. So we have, at least,reduced the effect of the noise by using the min-imal variation center.

The next two examples are based on patternsA and B. The divergence angles in patterns A

and B vary by several degrees. We interpretthese configurations as the location of primor-dia in a pair of phyllotactic patterns that do notconform to a mathematical ideal. We continueto interpret the noise in patterns A and B as anatural deviation plants make from a mathemat-ical ideal form. We add more noise to patterns A

and B which we interpret as experimental errorsin the measurement of primordia position. Weadd Gaussian noise with a standard deviation of2 to pattern A to get pattern A and we add

Gaussian noise with a standard deviation of 4 topattern B to get pattern B3. The divergenceangles are shown in figures 7 and 8.

We can see in figures 7 and 8 that the se-quences of divergence angles in patterns A andB computed with the minimal variation centerdoes follow the original sequence more closelythan the sequences computed using the centerof mass. Not every divergence angle computedwith the minimal variation center of pattern A

is closer to its corresponding value in pattern A

but there clearly is an overall tendency for thecorresponding values to be closer when the min-imal variation center is used to compute the di-vergence angles. The same things holds for pat-tern B.

3 Data files for patterns A,A,A,A,B,B,B,B,C,and C are at http://math.smith.edu/phyllo/Research/findcenter/findcenter.html

Distance Min. Var. CM

Pattern A 1.17o 2.33o

Pattern B 1.14o 2.60o

Pattern A 0.98o 1.63o

Pattern B 0.86o 1.76o

Table 3: Each of the four rows refers to a pat-tern that has been perturbed to simulate the ef-fect of experimental errors (see text for a de-scription of each pattern). The table shows thedistance between the sequence of divergence an-gles in the perturbed pattern and the sequenceof divergence angles in the corresponding unper-turbed pattern. In the second column the min-imal variation center has been used to computethe divergence angles in the perturbed pattern.In the third column the center of mass has beenused to compute the divergence angles in the per-

turbed pattern.

The distances (as defined by (1) in section 6)between the original sequences and the sequencesobtained from the perturbed patterns are shownin table 3. For each of the patterns the tableclearly shows that the minimal variation centerprovides us with the sequence of divergence an-gles which is closer to the original sequence.

We can expect similar results with naturallyoccurring phyllotactic patterns whose divergence

angles deviate less about their mean values thanthe divergence angles of patterns A and B do.We can see in figure 4 that patterns A and B

deviate quite noticeably from a mathematicallyideal form. They are not unrealistic but morewell formed patterns are fairly common in na-ture. The minimal variation center can be auseful tool to uncover systematic variations inthe divergence angles which may arise in capit-ula that are slightly imperfect.

8. Phyllotaxis in Three Dimensions

Although the shape of capitula tend to be wellapproximated by a flat disk most apices are domeor conical shaped. In these cases the primordiaform a three dimensional configuration. The ad-dition of an extra dimension means that the cen-ter of the pattern in space is no longer merely

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L

P

P

1

2

1

Figure 9: Given two points, P1, P2 and a line Lthere is a unique plane containing L and P1 and aunique plane containing L and P2. We take thedihedral angle between the planes as the anglebetween P1 and P2 relative to L. This angle islabeled 1.

a point but a curve. This curve is known asthe axis of the shoot. We do not want the axisto b e curved any more than necessary. Math-ematically speaking we want the curvature at

each point of the axis to remain within a cer-tain bound and for the integral curvature to besmall. In this paper we will focus mainly on thecase where the axis is straight (i.e. when thecurvature is zero everywhere).

The divergence angle between consecutive pri-mordia needs to be calculated relative to theaxis. Computing divergence angles is much eas-ier when the curve is a straight line where wecan make use of the dihedral angle between twoplanes that meet at the axis (see figure 9). Whenthe axis is not straight we need to generalize theconcept of a dihedral angle. To do this for anaxis that is sufficiently smooth we can make useof the Serret-Frenet frame of the curve but weleave this to be a topic for a future paper.

As in the planar cases the divergence anglesin many three dimensional phyllotactic patternstend to vary only slightly about some particular

value (e.g. 137.5o). Some phyllotactic patternsshow greater variation in their divergence anglesthan others. Computing these divergence anglesrequires a good estimate for the location of theaxis. Like in the planar case we dont want thevariation in the divergence angles to be an arti-fact of the choice for the shoots axis. We wouldlike to choose a curve that minimizes the varia-tion in the divergence angles. However for anyfinite configuration of points we can find a suffi-ciently winding curve so that the divergence an-gles computed relative to this curve will all bezero. Such a curve would be a poor choice forthe axis of most any configuration. We need tostrike a balance between the amount of curvaturein the axis and the amount of variation in the di-vergence angles. For now we will only consideraxis that are straight. Formally

We provisionally define the minimal varia-tion axis of a three dimensional phyllotactic pat-tern to be the line passing through the patternthat gives the smallest value for the standarddeviation of the divergence angles as computedfrom this line.

With this definition for the axis of a three di-mensional phyllotactic pattern we can constructan algorithm to find it. The algorithm will be

based on gradient descent. We need to estab-lishing a coordinate system to work in. Firstwe have to decide where the origin is going tobe. The location of the origin is only a matter ofconvenience and a convenient spot is near the tipof the shoot. This is also a conventional choicein making measurements for the location of pri-mordia of three dimensional patterns. It wouldbe most convenient if the minimal variation axiswere to pass through the origin but that is toomuch to expect before we have found it. We

have to be content with putting the origin closeto where the minimal variation axis meets thetip of the apex.

We will choose our initial estimate for theminimal variation axis to be the vertical coor-dinate axis which we will call the w-axis. Thepositive direction will point up. The remainingtwo coordinate axes will be called the u-axis and

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u v

w

Figure 10: A unit vector is shown as the thickstraight line. Its position is given in terms of(, ). These two angles are shown as thick arcs.The u-axis is the pole, gives the angle be-tween the vector and the positive u-axis, and gives the angle of rotation about the u-axis fromthe horizontal direction to the vector.

v-axis. We place them so that the u,v,w-axesform a right handed orthogonal coordinate sys-tem. As we iterate the algorithm our estimate forthe location of the minimal variation axis will beupdated but we will keep the coordinate systemfixed.

To specify the location of a line we need avector pointing in the direction of the line andthe location of some point that the line passesthrough. It is convenient to use unit vectors to

specify the direction of a line. The set of allunit vectors forms a sphere of radius 1 aboutthe origin. We can use spherical coordinates tospecify a unit vector on this sphere (see figure10). Each pair of angles (, ) will specify theunit vector

(cos(), sin()cos(), sin()sin())

This is slightly different from the usual form forspherical coordinates. We choose this form sothat the coordinate singularities4 are at (1, 0, 0).

These are perpendicular to the estimated direc-tion of the minimal variation axis so that thealgorithm is not likely to venture near them.

We let (m, m) specify the direction of theminimal variation axis and (0, 0) specify an es-timate for the direction of the minimal variation

4Coordinate singularities are like the north and southpoles where the Earths lines of longitude meet

L

(u ,v ,w )

u v

w

x

y

0

0 0 0

0

0

0

0

Figure 11: The vectors 0, 0 form an orthonor-mal basis for the plane which is perpendicular tothe line L0 and which passes through the origin.The point where L0 passes through this plane isx00 + y00.

axis. The initial estimate for the direction of theminimal variation axis is (90o, 90o) but as we it-erate the algorithm we will update the values for(0, 0) so that they become closer to (m, m).The corresponding unit vectors are

(um, vm, wm) =

(cos(m), sin(m)cos(m), sin(m)sin(m))

(u0, v0, w0) =

(cos(0), sin(0) cos(0), sin(0)sin(0))

The point in the line that we will use to spec-ify it is where it meets the plane which passesthrough the origin and which is perpendicular tothe line. The vectorsm = ( sin(m), cos(m)cos(m), cos(m)sin(m)) m = (0, sin(m), cos(m))

form an orthonormal basis for the plane whichpasses through the origin and which is perpen-dicular to (um, vm, wm). Wherever this plane

meets the minimal variation axis there is a uniquepair of numbers (xm, ym) such that the intersec-

tion point can be expressed as xmm + ym

m.The minimal variation axis is the set of points

Lm = {(um, vm, wm)t + xmm + ym

m : t R}

The minimal variation axis is completely speci-fied by the numbers (m, m, xm, ym).

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Similarly the vectors

0 = ( sin(0), cos(0)cos(0), cos(0)sin(0) 0 = (0, sin(0), cos(0))

form an orthonormal basis for the plane throughthe origin that is perpendicular to (u0, v0, w0).

There is a unique pair of numbers (x0, y0) suchthat this plane meets our estimate for the min-

imal variation axis at x00 + y0

0 (see figure11). The estimate for the minimal variation axisis the set of points

L0 = {(u0, v0, w0)t + x00 + y0

0 : t R}

This line is completely specified by the numbers(0, 0, x0, y0).

The algorithm we are constructing will take

the estimate (0, 0, x0, y0) and produce an newestimate that will be closer to (m, m, xm, ym).We do this by applying gradient descent to thesum of squares, S, of the divergence angles com-puted relative to the line L0.

Assume the positions of the sequence of pointsP1, . . . , P J of the three dimensional configurationhave been measured and recorded as (u1, v1, w1),. . . , (uJ, vJ, wJ) respectively. To compute the di-vergence angles relative to L0 from (u1, v1, w1),. . ., (uJ, vJ, wJ) we can orthogonally project thespatial configuration into a plane perpendicularto L0 (see figure 12). The computation of the di-vergence angles in a spatial configuration is thenreduced to the computation of divergence anglesof a planar configuration.

Let us denote the projection of (uj, vj, wj)

into the plane by (xj , yj). Since0,

0 form anorthonormal basis for the plane containing theorigin and perpendicular to L0 we get explicitexpressions for xj and yj by using the dot prod-uct.

xj = (uj , vj , wj) 0 =

uj sin(0) + cos(0)(vj cos(0) + wj sin(0))

yj = (uj , vj , wj) 0 = vj sin(0) + wj cos(0)

The computation of the divergence angles 1, . . . ,J1 of (x1, y1), . . . , (xJ, yJ) relative to (x0, y0)proceeds just as it did in the planar case. From

P

P

P L

1

2

3 0

1

Figure 12: The angle between points relative toL0 is preserved when we orthogonally project thepoints into any plane that is perpendicular toL0. (The plane in this figure is not the one thatpasses through the origin.)

1, . . . , J1 we can compute

S =J1j=1

(j )2

S is now a function of (0, 0, x0, y0). Gradientdescent can be expressed as

(0, 0, X0, Y0)

= (0, 0, x0, y0) S(0, 0, x0, y0)

where is a small number that scales down thelength of the gradient vector. We then replace(0, 0, x0, y0) with (0, 0, X0, Y0) and repeat.For a detailed computation of the gradient vectorsee supplementary appendix D,1

With the minimal variation axis we can re-cover the sequence of divergence angles in three

dimensional phyllotactic patterns. Experimentalerrors in the measurement of primordia positionswill limit the precision with which we can recoverthe angles. The arguments of section 6 (andsupplementary appendix C) require very littlemodification for three dimensional patterns. Solong as the standard deviations are small andthe number of primordia is large approximation

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(4) of section 6 remains valid and using the mini-mal variation axis to compute the divergence an-gles reduces the impact of experimental errors inthree dimensions just as the minimal variationcenter reduces their impact in two dimensions.The possibility exists that the impact of theseexperimental errors could be further reduced byusing an appropriately chosen curve instead of aline to compute the divergence angles from.

A program which implements this algorithmcan complete the calculations in a matter of min-utes on a modern desktop computer5

9. Summary

The field of phyllotaxis has been challenged bythe diversity of shapes that shoot apices can have.This paper has introduced the concept of the

minimal variation center to help deal with thisdiversity. The minimal variation center providesus with an objective way to compute divergenceangles in capitula whose primordia do not con-form p erfectly to a spiral lattice. It also helpsto mitigate the effect experimental errors in themeasurement of the primordia positions have onthe computed values of the divergence angles.This helps lay the groundwork to objectively com-pare theory against experiment for a more gen-eral class of phyllotactic patterns6.

Acknowledgment

I wish to thank Roger Meicenheimer for manyuseful discussions about bridging the gap betweentheory and experiment. Id also like to thankthe Department of Mathematics & Statistics ofMiami University and the Mathematics Depart-ment of Smith College for the use of their com-puters.

REFERENCES


6 Supplementary appendices are available atdoi:00000000000000000000000000000.

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641-676.

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7 pp. 42-110, Essai sur la disposition symmetri-

que des inflorescences. Ann. Sci. Nat. 7 pp.

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Devaney, R. (1989) An Introduction Chaotic Dy-namical Systems2nd. ed., Boston: Addison-Wes-

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Douady S. & Couder Y. (1996), Phyllotaxis asa Self Organizing Iterative Process, Part I, J.

Theor. Biol. 178 pp. 255-274.

Dumais, J. and Kwiatkowska, D. (2002), Anal-ysis of surface growth in shoot apices, Plant Jour-

nal31 pp. 229-241.

Hernandez, L.F., Havelange, A., Bernier, G. &Green, P.B. (1991), Growth behavior of single epi-

dermal cells during flower formation: Sequential

scanning electron micrographs provide kinematic

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Hill, J.P. (2001) Meristem development at the sporo-phyll pinna apex in Ceratopteris richardii, Int. J.

Plant Sci. 162 pp. 235-247.

Hofmeister W. (1868), Allgemeine Morphologie derGewachse, in Handbuch der Physiologischen Botan-

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Xanthium Shoot Apices, Amer. J. Bot. 64 pp.

33-44.

Matkowski A., Karwowski R. & Zagorska-MarekB. (1998), Two Algorithms of Determining theMiddle Point of the Shoot Apex by Surrounding

Organ Primordia Positions and their Usage for

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Meicenheimer R.D. (1986), Role of Parenchyma inLinum Usitatissimum Leaf Trace Pattern, Amer.J. Bot. 12 pp. 1649-1664.

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meristem. Protoplasma 147 pp. 77-79

Williams R.F. (1975), The Shoot Apex and LeafGrowth, Cambridge: Cambridge University Press.

Appendix A

In this appendix we derive formulas for the pa-

rameters of an equiangular spiral lattice fromthree consecutive lattice points. Let P1, P2, P3be three consecutive lattice points where P1 isthe outermost and P3 is the innermost latticepoint. Let C be the center of the spiral lattice(see figure 13). Recall that the coordinate ofP1, P2, P3 are (X1, Y1), (X2, Y2), (X3, Y3) respec-tively and the coordinates of C are (xc, yc).

Figure 13 shows that value of a equals thelength ofP1P2 divided by the length ofP2P3. Incoordinates this is

a =

(X2 X1)2 + (Y2 Y1)2

(X3 X2)2 + (Y3 Y2)2

To obtain we extend P1P2 to P1A. Figure14 shows that equals AP2P3. The angle thatP1A makes with a horizontal line is the sameas the angle P1P2 makes with a horizontal line.This angle is given by atan2(Y2 Y1, X2 X1).The angle that P2P3 makes with a horizontal lineis atan2(Y3 Y2, X3 X2). The difference be-

tween these two is : = atan2(Y3 Y2, X3 X2)

atan2(Y2 Y1, X2 X1)

The atan2 function is explained in appendix B.We can apply the law of cosines to P2CP3

to obtain the value of R. Looking at figure 13we see the law of cosines in this case gives:

Q2 = R2 + (aR)2 2(R)(aR)cos()

We know Q2

= (X3X2)2

+ (Y3Y2)2

. Solvingfor R gives:

R =

(X3 X2)2 + (Y3 Y2)2

1 + a2 2a cos()

Point C lies at distance R from P3 and it liesat distance aR from P2 (see figure 15). Or in

C

P1

P2

P3Q

aQ

R

aR

a R2

Figure 13: The length of P2C is a times thelength of P3C. The length of P3C is denoted byR in the figure. The length ofP1C is a times thelength of P2C. Since = P1CP2 = P2CP3the triangles P1CP2 and P2CP3 are similar.Therefore the length ofP1P2 is a times the lengthof P2P3. The length of P2P3 is denoted by Q inthe figure.

other words it must be one of the places wherethe circle with radius R about P3 intersects thecircle with radius aR about P2. The equationsfor these two circles are

(aR)2 = (xX2)2 + (y Y2)

2

R2 = (xX3)2 + (y Y3)

2.

The coordinates for the intersection points of

these two circles are given by those values of(x, y) which satisfy both of these equations. Solv-ing these two equations is tedious. We only pro-vide the solution here. First define the followingfour quantities:

x =(a cos() 1)(X3 X2) a sin()(Y3 Y2)

1 + a2 2a cos()

y =(a cos() 1)(Y3 Y2) a sin()(X3 X2)

1 + a2 2a cos()

The intersection points are

((x+) + X3, (y) + Y3)

((x) + X3, (y+) + Y3)

Point C must be one of these two points. Theseformulas for the center are similar to the onespresented in (Meicenheimer, 1986). With the

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A

C

P1

P2

P3

Figure 14: The sum of the angles in P1CP2is 180o and the p oints P1, P2, A are collinear.This gives us the equation P1P2C+CP1P2 +P1CP2 = P1P2C+CP2P3 +AP2P3. SinceP1P2C appears on both sides we can cancelthose terms. And since P1CP2 is similar toP2CP3 the angles CP1P2 and CP2P3 areequal. We can cancel them from the equation aswell. This leaves P1CP2 = AP2P3.

formulas here it is easier to see how the num-ber of intersection points between the two circlesdepends on the divergence angle. Except when = 0o or = 180o there are two distinct inter-section points. When = 0o or = 180o thecircles intersect in a single point which is C sono choice has to be made.

For the case = 0

o

and = 180

o

a choiceneeds to be made for C. Since C must lie on thecircle with radius a2R about P1 a rigorous way todetermine which of these two intersection pointsis actually C is to plug each into the equation:

(a2R)2 = (xX1)2 + (y Y1)

2

The correct choice for (xc, yc) will satisfy thisequation while the incorrect choice will not.

Appendix B

The tangent function has a period of 180o i.e.tan( + 180o) = tan(). Consequently the in-verse of the tangent function is not uniquely de-fined. However if we restrict the domain of tanto the interval between 90o and 90o it becomesa continuous one to one map onto the real line.This makes it convenient to restrict the range of

C

P1

P2

P3R

aR

a R2

Figure 15: The point C lies on a circle of radiusR about P3. It also lies on a circle of radius aRabout P2 and on a circle of radius a

2R aboutP1. The location of C can be determined fromthe fact that these three circles only intersectsimultaneously in a single point.

the inverse function to the interval between 90o

and 90o. This gives a continuous function thatis one to one on the whole real line. This inverseis called arctan.

Even though arctan is a convenient functionit cannot give us the angular coordinate for some

points in plane. For example consider the points(1, 1) and (1,1). They are collinear with theorigin and the slope of this line is 1. We can sub-stitute this slope into the arctan function to getthe angular coordinate of (1, 1) (i.e. the anglethat the ray starting at the origin and passingthrough (1, 1) makes with the positive x-axis).This angle is 45o = arctan(1). However the an-gular coordinate of (1,1) is 135o (or 225o)which is outside of the range of arctan.

One way to avoid this problem is to define afunction which makes use of both the horizontal

and vertical coordinates of the point. That is thepurpose of the atan2 function. Its range is theinterval between 180o and 180o along with thevalue 180o. The atan2 function can be defined

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in terms of arctan as follows:

atan2(y, x) = arctan(y/x) ifx > 0

atan2(y, x) = arctan(y/x) + 180o ifx < 0, y 0

atan2(y, x) = arctan(y/x) 180o ifx < 0, y < 0

atan2(y, 0) = 90o ify > 0

atan2(y, 0) = 90o ify < 0

atan2 is left undefined for the point (0, 0). Notethat it is conventional to put the y coordinatebefore the x coordinate in atan2.

An analysis of a phyllotactic pattern in theearly stages of development begins with measure-ments for the position of the primordia. Let theprimordia be labeled in ontological order so thatoldest one is labeled 1, the second oldest is la-beled 2, and so on up to the newest one whichis labeled J. Let the horizontal and vertical

coordinates measured for the jth primordium be(xj, yj).

If the origin of the coordinate system is usedas the center of the pattern then the angular co-ordinate of (xj , yj) would simply be atan2(yj, xj).However we want the angle that a ray starting atour initial estimate for the center, (x0, y0), andpassing through (xj, yj) makes with a horizontalray starting at (x0, y0). To do this we need totake the vector difference between (xj , yj) and(x0, y0) before using the atan2 function. In a

sense we are translating our coordinate systemso that (x0, y0) becomes the origin and (xj x0, yj y0) are the coordinates for the j

th pri-mordium. We denote the angular coordinate ofthe jth primordium by j. It is given by

j = atan2(yj y0, xj x0).

We denote the divergence angle between the jth

and j + 1th primordium by j. There are J 1divergence angles altogether. They are given by

j = j+1 j.

The formula for gradient descent is

(X0, Y0) = (x0, y0)

S

x0,

S

y0

(5)

where (X0, Y0) is the new estimate for the centerof the pattern. A convenient and well known

formula for computing the sum of squares is

S(x0, y0) =J1j=1

2j

J1j=1 j

2J 1

The dependence of j on (x0, y0) has been sup-

pressed in the notation. We now compute thegradient of this function. Its x component is

S

x0= 2

J1j=1

jjx0

2

J1j=1 j

J1j=1

jx0

J 1

(6)

and its y component is

S

y0= 2

J1j=1

jjy0

2

J1j=1 j

J1j=1

jy0

J 1

. (7)

We need the partial derivatives for the j. Theycan be computed from the formulas above for j.

jx0

=j+1

x0

jx0

(8)

jy0

=j+1

y0

jy0

These partial derivatives for the j can be com-puted from the formulas above for j.

jx

0

=

x0

atan2(yj y0, xj x0)

jy0

=

y0atan2(yj y0, xj x0)

We now need the partial derivatives of atan2.For x = 0 they can be computed by using thechain rule with arctan(y/x), arctan(y/x)+180o,and arctan(y/x) 180o. Of course the additionof a constant has no affect on the derivative.

xatan2(y, x) =

y

x2 + y2

yatan2(y, x) =

x

x2 + y2

We can extend these formulas by continuity tothe cases where x = 0 and y = 0 so that atan2 isdifferentiable over the whole plane except at the

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origin. Now applying the chain rule with theseformulas gives

jx0

=yj y0

(xj x0)2 + (yj y0)2(9)

jy0

= xj x0

(xj x0)2 + (yj y0)2

We now have all the formulas needed to im-plement the gradient descent algorithm. We usethe measured values, (xj, yj), for the location ofthe primordia in equation (9) to give us the par-tial derivatives of the angular coordinates j. Wesubstitute this into equation (8) to get the par-tial derivatives of the divergence angles j . Wesubstitute this into equations (6) and (7) to getthe partial derivatives of the sums of squares S.And finally we substitute this into equation (5).

These calculations are too much to do byhand but they are actually quite easy to do on adesktop computer7.

Appendix C

In this appendix we estimate the error of approx-imations (2), (3), and (4) in section 6. We let be the mean of 1, . . . , J1 and s be the stan-

dard deviation. Similarly

denotes the mean of1, . . . ,

J1 and s is the standard deviation.

Since divergence angles are differences be-

tween angular positions the scale of the differ-ences between corresponding angular positionsis about the same as the scale of the differencesbetween corresponding divergence angles. Thereare at most a few hundred primordia in an apexand there is very little chance that correspond-ing divergence angles would differ by more thanfive times the sum of the standard deviations ofthe two sequences of divergence angles. So it isvery unlikely that experimental error in the mea-surement of a primordiums position would cause

the angular position to differ by more than 5(s +s). In most cases it will differ by much less buthere we will make a very conservative estimate.The differences between the sums

J1j=1 j and


J1j=1

j depends only the difference between theangular positions of (x1, y1), (x

1, y

1) and the dif-ference between the angular positions of (xJ, yJ),(xJ, y

J). This gives us

J1

j=1

j J1

j=1

j 10(s + s) (10)The symbol is meant to indicate that it isvery likely for the left hand side to be less thanthe right hand side.

A straight forward algebraic computation givesthe equalityJ1

j=1 (

j )2

J 1

J1j=1 (

j

)2

J 1

= (

)

2

=

1

(J 1)2J1

j=1

j

J1j=1

j

2

Combining this computation with inequality (10)just above gives

J1j=1 (

j )2

J 1

J1j=1 (

j

)2

J 1

(11)

100(s + s)2

(J 1)2

This is our estimate for the error of approxima-

tion (2) in section 6. Keeping s, s

small and Jlarge keeps this error small.

Another straight forward algebraic computa-tion shows thatJ1

j=1 (

j j)2

J 1

J1j=1 (

j )2

J 1

=1

J 1

J1j=1

((j j) + (

j ))( j)

Taking the absolute value of both sides and ap-

plying the triangle inequality in a straight for-ward manner several times gives us

J1j=1 (

j j)2

J 1

J1j=1 (

j )2

J 1

1

J 1

J1j=1

(|j j| + |

j |)| j |

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Another application of the triangle inequality givesus

|j | |

j

| + |

|

Therefore

J1

j=1 (

j j)

2

J 1J1

j=1 (

j )

2

J 1

1

J 1

J1j=1

(|j j|+ |

j

| + |

|)| j |

As we have already stated it is very unlikely forj and j to differ by more than 5(s + s

). It is

also very unlikely for j and to differ by morethan 5s or for j and to differ by more than5s. And using inequality (10) above gives

|

| 10(s + s)

J 1

ThereforeJ1

j=1 (

j j)2

J 1

J1j=1 (

j )2

J 1

1

J 1

J1j=1

5(s + s) + 5s +

10(s + s)

J 1

5s

Since there are J 1 terms in the sum this sim-

plifies toJ1

j=1 (

j j)2

J 1

J1j=1 (

j )2

J 1

(12) 25s

s + 2s +

10(s + s)

J 1

This is an estimate for the error of approximation(3) in section 6.

Combining the triangle inequality with in-equalities (12) and (13) we get

J1j=1 (

j j)2

J 1

J1j=1 (

j )2

J 1

(13)

100(s + s)2

(J 1)2+ 25s

s + 2s +

10(s + s)

J 1

The left hand side of this inequality contains thetwo terms in expression (4) of section 6. We see

that the smaller s and s are the more confidentwe can be that the left and right hand sides of(4) are close to each other. The coefficients mayseem large but remember that we are being veryconservative here. The key point is that the dif-ference between the left and right hand sides of(4) has very little chance of being larger than thishomogeneous quadratic polynomial in s and s.Therefore the smaller we can make s and s bethe more confident we can be that the left andright hand sides of (4) are close to each other.

If we could be lucky enough to get s = s = 0then the left and right hand sides of (4) wouldbe equal. Further if s = 0 the right hand sideof (4) would be zero. Therefore the left handside would be zero as well. This means the twosequences 1, . . . ,

J1 and 1, . . . , J1 would bethe same.

With experimental errors in the measurementsof naturally occurring phyllotactic patterns it isvery unlikely we would get s = s = 0 but usingthe minimal variation center to keep these stan-dard deviations as small as we can reduces theamount of separation between the sums

J1j=1 (

j

j)2 and

J1j=1 (

j

)2. Since the minimal vari-

ation center minimizes the sumJ1

j=1 (

j

)2

it approximately minimizes the sumJ1

j=1 (

j

j)2. This means we have reduced the distance

between the two sequences of divergence angles1, . . . ,

J1 and 1, . . . , J1.

Appendix D

In this appendix we compute the partial deriva-tives of the function S(0, 0, x0, y0) in section8. The formula for gradient descent is

(0, 0, X0, Y0)

= (0, 0, x0, y0) S

0,

S

0,

S

x0,

S

y0

Since the values of (xj, yj) as defined in section8 do not depend on x0 or y0 the computation ofthe partial derivatives with respect to x0 and y0proceeds just as it did in appendix B. We onlyneed to compute the partial derivatives with re-

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spect to 0 and 0. We begin with

S

0= 2

J1j=1

jj0

2

J1j=1 j

J1j=1

j0

J 1

S

0= 2

J1

j=1

jj0

2

J1j=1 j

J1j=1

j0

J

1

We needj0

andj0

. Like in appendix B

j0

=j+1

0

j0

j0

=j+10

j0

Applying the chain rule to

j = atan2(yj y0, xj x0)

gives

j0

=(xj x0)

yj0

(yj y0)xj0

(xj x0)2 + (yj y0)2

j0

=(xj x0)

yj0

(yj y0)xj0

(xj x0)2 + (yj y0)2

We compute xj/0, yj/0, xj/0, yj/0using the basic rules for finding derivatives.

xj0 = uj cos(0) sin(0)(vj cos(0) + wj sin(0))yj0

= 0

xj0

= vj cos(0)sin(0) + wj cos(0)cos(0)

yj0

= vj cos(0)wj sin(0)

These formulas allow us to compute S/0 andS/0

8.


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ARTICLE - Finding the Center of a Phyllotactic Pattern - Scott Hotton (Journal Article)

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