Do Fibonacci numbers reveal the involvement of geometrical...

Botanical Journal of the Linnean Society

2006

150

3ndash24 With 10 figures

copy 2006 The Linnean Society of London


2006

150

3ndash24

3

Blackwell Science LtdOxford UKBOJ


0024-4074The Linnean Society of London 2006January 2006

150

1324

Original Article

FIBONACCI NUMBERS IN PHYLLOTAXIST J COOKE

Corresponding Author E-mail tc23umailumdedu

Donald Kaplanrsquos Legacy Influencing Teaching and Research

Guest edited by D A DeMason and A M Hirsch

Do Fibonacci numbers reveal the involvement of geometrical imperatives or biological interactions in phyllotaxis

TODD J COOKE

Department of Cell Biology and Molecular Genetics University of Maryland College Park MD 20742 USA

Received December 2004 accepted for publication April 2005

Complex biological patterns are often governed by simple mathematical rules A favourite botanical example is theapparent relationship between phyllotaxis (ie the arrangements of leaf homologues such as foliage leaves and floralorgans on shoot axes) and the intriguing Fibonacci number sequence (1 2 3 5 8 13 ) It is frequently alleged thatleaf primordia adopt Fibonacci-related patterns in response to a universal geometrical imperative for optimal pack-ing that is supposedly inherent in most animate and inanimate structures This paper reviews the fundamentalproperties of number sequences and discusses the under-appreciated limitations of the Fibonacci sequence fordescribing phyllotactic patterns The evidence presented here shows that phyllotactic whorls of leaf homologues arenot positioned in Fibonacci patterns Insofar as developmental transitions in spiral phyllotaxis follow discernibleFibonacci formulae phyllotactic spirals are therefore interpreted as being arranged in genuine Fibonacci patternsNonetheless a simple modelling exercise argues that the most common spiral phyllotaxes do not exhibit optimalpacking Instead the consensus starting to emerge from different subdisciplines in the phyllotaxis literature sup-ports the alternative perspective that phyllotactic patterns arise from local inhibitory interactions among the exist-ing primordia already positioned at the shoot apex as opposed to the imposition of a global imperative of optimalpacking copy 2006 The Linnean Society of London


2006

150

3ndash24

ADDITIONAL KEYWORDS

auxin ndash golden ratio ndash number sequence ndash optimal packing ndash spiral phyllotaxis

ndash whorled phyllotaxis

INTRODUCTION

Western philosophy has two major complementaryintellectual traditions (1) Platonic idealism in whichan overarching theory is used to integrate existingobservations and to predict new observations and (2)Aristotelian empiricism in which individual observa-tions are used to construct a unifying theory Phyllo-taxis which is broadly defined as the arrangement ofleaf homologues (ie lateral determinate organs) onshoot axes has perhaps attracted wider attentionthan most other botanical subjects in part because it

appeals to the proponents and practitioners of bothtraditions Those scientists interested in theoreticalapproaches including idealistic morphologists andtheoretical biophysicists such as Goethe BraunThompson and Green have attempted to construct anappealing synthetic theory for explaining phyllotacticpatterning and then search for compelling botanicalexamples to support that theory By contrast thosefavouring empirical approaches including compara-tive morphologists and molecular geneticists such asHofmeister Kaplan Meyerowitz and Kuhlemeierhave studied phyllotactic patterns in a range of differ-ent plants or genetic variants in the hope that thiscomparative approach might reveal fundamental

4

T J COOKE



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principles or underlying mechanisms What has thepotential to tie these disparate approaches together isthe widespread recognition that phyllotaxis displayssome truly remarkable and quite seductive mathemat-ical properties Moreover judging from the popular lit-erature the mathematics of phyllotaxis has allowedthis problem to transcend specialized scientific inter-ests so that it has also captured the imagination of theeducated public

It is often asserted that geometrical patterns in bio-logical structures are likely to result from simplephysical processes such as surface tension mechani-cal stress and fluid dynamics that are intrinsic tomatter itself (eg Thompson 1942 Ball 1999 Stew-art 2001) According to this perspective biologicalpattern is seen as the unavoidable consequence ofwhat might be called geometrical imperatives thatoperate on both inanimate and animate structures Noother phenomenon in plant morphology seems a morelikely candidate for arising from the action of a geo-metrical imperative than does phyllotaxis In seedplants despite an infinite number of conceivablearrangements the leaves are arranged in two basicpatterns spiral patterns composed of one leaf pernode and whorled patterns composed of two or moreleaves per node It is repeatedly claimed that thesepatterns can be described with reference to a simpleapparently universal and incredibly intriguing num-ber sequence known as the Fibonacci sequence

As the expression says fools rush in where angelsfear to tread and thus my colleague Wanda Kelly andI are proceeding in accordance with our naiumlve beliefthat the disciplines mentioned above have alreadymade the crucial discoveries for establishing the con-ceptual framework needed to solve phyllotaxis as ascientific problem In our opinion it has been the fail-ure of each discipline to take the discoveries from theother disciplines into account that has prevented thebotanical community as a whole from recognizing thisgreat achievement We are currently making selectedobservations designed to integrate those discoveriesfrom various disciplines into a coherent framework(for our first contribution see Kelly amp Cooke 2003)The present paper attempts to perform a clear-sightedanalysis of the Fibonacci sequence and its relationshipto plant phyllotaxis in an effort to separate botanicalessence from the Pythagorean mysticism plaguingmany scientific and popular expositions In particularthe objectives of this paper are (1) to describe the fun-damental properties of number sequences (2) to inter-pret the Fibonacci number sequence with respect tothese properties in order to illustrate its inherent lim-itations for describing certain phyllotactic arrange-ments and (3) to evaluate whether the phyllotaxesexhibiting genuine Fibonacci-based patterns arisefrom the universal geometrical imperative of optimal

packing or whether they are generated as the con-sequence of the underlying biological interactionsspecifying leaf position

FUNDAMENTAL PROPERTIES OF FIBONACCI SEQUENCES

A

PRIMER

ON

NUMBER

SEQUENCES

This section provides an elementary description ofthe critical features of number sequences A numbersequence is defined as any set of numbers that arearranged in a prescribed order Of particular interestare certain sequences known as recursive sequenceswhere each term is defined as a function of the pre-ceding term(s) A class of related recursive numbersequences is fundamentally defined by the mathemat-ical

formula

or rule used to generate each sequence inthat class For instance the number sequence

1 2 4 8 16 32 64 128 256 512

is generated by doubling the preceding term to pro-duce the succeeding term Its formula can be symbol-ized as

2

x

n

minus

1

=

x

n

where

x

n

minus

1

and

x

n

represent the values of the preced-ing term (

n

minus

1) and succeeding term (

n

) respectivelyUsing the same formula it is possible to generateanother number sequence in this class as

3 6 12 24 48 96 192 384 768

It is seen from these two examples is that one featurecapable of distinguishing between two particularsequences within a class is the

initial term

(or initialterms in those classes where the formula acts on morethan one preceding term to generate the succeedingterm) Indeed the formula and the initial term(s) areentirely sufficient together to define a unique recur-sive sequence (Vorobyov 1963 Koshy 2001)

Another feature that can help to characterize anumber sequence is its

limit

which represents thenumber that is ultimately approached by a sequencecomposed of many numbers Many sequences such asthe two listed above diverge without any finite limit sothat they can be said to reach a limit of infinity Farmore interesting are those sequences that converge ona specific number such as

which is generated by the formula

x

n

minus

1

=

2

x

n

Of course the limit for this sequence is zero It isworth noting for future reference that the limit of aconverging sequence may have unique properties thatare not shared by the actual numbers in thatsequence In this particular example all the numbersno matter how infinitesimally small in this sequence

1 12

14

18

116

132

164

FIBONACCI NUMBERS IN PHYLLOTAXIS

5



2006

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have definite values whereas zero has none By con-trast the sequence of

converges on the unexceptional limit of 2Ironically the least informative characteristic of a

number sequence may be a

small set

of consecutiveterms in the sequence Frequently a given small setcan be generated by several different formulaethereby signifying that the set belongs to differentclasses of number sequences As a trivial example thesmall set of 1 2 and 4 can be a part of the first numbersequence given above or a part of another sequenceconsisting of

1 2 4 7 11 16 22 29 37 46


x

n

minus

1

+

p

n

minus

1

=

x

n

where

p

n

minus

1

represents the number corresponding tothe position of the preceding number in the sequenceFor readers interested in small integer sets and thelarger sequences including those sets the query toolavailable at Sloane (2004) is quite informative

An understanding of all these characteristicsincluding formula initial terms limit and small setwill be critical to our analysis of the Fibonacci numbersequence and its relationship to phyllotacticpatterning

S

ALIENT

FEATURES

OF

PRIMARY

F

IBONACCI

NUMBER

SEQUENCES

Fibonacci number sequences represent a special classof recursive number sequences that is defined by theformula that the sum of the two preceding numbersgenerates the succeeding number or

x

n

minus

2

+

x

n

minus

1

=

x

n

(for general references see Coxeter 1953 Vorobyov1963 Hoggatt 1969 Vajda 1989 Dunlap 1997Koshy 2001) The most familiar Fibonacci sequencestarts off with 1 and 2 (or the equivalent of 1 1 and 2)as its initial terms The Fibonacci formula performedon these initial numbers generates the so-called pri-mary Fibonacci sequence or

1 2 3 5 8 13 21 34 55 89 144 233 377

Using different initial terms this formula can gener-ate an infinite number of accessory Fibonaccisequences (Fig 1) such as

1 3 4 7 11 18 29 47 76 123 199 322 521 1 4 5 9 14 23 37 60 97 157 254 411 665

1 5 6 11 17 28 45 73 118 191 309 500 809 etc

Obviously all these Fibonacci sequences mustmonotonically increase toward their limits of infin-ity Other classes of Fibonacci-related sequences aredescribed in the mathematics literature (see Sloane2004)

1 1 1 1 1 1 112

34

78

1516

3132

6364

Of particular interest are the primary fractionalFibonacci sequences which are produced by applyingthe Fibonacci formula to the numerators and to thedenominators of successive fractions of primaryFibonacci numbers or

The primary fractional Fibonacci sequences are repre-sented by the numbers

and their reciprocals

These sequences do not approach infinity as their lim-its but rather they converge on the never-repeatingnever-ending irrational numbers 16180339887 and 06180339887 respectively In Euclidean geo-metry certain irrational numbers are recognized tohave singular properties and thus they are assignedtheir own symbols for example

π

is used to represent31415926535 or the ratio of the circumference of acircle to its diameter Thus 16180339887 is sym-bolized as

φ

and its reciprocal as 1

φ

It turns out that

φ

also represents what the Greeks called the lsquoextremeand mean ratiorsquo which corresponds to a division of aline such that the ratio of the line to the larger seg-ment is equal to the ratio of the larger segment tosmaller segment (Fig 2) As is illustrated in thisdrawing

Over the next 2400 years

φ

has repeatedly appeared insuch diverse endeavours as mathematics art archi-tecture music nature and philosophy and conse-quently it has acquired the colloquial name of the

xx

xx

xx

n

n

n

n

n

n

-

-

-

- -+ =2

3

1

2 1

21

32

53

85

138

2113

3421

5534

8955

12

23

35

58

813

1321

2134

3455

5589

f = =ABAC

ACCB

Figure 1

Initial terms in the primary and initial acces-sory Fibonacci sequences plus the multiples of the termsin the primary and 1st accessory sequences

Primary sequence 1 2 3 5 88

13 21216

1 8 9 17

1 6 11 1751 4 9 145

1 6 13 207

1 3 4 11 187

11 17 197

1 9 1910

2 13 17 195 10

1 87 15

4 9 12 14 16 18 2010 15

6 8 9 12 14 16 18 2015 21

Sequence multiplesUnrelated numbers

1st accessory sequence

2nd accessory sequence3rd accessory sequence4th accessory sequence5th accessory sequence6th accessory sequence7th accessory sequence


6

T J COOKE



2006

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Golden Ratio (for informative essays see Huntley1970 Dunlap 1997 Kappraff 2002 Livio 2002) Thereader is referred to the Appendix for additional infor-mation about the fascinating numerical properties of

φ

The golden ratio derived from the subdivision of a

line leads to many other geometrical relationshipsthat comprise what might be called lsquogolden geometryrsquo(Coxeter 1953 Vorobyov 1963 Hoggatt 1969 Hunt-ley 1970 Vajda 1989 Koshy 2001 Kappraff 2002Livio 2002) For instance a golden rectangle can bedrawn from the golden segment (ACB) such that theratio of the longer side (AB) over the shorter side (BD)is also equal to

φ

(Fig 2) If the largest possible squareis drawn within the golden rectangle then the remain-ing rectangle must have the same proportions as theoriginal rectangle meaning that the aspect ratio of thenew longer side (BD) over the new shorter side (BC) isonce again equal to

φ

In terms of an equation

The golden rectangle can thus be said to exhibit theproperty of self-regeneration in that a larger goldenrectangle can be subdivided to generate a square anda smaller golden rectangle This subdivision processcan be continuing

ad infinitum

with the result thateach subdivision results in an even smaller rectanglewith an aspect ratio of

φ

This process of repeated sub-divisions leaves a never-reachable point of free spacewhich is referentially known as the lsquoEye of Godrsquo nearthe centre of the golden rectangle The golden rectan-gle is the only rectangle with the property whereby thecutting off of the largest possible square produces a

f = =ABBD

BDBC

smaller rectangle with an identical shape as the orig-inal rectangle

Similarly it is possible to divide a circle into twogolden angles which exhibit the following relation-ships (Fig 2)

where

θ

l

and

θ

s

represent the larger and smallergolden angles of the circle respectively Rearrangingthis equation to solve it for the angles

and

θ

s

is the so-called golden or ideal angle often proposedto represent the optimal displacement of leaf primor-dia on shoot apices as is examined in a later sectionAn alterative method for calculating the ideal angleinvolves the limit of the reciprocal primary fractionalseries starting with the initial terms of 13 25 and38 This limit is equal to

φ

minus

2

as is shown in AppendixTable A2 Then

U

NDER

-

APPRECIATED

MATHEMATICAL

CONSTRAINTS

ON

THE

APPLICATION

OF

F

IBONACCI

SEQUENCES

TO

BIOLOGICAL

PHENOMENA

The Fibonacci literature has unbridled enthusiasm foridentifying the putative involvement of the Fibonaccisequence in biological and especially botanical phe-nomena (eg Coxeter 1953 Huntley 1970 Garland1987 Koshy 2001 Britton 2003 Knott 2004) Itmakes one almost forget that the Fibonacci sequencewas first devised as the solution to a

hypothetical

mathematical problem about rabbit populationgrowth I believe that we botanists are well advised toexpress greater scepticism toward any alleged exam-ple of the botanical manifestation of the Fibonaccisequence The following questions can be used toinform our thinking on this issue

(1) Does an individual grouping of biological objectsas a primary Fibonacci number provide compellingevidence for the underlying participation of theFibonacci sequence The numbers 2 3 and 5 (and theirmultiples) are frequently alleged to disclose theinvolvement of the Fibonacci sequence in a given pro-cess because they are taken to represent uniqueFibonacci numbers as opposed to other lsquonon-Fibonaccirsquonumbers It follows from this allegation that any

fq

qq

= =360

1

1infins

qfl = =360

222 492infin infin

q qfs

l= = 137 507 infin

q fs = ( ) =360 137 507infin infin-2

Figure 2 Several examples of golden geometry derivedfrom the golden ratio (φ) which was first recognized as thedivision of a line such that the ratio of the line to the largersegment is equal to the ratio of the larger segment tosmaller segment

FIBONACCI NUMBERS IN PHYLLOTAXIS 7

copy 2006 The Linnean Society of London Botanical Journal of the Linnean Society 2006 150 3ndash24

structure appearing in a group of 5 such as the digitson the human hand or the petals of a rose flowercan be interpreted as being a manifestation of theFibonacci sequence This argument is easily refuted byre-examining Figure 1 It is worth noting that the firstsix positive integers are either components or multi-ples of the primary Fibonacci sequence thus a smallgroup must be composed of at least 7 units before itappears to be unrelated to the primary Fibonaccisequence Furthermore of the first 21 integers 7 inte-gers are included in the primary sequence and 12 inte-gers are multiples of those 7 integers (Fig 1) Twonumbers 8 and 21 are both components and multi-ples of the primary sequence Thus almost everysmall group of biological objects must unavoidably bequantified in terms of Fibonacci numbers It mightinstead be argued that the only meaningful groupingof biological objects might be those groups of 7 11 17or 19 that have no obvious relation to the primaryFibonacci sequence

In fact being a component of Fibonacci sequencesis an intrinsic property of all positive integers Ifwe restrict our attention to only those Fibonaccisequences starting with an initial term of either 1 or 2then 3 is a term in two non-redundant sequencesnamely the primary and first accessory Fibonaccisequences (Fig 1) whereas 4 is a part of three non-redundant sequences namely the first accessorysequence plus two other sequences

1 4 5 9 14 2 4 6 10 16

All integers (n) greater than 4 belong to at least fournon-redundant Fibonacci sequences starting with theinitial terms of 1 or 2 as follows

1 n n + 1 2n + 1 1 n minus 1 n 2n minus 1 2 n n + 2 2n + 2 2 n minus 2 n 2n minus 2

In addition all odd integers 7 or above belong to atleast one additional non-redundant sequence given as

Similarly all even integers 8 or above belong to atleast one additional non-redundant sequence given as

These considerations show that all positive integerscan be considered as being Fibonacci numbers It fol-lows that a single number by itself does not allow us todiscriminate between a genuine Fibonacci relation-ship and other arrangements having nothing to dowith Fibonacci sequences No credibility can beassigned to any claim that a particular number dis-closes the involvement of Fibonacci sequences

1 1 1 3 112

12

12 n n n n-( ) +( ) +( )

2 2 2 3 212

12

12 n n n n-( ) +( ) +( )

(2) Can the groupings of biological objects in smallsets exhibiting consecutive numbers such as 2 3 5and 8 or 3 4 7 and 11 be exclusively attributed to theoperation of a Fibonacci sequence In other words isthe appearance of biological objects in 2s 3s and 5ssufficient to reveal the involvement of the primaryFibonacci sequence An earlier section devoted to aprimer on number sequences demonstrated that nosmall set should be assumed to represent only onenumber sequence and this warning most certainlyapplies to small sets taken from Fibonacci sequences

Sloane (2004) provides a query tool that allows thescreening of a database of c 100 000 sequences inorder to identify all sequences containing a specifiedsmall number set Table 1 shows that a miniscule pro-portion of the number sequences including the shortsequence of 1 2 3 and 5 are related to Fibonaccisequences Even the addition of 8 and 13 to this shortsequence makes only 52 of the identified sequencesrelated to Fibonacci sequences Therefore identifyinga small set of consecutive numbers as belonging to aFibonacci sequence is a necessary but not sufficientcriterion for establishing the operation of theFibonacci sequence in the biological pattern underinvestigation

(3) Does the primary fractional Fibonacci sequence(21 32 53 85 etc) have unique mathematicalproperties that arise from its limit of φ Perhaps spe-cial consideration should be granted to the numbers inthe primary Fibonacci sequence as opposed to thenumbers in other Fibonacci sequences I have alreadyindicated above that the fractional sequences com-posed of primary Fibonacci numbers result in goldenratios of φ and φminus1 as their limits and therefore itmight seem reasonable to propose that the primaryfractional sequences might have unique featuresattributable to their limits

However one must also be disabused of this appeal-ing notion because a fractional Fibonacci sequence

Table 1 The results from querying the on-line search toolavailable at Sloane (2004) for the number of integersequences containing specified short sequences derivedfrom the primary Fibonacci sequence Maximum numberof sequence matches provided in response to a given queryis 100

Query sequence

Total matches

Fibonacci-relatedsequences

1235 100 9 (9)12358 100 37 (37)1235813 79 41 (52)123581321 40 26 (65)12358132134 26 22 (85)

8 T J COOKE


constructed from any two initial numbers chosen atrandom will inevitably converge on either φ or φminus1 as isnoted by several authors including Thompson (1942)Huntley (1970) and Livio (2002) For example using 4and 87 as the initial numbers the resulting fractionalFibonacci sequences are

and the reciprocal

The 8th term is equal to 16156 in the first frac-tional sequence and to 06189 in the reciprocalsequence which illustrates just how rapidly fractionalFibonacci sequences (with an initial term of xaxb) con-verge on φ (in the case of xa gt xb) or φminus1 (in the case ofxa lt xb) Moreover all fractional Fibonacci sequencesapproach the powers of φ as their limits following thesame formulae as shown for the primary fractionalsequences (Appendix Table A2) It should be obviousthat specific numbers even those in the primaryFibonacci sequence have no special mathematicalrelationship with φ or φminus1 but rather these limits arethe inevitable outcome of the fractional Fibonacciformula

The mathematical relationships described abovehave profound implications for any attempt to relate aset of grouped objects exhibiting some numbers from aprimary fractional Fibonacci sequence to the underly-ing mechanism generating the biological patternFirst of all it underscores the concept from the primersection that the formula is critical for defining anyclass of number sequences including Fibonaccisequences More specifically it establishes that thelimit φ and the mathematical properties associatedwith it are solely attributable to the operation of thefractional Fibonacci formula as opposed to being asso-ciated with the trivial numbers comprising any givenfractional sequence Therefore the operation of aFibonacci sequence can only be visualized in a biolog-ical pattern exhibiting two characteristics (1) thebiological objects are arranged in various groupingsexhibiting different Fibonacci numbers and (2) devel-opmental transitions to other groups of different num-bers must follow a discernible Fibonacci formula Onlyif the pattern expresses both characteristics can aninvestigator argue for the likely involvement of aFibonacci sequence

What the reader needs to retain from this ratherbelaboured discussion is that just because some bio-logical objects are grouped in a specific number foundin the primary Fibonacci sequence it does not meanthat these objects are being arranged in accordancewith the Fibonacci sequence For example let us say

874

9187

17891

269178

447269

716447

1163716

18791163

487

8791

91178

178269

269447

447716

7161163

11631879

that an organism is usually observed to produce astructure composed of five units If this organism orrelated organism can also develop the same structurewith either three or eight units then we have muchstronger evidence that the structure depends on theoperation of a Fibonacci-based mechanism Howeverif the occasional smaller and larger structures arecomposed of four and six units respectively then thisstructure is constructed without the apparent involve-ment of the Fibonacci sequence We are now preparedto evaluate the question of whether phyllotactic pat-terning in plants can be ascribed to the operation ofFibonacci sequences

FIBONACCI NUMBERS AND PHYLLOTACTIC PATTERNS

In the phyllotaxis literature it is often asserted thephyllotactic patterns result from the operation of thegeometrical imperative of optimal packing or itsequivalent This assertion can be deconstructed intothree sequential propositions

1 Are the primordia of leaf homologues arrangedaccording to the numbers composing the Fibonaccisequence

2 Do the arrangements exhibiting Fibonacci numbersreveal the underlying operation of the Fibonacciformula

3 Do the arrangements following the Fibonacci for-mula generate optimal packing

In this section the first two questions will be used toevaluate the organization of leaf primordia in the twoprincipal types of phyllotactic arrangements observedin seed plants The third question is deferred until thefollowing section

PHYLLOTACTIC WHORLS

One common phyllotactic pattern is the whorl wherea group of leaf homologues such as foliage leaves orfloral organs arise at the same node of a shoot axisMany aquatic angiosperms such as Myriophyllumspicatum L Anacharis canadensis (Michx) Planchand Ceratophyllum demersum L as well as some ter-restrial plants are observed to develop foliage leavesin whorls of 3 4 and 5 Most angiosperm flowers pro-duce petals and other floral organs in whorls of 2 3and 5 or their multiples Just to cite a few examplesalmost all species in the Ranunculaceae and Rosaceaehave 5 petals whereas many species in the Liliaceaeare characterized by 3 or 6 petals Do these numbersdisclose the role of the Fibonacci sequence in specify-ing the number of leaf homologues in each whorl as isargued in the botanical literature (eg Church 1920Endress 1987) It should be clear from the previous



section that the critical evidence for evaluating thisclaim lies in the transitions to other whorls with dif-ferent numbers of leaf homologues

The evidence available from those plants withwhorled foliage leaves is incontrovertible Vegetativeshoots are indeterminate structures with many nodesof foliage leaves so that it is relatively easy to identifyand characterize whorled plants with different leafnumbers at their nodes For example McCully amp Dale(1961) studied the heteroblastic changes in leaf num-ber in successive whorls in the angiosperm Hippurissp L which exhibits whorls ranging from 2 to 16leaves Their observations demonstrated that thenumber of leaves in successive whorls change by smallincrements of one or two leaves with the leaf numberbeing strongly correlated with the diameter of theshoot apex at the time of whorl initiation (Fig 3) The

whorled shoots of several species of the sphenopsidEquisetum L exhibit similar changes in leaf numberthat are also related to apex diameter (Bierhorst1959) These studies establish that leaf numbersin vegetative whorls do not undergo heteroblasticchanges in accordance with a discernible Fibonacciformula Therefore the Fibonacci sequence plays noapparent role in the generation of whorled phyllotaxison vegetative shoots

By contrast flowers are determinate structures thatare frequently composed of single whorls of each typeof floral organ therefore it is generally impossible toobserve developmental transitions in floral organwhorls such as those observed in foliage leaf whorls onvegetative shoots However there are two reasons forconcluding that the Fibonacci sequence is also unin-volved in the specification of whorled phyllotaxis inflowers One ever since Goethe (1790) plant morphol-ogists have recognized that all determinate lateralorgans such as foliage leaves and floral organs arehomologueous structures It is noteworthy that thismorphological concept has received molecular confir-mation insofar as triple mutations in the ABC classgenes cause the floral organs to revert to leaf-like phe-notypes (Coen amp Meyerowitz 1991) Thus one mightreasonably hypothesize that phyllotactic arrange-ments of whorled floral organs are mediated by non-Fibonacci mechanisms related to those operatingin leaf whorls Two several Arabidopsis mutantsexhibit altered numbers of floral organs as comparedwith wild-type plants Wild-type Arabidopsis flowersdevelop concentric whorls of 4 sepals 4 petals 6 sta-mens and 2 carpels whereas these mutant flowersdevelop more or fewer organs in several whorls(Table 2) For example wus flowers tend to have 3 or 4sepals 3 or 4 petals and 0ndash3 stamens (Laux et al1996) By contrast pan flowers often develop 5 andsometimes 6 organs in the three outer whorls (Run-ning amp Meyerowitz 1996) One cannot assign the

Figure 3 Relationship between the number of leaf pri-mordia in the youngest whorl and the diameter of theapical dome The solid circles and dotted line represent theobservations on aerial shoots the stars and solid line rep-resent the observations on submerged shoots The linesconnect the mean diameters correlated with each leaf num-ber Redrawn with permission from McCully amp Dale (1961)

Table 2 The number of sepals petals and stamens in wild-type and mutant flowers of Arabidopsis thaliana

Mutant name orTAIR number

Floral organ number

ReferenceSepals Petals Stamens

wild-type 4 4 6CS2310 3ndash4 3 3ndash4 TAIR (2004)petal loss (ptl) 4 0ndash3 6 Griffith et al (1999)wuschel (wus) 3ndash4 3ndash4 0ndash3 Laux et al (1996)perianthia (pan) 5 5 5 Running amp Meyerowitz (1996)CS2292 4ndash5 4ndash5 TAIR (2004)CS2289 5ndash6 6ndash7 TAIR (2004)clavata1 (clv1) 4ndash6 4ndash6 6ndash10 Leyser amp Furner (1992) Clark et al (1993)clavata3 (clv3) 5ndash6 5ndash6 9ndash11 Clark et al (1995)

10 T J COOKE


observed differences between organ numbers in wild-type vs mutant flowers to the operation of any obviousFibonacci formula Moreover the changes in floralorgan number are directly correlated with floral mer-istem size in certain mutants (wus Laux et al 1996clv1 Clark Running amp Meyerowitz 1993 clv3 ClarkRunning amp Meyerowitz 1995) but not in others (panRunning amp Meyerowitz 1996 ptl Griffith da SilvaConceiccedilao amp Smyth 1999) so that a related mecha-nism may be partially responsible for specifying whorlnumber in both foliage leaves and floral organs

The unrestrained tendency to visualize theFibonacci sequence in botanical patterns has led tosome rather ill-conceived interpretations about howvarious flowers produce their petals in whorls of pri-mary Fibonacci numbers ranging from 1 to 89 as arecommonly cited in the mathematics literature (egHuntley 1970 Koshy 2001) and in popular publica-tions (eg Garland 1987 Britton 2003 Knott 2004)These exuberant claims do not pass close scrutiny forseveral reasons not the least of which is that thestructures cited are often not petals at all For exam-ple Britton (2003) illustrates the calla lily as an exam-ple of a flower with a single petal it turns out that thisstructure is an enlarged bract known as the spathethat grows around the condensed inflorescence com-posed of many small flowers Various members of theAsteraceae are almost universally cited as havingpetal numbers equal to the primary Fibonacci num-bers of 8 13 21 34 55 and 89 Of course these so-called petals are more properly referred to as ray flo-rets which do not arise in true whorls but rather incompressed spirals called pseudowhorls Nor do theray florets of the Asteraceae appear to meet any rig-orous standard for exhibiting the operation of theFibonacci formula As an initial survey I counted thenumber of ray florets on 100 inflorescences of severalAsteraceae species readily available in Spring SilverMD (Fig 4) In a clone of Rudbeckia fulgida Ait lsquoGold-strumrsquo growing in my back garden the mean numberof ray florets per capitulum for 100 capitula was 1282which happens to fall quite close to the primaryFibonacci number of 13 as reported by Britton (2003)However Figure 4 illustrates that the ray florets onindividual capitula ranged from 10 to 15 in numberBy contrast 100 capitula of a large Chrysanthemummorifolium L plant purchased from a local nurseryexhibited a mean number of ray florets per capitulumof 2568 and a range of 20ndash36 ray florets on differentcapitula A population of Cichorium intybus L grow-ing along an exposed roadside displayed a mean of1652 ray florets per capitulum ranging from 13 to 20florets on different capitula It is clear from this smallsample that different Asteraceae species exhibit anormal distribution of ray florets in their capitulawith the means apparently approaching a primary

Fibonacci number in certain species However there isno cogent evidence from Figure 4 that such occasionalcoincidences have any biological significance and thusit appears that the Fibonacci sequence does not par-ticipate in the regulatory mechanism specifying rayfloret number

In conclusion the evidence on whorled phyllotaxispresented here can be used to address the threepropositions stated at the beginning of this sectionWhorled phyllotaxes do satisfy the first propositioninsofar as the whorls on both vegetative and reproduc-tive shoots are often composed of a primary Fibonaccinumber of leaf homologues However the evidencedoes not satisfy the other two propositions Develop-mental transitions of foliage leaf whorls and geneticmanipulations of floral organ whorls do not follow dis-cernible Fibonacci formulae Therefore the whorledarrangements of foliage leaves and of floral organs donot depend on a Fibonacci-based mechanism Conse-quently whorled phyllotaxis cannot result from theoperation of a hypothetical geometrical imperative foroptimal packing

PHYLLOTACTIC SPIRALS

In many terrestrial seed plants the foliage leaves onvegetative shoots are routinely observed to develop inopposing clockwise and anticlockwise spirals calledparastichies If the leaves are assigned a numberin the order of their origin then the intervals in thenumbers between successive leaves in these spiralpairs are typically related to the primary Fibonaccisequence (for illustrations see Williams 1975) Forexample a shoot apex producing leaf primordia in twoopposing parastichies with primordium intervals ofn + 2 and n + 3 is said to exhibit the (23) phyllotaxisThis arrangement is roughly equivalent to the 25phyllotactic fraction of mature shoots where the gen-

Figure 4 Distribution of the number of ray florets in 100capitula of three Asteraceae species Rudbeckia fulgida(mean of 1282 florets per capitulum) Cichorium intybus(mean of 1652 florets per capitulum) and Chrysanthemummorifolium (mean of 2568 florets per capitulum)



erative spiral is seen to complete two circuits aroundthe stem for every five leaves

Frequently the parastichies used to characterizespiral phyllotaxis are the so-called contact paras-tichies or those derived from drawing spirals throughadjacent primordia in direct contact Fujita (1938)surveyed the distribution of spiral phyllotaxis in thevegetative and reproductive axes of seed plants Inangiosperms c 80 of all spiral phyllotaxes arereportedly characterized by contact parastichies in the(23) pattern (Table 3) Most other spiral phyllotaxeson vegetative shoots exhibit either the (12) or the (35)arrangement of contact parastichies although thecommon distichous (11) phyllotaxis was apparentlyexcluded from this survey Thus Fibonacci spirals rep-resent the predominant pattern among all possiblespirals in this survey as well as in other surveys(Church 1920 Jean 1994) One cautionary note isthat contact parastichies are dependent on primordialshape and thus they may not provide an accuratemeasure of relative primordial position Richards(1948 1951) quite rightly emphasized that the posi-tion of successive primordia is completely specified inthe transverse plane by the divergence angle and theplastochron ratio ie the relative radial distances oftwo successive primordia In Richardrsquos analysis pri-mary attention is granted to those pairs now known asconspicuous parastichy pairs (Adler 1974 Jean 1994)whose intersection most closely approaches a 90degangle It turns out that these conspicuous parastichypairs also exhibit adjacent Fibonacci numbers andmoreover they will usually but not always coincide

with the more obvious contact parastichy pairs (fordiscussion see Williams 1975 Jean 1994) Irrespec-tive of the approach used to identify the parastichypairs it is inescapable that the spiral phyllotaxes ofvegetative shoots are overwhelmingly characterizedby low Fibonacci numbers

Reproductive shoots display spiral patterns on twodifferent morphological levels namely floral organsin individual flowers and flowers in inflorescences(Fujita 1938 Endress 1987) In comparison with veg-etative shoots reproductive shoots show a muchgreater distribution of spiral phyllotaxes rangingfrom (23) to (3455) patterns with the mode being(35) (Table 3) Such flowers as water lilies and mag-nolias with high numbers of floral organs tend todevelop their organs in spiral patterns exhibiting pri-mary Fibonacci numbers for example the flowers ofMagnolia obovata Thunb exhibit (1321) patterns ofstamens and of carpels (Fujita 1938) Because floralorgans are presumably homologous to foliage leavesthese observations suggest that spiral phyllotaxis ofboth organ types may depend on related patterningmechanisms However the floral organs of certainflowers including Michelia fuscata (Andr) Blume(Tucker 1961) exhibit spiral patterns that do not fol-low the primary Fibonacci sequence (Table 3) A plau-sible explanation of these divergent patterns lies inthe much higher rate of floral organ initiation whichmay also account for the occasional appearance of cha-otic arrangements (Endress 1987)

Lastly the flowers on the inflorescences of mostangiosperms such as Capsella bursa-pastoris (L)

Table 3 Distribution of spiral phyllotaxes in angiosperms Phyllotactic patterns were measured as contact parastichiesin apical cross-sections The divergence angles calculated for the contact parastichies assume an orthogonal arrangementof those parastichies The data for reproductive shoots were compiled from the arrangements of floral organs in individualflowers and those of flowers in inflorescences nd no data collected for these spirals Adapted from Fujita (1938) astabulated by Williams (1975)

Phyllotactic patterns Divergence angles (deg) Vegetative shoots Reproductive shoots

Primary Fibonacci spirals(11) 180 nd nd(12) 120 45 ndash(23) 144 335 35(35) 135 53 43(58) 13846 4 25(813) 13714 1 12(1321) 13765 ndash 11(2134) 13745 ndash 2(3455) 13753 ndash ndash

Accessory Fibonacci spirals 1 29Bijugate spirals ndash 8Total shoots 439 166Species represented 411 121

12 T J COOKE


Medic and Antirrhinum majus L are usually posi-tioned in spiral patterns exhibiting low Fibonaccinumbers (Table 3 Fujita 1938) It is quite likely thatthe mechanism specifying the position of individualflowers may also be related to those operating in foli-age leaf and floral organ phyllotaxis It turns out thatflowers tend to arise in the axils of leaf-like bractswhich are also considered as being leaf homologuesBecause these bracts are usually arranged in spiralpatterns the result is that the entire inflorescencetends to display spiral phyllotaxis It is worth pointingout that the phyllotaxis literature tends to grant dis-proportionate attention to the few extraordinary casesof reproductive structures displaying high Fibonaccinumbers such as the ovulate cones of various conifersthe multiple fruit of the pineapple Ananas comosus(L) Merr and the disc flowers on the capitula of theAsteraceae For example pineapple fruits are typi-cally characterized by either (813) or (1321) paras-tichies It is obvious that the spiral organization ofconifer cones and pineapple fruits reflects the position-ing of the evident bracts subtending the individualunits in these reproductive structures The extreme(3455) phyllotaxis reported in Table 3 is exhibited bydisc florets on the capitulum of the sunflower Helian-thus annuus L (Fujita 1938) The capitula of theAsteraceae are traditionally interpreted as beingcondensed shoot systems and it is therefore expectedthat their organization is dependent on the samedevelopmental mechanisms operating in vegetativeshoots (Burtt 1978) Indeed many Asteraceae speciesincluding Helianthus annuus and other members ofthe tribe Heliantheae have retained a subtendingbract called the palea or receptacular scale at the baseof each floret (P K Endress pers comm) which ispresumably involved in the positioning of the floretson the capitulum (The palea may be reduced to formreceptacular bristles or is completely missing in otherAsteraceae species but it is unlikely that these specieswould have evolved novel mechanisms for positioningtheir florets) In conclusion it seems quite reasonableto make the broad generalization that the spiral phyl-lotaxes of vegetative shoots flowers and inflorescencesare all generated by related mechanisms acting tospecify the positions of leaf homologues

Even though spiral phyllotaxes are routinely char-acterized by Fibonacci numbers one must also showthat developmental transitions to other spirals followa Fibonacci formula in order to confirm the operationof Fibonacci-based mechanisms in spiral phyllotaxisThe vegetative shoots of most plants exhibit a stablecharacteristic spiral phyllotaxis following the initia-tion of the first few foliage leaves however certainplants do undergo phyllotactic transitions followingthe Fibonacci formula throughout vegetative growthJust to cite one example the vegetative shoot of

Linum usitatissimum L undergoes a heteroblasticincrease in the numbers of its Fibonacci spirals(Williams 1975) The 4-day-old seedling exhibits adecussate pattern that is originally established inthe embryo (Fig 5) Subsequent leaf primordia arearranged in a (35) phyllotaxis in the apices of 8- and15-day-old plants Then the shoot apex starts produc-ing new primordia at a much higher rate resulting ina (58) phyllotaxis in 22-day-old apices In the apices ofthe 50-day-old plants with over 200 leaves the contactparastichies are still arranged in the (58) pattern butthe conspicuous parastichies are seen to approach the(813) pattern (Fig 5) Various species in the Magno-liaceae exhibit stepwise transitions following theFibonacci formula in the spiral phyllotaxes of stamensvs carpels (Fujita 1938) For instance the stamens ofMagnolia grandiflora L arise in an (813) phyllotaxisbut its carpels change to a (1321) pattern Bycontrast the reproductive organs of Liriodendrontulipifera L undergo the opposite transition in paras-tichy numbers Comparable Fibonacci-based transi-tions are also seen in inflorescences such as sunflowercapitula where the transitions depend on capitulumsize and flower position Although the disc flowers aretypically observed to arise in a (3455) pattern in theouter regions of normal-sized sunflower capitulasmall capitula exhibit either (1321) or (2134) pat-terns and larger capitula exhibit higher Fibonacci spi-rals in step-wise increases to a maximum of the(144233) pattern (Jean 1984) It is also observed thatthe disc flowers on a normal capitulum proceed froma (3455) phyllotaxis at the periphery to a (2134)pattern in the intermediate region and then tolower Fibonacci spiral phyllotaxes near the centre(Thompson 1942 Richards 1948 Williams 1975) Inoilseed sunflower hybrids large capitula displayingthe peripheral (89144) phyllotaxis are also seen toundergo step-wise Fibonacci decreases toward theircentres (Palmer 1998) In marked contrast to whorledphyllotaxis the evidence presented here means thateven this skeptical author cannot cogently argueagainst the characterization of spiral phyllotaxis ofboth vegetative and reproductive shoots in terms ofthe formula for the primary Fibonacci numbers

GEOMETRICAL IMPERATIVE OF OPTIMAL PACKING

However there remains the question of whether or notsuch spiral arrangements are attributable to the leafprimordia being positioned in optimal packing Sev-eral mathematical models have employed close pack-ing contact pressure or their equivalents as the causalmechanism for generating spiral patterns exhibitingFibonacci numbers (eg van Iterson 1907 Erickson1973 Adler 1974 Ridley 1982a) In general these



Figure 5 Transverse sections of shoot apices of Linum usitatissimum at different developmental stages For each stagethe top drawing indicates the number of each leaf primordium on the apex starting with the first epicotylar primordiumas number 1 and the bottom drawing shows the corresponding contact parastichies superimposed on the apex Day 4 apexexhibits a decussate pattern that is originally established in the embryo the stippled structures represent lateral budsthat have developed in the axils of the cotyledons Subsequent leaf primordia on the day 8 and 15 apices are initiated ina (35) phyllotaxis but younger leaf primordia arise in a (58) phyllotaxis on the day 22 apex On the day 50 apex thecontact parastichies are still arranged in a (58) pattern but the conspicuous parastichies approach an (813) patternRedrawn with permission from Williams (1975)

14 T J COOKE


models are designed to evaluate the relationshipbetween the angular divergence of successive units ofuniform size and the packing efficiency of the overallstructure This research has convincingly shown thata generative spiral with a divergence angle equal tothe so-called ideal or Fibonacci angle of 1375deg resultsin optimal packing Moreover some efforts have suc-cessfully generated realistic models of sunflower capit-ula that can even show decreased Fibonacci numberstoward the centre (eg Vogel 1979 Rivier et al 1984)This work has sparked renewed interest in applyingcrystallographic approaches to phyllotaxis (Rivieret al 1984 Jean 1994 Mackay 1998 Selvan 1998)Lastly a modified version of an optimal packing argu-ment is sometimes used as a deus ex machina toexplain what appears inexplicable by even those work-ers whose research does not emphasize Fibonaccinumbers For example Green (1999 1064ndash1065)invoked relative packing as a rather contrived ratio-nale to account for the switch between spiral andwhorled patterns Thus it seems entirely appropriatehere to attempt a critical analysis of the putative roleof optimal packing in spiral phyllotaxis

A SIMPLE MODEL

Underlying most proposed packing mechanisms is theimplicit assumption that golden geometry expressedin the form of the Fibonacci angle of 1375deg is operat-ing in phyllotactic patterning Both theoretical con-siderations and direct observations invalidate thatassumption For instance as a simple graphical exer-cise let us examine the relative packing in a subdi-vided golden rectangle vs other subdivided rectangleswith the aspect ratios corresponding to the commoncontact parastichies observed in spiral phyllotaxis(11 12 23 35 58 and 813) and the resulting diver-gence angles (180deg 120deg 144deg 135deg 13846deg and13714deg) (Table 3) It is assumed in the initial presen-tation of this exercise that the contact parastichies canbe used to estimate the divergence angles of actualleaf primordia arising on the shoot apex The limita-tions of this assumption are addressed in the followingsection

As described earlier a unique property of a goldenrectangle (with the aspect ratio of 1φ) is that can besubdivided into a square and a smaller golden rectan-gle ad infinitum with each successive rectangleexhibiting the same proportions as the previousrectangle It turns out that if circles are inscribed inthe squares then a subdivided golden rectangle asillustrated in Figure 6 appears quite reminiscent oftwo-dimensional projections of genuine shoot apicesFirst of all the ability of the golden rectangle toundergo repeated subdivisions is highly suggestive ofthe indeterminate growth of most vegetative and

Figure 6 Modelling results from one process of subdivid-ing the golden rectangle and other rectangles with aspectratios corresponding to the most common contact paras-tichies The subdivision process illustrated in this figureinvolved first cutting off the largest possible square in theoriginal rectangle and then repeating the process in theremaining portion of the rectangle until the entire rectan-gle is occupied by the squares The subdividing lines aremarked by lower-case letters in the order of their insertionCircles representing leaf primordia (grey shading) areinscribed in the squares The space between the squaresand the circles is defined as inscribed free space (unshadedareas) After six subdivisions the golden rectangle containsan unsubdivided centre (black shading) in the shape of agolden rectangle that can further be subdivided ad infini-tum The dashed lines in the golden rectangle converge onthe lsquoEye of Godrsquo The other rectangles can undergo only afinite number of these subdivisions until they are entirelyoccupied by the squares



reproductive shoots The resulting primordia drawn ascircles (or other realistic shapes) are seen to maintainthis shape as one proceeds from the lsquoolderrsquo ie largerand first-drawn primordia near the edges of thegolden rectangle to the lsquoyoungerrsquo ie smaller andlater-drawn primordia closer to its centre Even theexpression lsquoEye of Godrsquo seems a rather appropriatename for the apical dome at least to this botanist Ofcourse there are several noteworthy differences (1) asubdivided golden rectangle exhibits a divergenceangle of 90deg as opposed to the larger angles observedin the generative spirals of most plants and (2) thecentral region of a subdividing golden rectangle is notrestored to its original size following each subdivisionas is the apical dome of a real shoot apex Neverthe-less a subdivided golden rectangle is realistic enoughto allow us to evaluate the packing efficiencies of two-dimensional projections of actual apices expressingdifferent contact parastichies

A subdivided golden rectangle has several otheradvantages as a model for phyllotactic patterningThis model provides an explicit definition of optimalpacking that is pertinent to actual phyllotaxis In par-ticular optimal packing can now be defined as havingtwo independent properties (1) self-regeneration ieeach subdivision of the golden rectangle results in theformation of a new square andor its inscribed formplus a smaller golden rectangle capable of anothersuch subdivision and (2) tight packing which isexpressed as no residual free space following each sub-division into the largest possible square and thesmaller golden rectangle Furthermore the model ofsubdivided rectangles offers the opportunity to deter-mine whether the optimal packing characteristic of asubdivided golden rectangle is also exhibited by othersubdivided rectangles constructed from the contactparastichies representing the most common phyllo-taxes In other words this model allows us to testwhether spirals exhibiting the fractional Fibonaccisequence have the same geometrical properties as dothe spirals arising from φ the limit of that sequence

The largest possible square drawn in the goldenrectangle depicted in Figure 6 will completely fill therectangle except for the remaining smaller goldenrectangle In Figure 6 this subdivision is repeated sixtimes which leaves an unsubdivided central regionthat retains the same proportions as the originalgolden rectangle Because each subdivision regener-ates a smaller rectangle with the same aspect ratioas the original rectangle this subdivision can berepeated ad infinitum with no residual free spacebeing left over within the original boundaries of thegolden rectangle Thus the golden rectangle meets thecriterion for tight packing given above If a more real-istic form is inscribed in the squares to represent leafprimordia then a second type of free space is located

between the boundaries of each inscribed form and itssurrounding square This free space is called inscribedfree space in order to differentiate it from any poten-tial residual free space associated with the initialdrawing of the largest squares For the sake of sim-plicity this paper uses inscribed circles to representleaf primordia In a subdivided golden rectangle theinscribed free space outside the circles but within thesquares is equal to the ratio of the areas of a circle andof a square which equals π4 or 2146 of the totalarea of the golden rectangle

Figure 6 also illustrates the results from drawingthe largest possible squares in other rectangles whoseaspect ratios (1 times 1 1 times 2 2 times 3 3 times 5 5 times 8 8 times 13)represent the most common spiral phyllotaxes Usingthe 2 times 3 rectangle as an example the first subdivisionis seen to cut off the largest possible square of 2 times 2dimensions and leave a 1 times 2 rectangle This smallerrectangle is in turn subdivided into two largest pos-sible squares of 1 times 1 dimensions with the result thatno residual free space is left within the original rect-angle Inscribing circles within the squares of this sub-divided rectangle results in 2146 inscribed freespace As this rectangle just like a subdivided goldenrectangle in Figure 6 has no residual free space itexhibits tight packing However unlike the goldenrectangle this rectangle does not exhibit the propertyof self-regeneration because a finite number of largestpossible squares consumes the entire rectangleAlthough the subdivided rectangles with other initialdimensions in Figure 6 undergo a variable number ofsuch subdivisions ranging from zero in the 1 times 1square to five in the 8 times 13 rectangle the subdivisionsof each rectangle consume the entire rectangle with noresidual free space and 2146 inscribed free space Itcan therefore be concluded that using the largest pos-sible square to subdivide any rectangle constructedfrom the fractional primary Fibonacci sequenceresults in tight packing without any capacity forself-regeneration The latter limitation means thatthese particular rectangles are unrealistic models ofactual apices and thus they will not be consideredfurther

By contrast the order of the steps in the subdivisionprocess can be reversed so that first a smaller rectan-gle of the same proportions as the original rectangle iscut off at a 90deg divergence angle and then the largestpossible square with its inscribed circle is drawn inthe remaining area This reversed order has no effecton the appearance of a subdivided golden rectangle sothat it continues to exhibit both tight packing andself-regeneration (Fig 7) However this reverse doesalter the appearances of the subdivided rectanglesthe dimensions of which are taken from the contactparastichies corresponding to the fractional primaryFibonacci sequence as is also illustrated in Figure 7

16 T J COOKE


Table 4 presents the calculations of residual andinscribed free space for these subdivided rectangles Inthe 2 times 3 rectangle every subdivision results in asmaller rectangle whose sides maintain the 2 times 3 pro-portions and thus this rectangle is capable of self-regeneration ad infinitum in a manner identical tothis process in the golden rectangle However thelargest possible square does not fill in the other part ofeach subdivision with the consequence that 1667residual free space is left within the boundaries of theoriginal rectangle Thus this rectangle does notexhibit tight packing Owing to the absence of tightpacking the inscribed free space of 1788 is less thanthe maximum of 2146 achievable in any rectangledisplaying tight packing

Figure 7 Modelling results from another process of subdividing the golden rectangle and other rectangles with aspectratios corresponding to the most common contact parastichies The subdivision process involved first cutting off the largestpossible rectangle with the same aspect ratio of the original rectangle and at a divergence angle of 90deg and then drawingthe largest possible square in the remaining space The procedure used to subdivide the 1 times 1 square is described in thetext Only the first four subdivisions are shown for each rectangle Circles representing leaf primordia (grey shading) areinscribed in the squares The space between the squares and the circles is defined as inscribed free space (unshaded areas)In each subdivision the space left over after drawing the largest possible square is defined as residual free space (girdshading) Because every subdivision regenerates a rectangle with the same aspect ratio as the original rectangle theunsubdivided centre (black shading) remaining in all rectangles can be subdivided ad infinitum The calculated values forfree space in the subdivided rectangles are presented in Table 4

Table 4 Calculated values for free space in the rectanglesdepicted in Figure 7 after being subdivided ad infinitumThe rectangle with an aspect ratio of 3455 was not illus-trated there For definitions see the legend to Figure 7

Rectangle(aspect ratio)

Residual freespace ()

Inscribed freespace ()

Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150



This same process can also be used to subdivide theother rectangles in Figure 7 so that each one exhibitsself-regeneration ad infinitum In the first step of eachsubdivision all the rectangles can be subdivided togenerate one and only one rectangle of the same pro-portions but an infinite number of possible squaresregenerating the 1 times 1 square can be drawn withinits original boundaries For illustrative purposesthe regenerating squares within the 1 times 1 square aredrawn with their dimensions being one-half thedimensions of the available space at each subdivisionThen the residual free space ranges from 6667 inthe 1 times 1 square as drawn to 250 in the 5 times 8 rect-angle (Table 4) Conversely the inscribed free space islowest in the 1 times 1 square at 715 and highest in the5 times 8 rectangle at 2092 Because the subdivisions ofthese rectangles illustrated in Figure 7 must inevita-bly produce residual free space they are not charac-terized by tight packing Other rectangles constructedfrom higher terms in the fractional sequence canapproach but do not achieve perfect tight packing forexample in the 34 times 55 rectangle (model not shown)the residual free space is equal to 005 of the totalrectangle In essence in the case of all rectangles withaspect ratios representing contact parastichies a sub-division process regenerating the original aspect ratiowill necessarily preclude tight packing It turns outthat this statement is also true for all other rectanglesexcept the golden rectangle (data not shown) Thusoptimal packing which is defined here as the simul-taneous expression of self-regeneration and tightpacking can only be achieved by those arrangementsmanifesting some type of golden geometry If the leafprimordia in spiral phyllotaxes are not positioned witha divergence angle of 1375deg then it follows from thisgraphical exercise that their arrangement is notattributable to the hypothetical operation of a globalgeometrical imperative of optimal packing

OTHER CONSIDERATIONS

Of course the above analysis assumes that contactparastichies are orthogonal to each other such that thedivergence angles can be calculated as shown inTable 3 This is true in only exceptional cases wherethe leaf primordia are initiated in superimposedorthostichies However the converse assumption thatthe primordia initiated in Fibonacci spirals arearranged in divergence angles equal to the goldenangle of 1375deg is also false Most apices with (11) or(12) phyllotaxis display divergence angles that aremuch closer to the expected values of 180deg and 120degrespectively (eg Williams 1975 30) Surprisinglythe literature contains few reliable measurementsof divergence angles in shoot apices with higherFibonacci numbers (for critical evaluation see Jean

1994 111ndash113 317ndash320) Maksymowych amp Erickson(1977) performed a meticulous study on the (23) phyl-lotaxis of vegetative apices of Xanthium pensylvani-cum Wallr They reported that the mean divergenceangles of leaf primordia on 8 apices was 1391deg with arange of 1355ndash1434deg The divergence angles withinindividual apices exhibited much greater ranges forexample the apex cited above with a low mean angleof 1355deg had individual angles ranging from 124deg to140deg Clearly these divergence angles did not corre-spond to the expected angle of 144deg However the pri-mordia were also not positioned according to theFibonacci angle of 1375deg so that they were not exhib-iting optimal packing

This interpretation that optimal packing can only beachieved by golden geometry is strongly supported byRidleyrsquos (1982b) effort to model sunflower capitulawith different divergence angles (Fig 8) The capitu-lum model constructed with the Fibonacci angle as itsdivergence angle resulted in a packed arrangementresembling prior efforts using the same constraint(Vogel 1979) However the capitula constructed withdivergence angles equal to either 13745deg or 13792degexhibited well-ordered but rather loosely packed mod-els thereby showing that even slight variation fromthe Fibonacci angle disrupted optimal packing (foranother example see Prusinkiewicz amp Lindenmayer1990 101) It is difficult if not impossible to imagineany biological system being capable of organizingitself with such discriminating accuracy as a directresponse to a hypothetical geometrical imperative foroptimal packing It seems more likely that the spiralphyllotaxes observed in the sunflower capitulum andother examples with higher Fibonacci numbers arethe outcome of some biological process the conse-quence of which is that such structures tend toapproach optimal packing

Lastly several workers have hypothesized thatplants position their leaves in response to the selectionpressure to maximize photosynthesis Spiral phyllo-taxes with Fibonacci numbers are thus proposed torepresent the optimal arrangement for minimizinghow much younger leaves might shade older leaves onthe same axis (eg Wright 1873 Leigh 1972 KingBeck amp Luumlttge 2004) These arguments are weakenedby the unrealistic assumptions that the sun is alwayslocated at its zenith (or the plants are growing per-pendicular to a fixed light direction) and that leavesare not capable of adjusting their relative positions fol-lowing their initiation as was noted by Thompson(1942) Even more decisive are the computer simula-tions of the capacity of model plants with differentphyllotactic fractions (and hence different divergenceangles) to absorb light (Niklas 1988 1998) His sim-ulations examined almost all realistic factors affectinglight reception including morphological features lat-

18 T J COOKE


itude season and time of day and they showed thatmost model plants except for rosette morphs with nar-row leaves could potentially compensate for any neg-ative effects of leaf overlap due to phyllotactic patternby altering leaf shape leaf orientation andor intern-ode length Finally these arguments about minimaloverlap suffer from the logical error of confusing theproximate cause (ie developmental mechanism gen-erating phyllotactic patterns) from the ultimate cause(ie selection pressures acting on those patterns)Among others Goodwin (1994) has emphasized thatnatural selection by itself does not generate biologicalform but rather acts to stabilize the most adaptedforms

In conclusion the considerations presented in thesetwo sections on optimal packing demonstrate that thecommon spiral phyllotaxes expressing low Fibonaccinumbers do not exhibit optimal packing whichimplies that a geometrical imperative related to opti-mal packing cannot be operating to specify primordialposition in such phyllotaxes The infrequent spiralphyllotaxes expressing higher Fibonacci numbers areseen to approach a state of optimal packing Howeverbecause it is likely that the same biological processesare specifying primordial position in all phyllotacticspirals the tighter packing observed in higherFibonacci spirals must be a secondary consequence ofthose processes

IS PHYLLOTACTIC PATTERN GENERATED AS THE CONSEQUENCE OF UNDERLYING

BIOLOGICAL PROCESSES

Of course a satisfying answer to this questionrequires the complete characterization of the biologi-

cal processes specifying phyllotactic pattern Thissubject is the focus of considerable theoretical andexperimental research which extends beyond thetopic of this paper (for reviews see Jean 1994 Lyn-don 1998) What I shall briefly discuss here is a prom-ising approach based on recent research in physicsmodelling physiology and molecular genetics

In my opinion Douady amp Couderrsquos (1992) effort tocreate a physical model of phyllotaxis represented amajor advance in this field They utilized tiny ferro-fluid drops of equal volume floating on a circular dishof silicon oil to mimic leaf primordia being displacedoff a shoot apex The dish was exposed to a magneticfield that caused the drops to act as small magneticdipoles capable of repelling each other with a forceproportional to dminus4 where d is the distance betweenany two drops The magnetic field was weakest at thecentre and strongest at the edge so that the dropsonce they were released onto the centre of the siliconoil floated toward the edge The spacing between thesuccessive drops on the silicon oil was regulated bychanging either the time interval between theirrelease or equivalently the strength of the magneticfield which affected their velocity toward the edge

The results from this physical model are absolutelystunning (Fig 9) If the drop release andor movementrates were tuned so that only two successive dropswere floating on the silicon oil at the same time thenthe second drop was repelled by the previous drop andthus they moved in opposite directions generating theequivalent of a distichous (11) phyllotaxis Smallincreases in drop rate produced steady (12) and(35) patterns as illustrated in Figure 10 Furtherincreases resulted in the most robust drop patternsexhibiting even higher Fibonacci numbers in a step-

Figure 8 Packing efficiencies of model sunflower capitula constructed with different divergence angles The models inpanels A B and C employed divergence angles equal to 13745deg the golden angle of 13751deg and 13792deg respectively Allthree capitula display well-ordered arrangements of individual units but only the capitulum constructed with the goldenangle exhibits tight packing of those units Redrawn with permission from Ridley (1982b)



wise fashion with the maximum reaching the (13 21)pattern In essence Douady amp Couder (1992) managedto create spiral phyllotaxis on the lab bench

Then they proceeded to link the fundamental pro-cess in their physical model ie the mutual repulsionof magnetized ferrofluid drops to the observations ofinhibitory interactions among young leaf primordia onthe shoot apex (Douady amp Couder 1996a b c) Com-puter simulations of their model were performedfollowing either Hofmeisterrsquos (1868) rule that theincipient primordium arises in the largest space avail-able at sequential intervals equal to the plastrochron(Douady amp Couder 1996a) or Snow amp Snowrsquos (1952)modification that the primordium arises at the first

permissible site to achieve a certain minimum space(Douady amp Couder 1996b c) In these simulations themovements of the repelling elements ie the modelprimordia were restricted to the radial direction inorder to make their behaviour resemble the displace-ment of real primordia off the shoot apex These sim-ulations were also able to display the spontaneousorganization of model primordia into well-definedspiral phyllotaxes exhibiting the Fibonacci numberscharacteristic of vegetative apices Thus their resultsare entirely consistent with earlier efforts to modelspiral phyllotaxis on the basis of the action of a singlediffusible inhibitor with a minimum threshold forpermitting primordial initiation (Thornley 1975Mitchison 1977 Veen amp Lindenmayer 1977) or theinteraction between a local autocatalytic activator ofprimordial initiation and a long-range diffusible inhib-itor (Meinhardt 1984 Meinhardt Koch amp Bernasconi1998)

Several lines of experimental evidence suggest thatthe hormone auxin is a plausible candidate for thisputative regulator of primordial positioning In gen-eral phyllotactic patterns are remarkably stable inresponse to experimental treatments neverthelessauxins and auxin regulatory compounds can pro-foundly alter vegetative phyllotaxis (for reviews seeLyndon 1998 Kuhlemeier amp Reinhardt 2001) Forinstance Snow amp Snow (1937) reported that auxinapplied to the shoot apices of Epilobium hirsutum Lcaused the origin of subsequent primordia to shifttoward the site of auxin application with the resultthat the phyllotaxis switched from the normal decus-sate to a spiral pattern The polar auxin transportinhibitor triiodobenzoic acid converted the normal(23) phyllotaxis of Chrysanthemum seedlings into adistichous pattern (Schwabe 1971) Other auxin reg-

Figure 9 Three different Fibonacci patterns of ferrofluid drops floating on a dish of silicon oil and exposed to a magneticfield The numbers show the sequence of the deposition of the drops starting with drop 0 being the first drop releasedonto the centre of the dish The drop patterns in panels A B and C correspond to (11) (12) and (35) arrangementsrespectively Reproduced with permission from Douady amp Couder (1992)

Figure 10 Hypothetical model for the role of polar auxintransport in phyllotaxis A shoot apex is depicted in longi-tudinal section through the sites of P1 (the youngest visibleprimordium) and I1 (the next incipient primordium) at anearly (panel A) and a later (panel B) stage of I1 initiationPolar auxin flux is indicated with arrows (A) P1 acts as anauxin sink to divert acropetal auxin flux thereby prevent-ing auxin accumulation above the P1 site but allowingauxin accumulation at the I1 site (B) The auxin accumula-tion at the I1 site promotes the formation of a new primor-dium which will in turn act as a new auxin sinkReproduced with permission from Reinhardt et al (2003)

20 T J COOKE


ulatory compounds were similarly able to mediate theswitch from decussate to spiral patterns on Epilobiumhirsutum apices (Meicenheimer 1981) In tomato api-ces polar auxin transport inhibitors suppressed theformation of new leaf primordia but subsequent local-ized applications of exogenous auxin induced leaf ini-tiation along the same radial line as the auxinapplications (Reinhardt Mandel amp Kuhlemeier 2000)

Finally recent molecular genetic studies have lentcompelling support to this concept that auxin is inti-mately involved in phyllotactic patterning Using Ara-bidopsis thaliana (L) Heynh as their experimentalplant Reinhardt et al (2003) confirmed that new pri-mordia arose at the sites of exogenous auxin applica-tion on the leafless apices of mutant pin1 plantsdeficient in the auxin efflux protein PIN1 PIN1 ispreferentially localized in the apical sides of the cellsin the outermost layers of the shoot apical meristem sothat Reinhardt et al (2003) deduced that auxin mustmove in these layers up toward the apical doneBecause existing primordia are apparently acting asauxin sinks highest auxin concentrations accumulateat the sites furthest from these primordia with theresult that these localized auxin accumulations canthen trigger the initiation of new primordia (Fig 10)Insofar as the gaps between existing primordia arethus determining the future sites of primordial initi-ation this model provides a satisfying explanation forreiterative features of phyllotactic patterning How-ever other plant hormones are also implicated inphyllotactic patterning For instance Giulini Wang ampJackson (2004) studied the abphyl1 mutant of Zeamays L that initiates its leaves in a decussate patternin contrast to the wild-type distichous pattern Thealtered phyllotaxis in this mutant is attributable to aloss-of-function mutation in a cytokinin-inducibleresponse regulator that affects the expansion of theshoot apical meristem An overview of recent progressin the genetic regulation of leaf initiation is presentedin Fleming (2005)

CONCLUDING REMARKS

The most important contribution of this paper is thatit establishes a rigorous criterion for evaluatingwhether or not a given phyllotaxis can be consideredto represent a Fibonacci pattern In particular thiscriterion specifies that the numbers used to character-ize the pattern are not sufficient by themselves to con-firm an underlying Fibonacci operation but rathersuch confirmation depends on whether or not the tran-sitions to different phyllotactic numbers follow a dis-cernible Fibonacci formula The evidence presentedhere documents that the whorled phyllotaxes of bothfoliage leaves and floral organs often coincidentallyexhibit Fibonacci numbers but these phyllotaxes do

not represent Fibonacci patterns By contrast spiralphyllotaxes display developmental transitions to dif-ferent numbers in the Fibonacci sequence so thatthese phyllotaxes can be classified as being genuineFibonacci patterns Nevertheless there is no compel-ling evidence to suggest that leaf primordia in spiralphyllotaxes are being positioned in accordance with aglobal geometric imperative for optimal packing

This interpretation is significant in light of our ear-lier paper on the relationship between apical geometryand phyllotactic patterns in aquatic angiosperms(Kelly amp Cooke 2003) Angiosperm lineages have re-invaded the aquatic environment around 200 times(Cook 1999) which has apparently resulted inaquatic angiosperms having the potential to express agreater range of phyllotactic patterns than their ter-restrial relatives It is significant that those aquaticangiosperms generating whorled phyllotaxes arealways characterized by unusual protuberant apicesthat initiate their leaf primordia on the lateral axisbelow the apical dome By contrast almost all aquaticplants retaining alternate ie spiral phyllotaxes oftheir terrestrial ancestors develop their leaves on theapical dome rice the sole exception to this generali-zation is more appropriately viewed as a whorl of one(Kelly amp Cooke 2003) It is conceivable that theaquatic plants exhibiting whorled phyllotaxis hadindependently and repeatedly evolved new mecha-nisms for specifying that particular phyllotactic pat-tern However it seems more reasonable to proposethat the same underlying mechanism for phyllotacticpatterning is acting in all aquatic angiosperms butthe positional constraints restricting leaf initiation tothe apical dome are no longer operating in thoseaquatic plants with protuberant apices with theresult that they can now initiate leaf primordia inwhorls arising on their lateral axes In other wordsthe same mechanism is apparently acting to generateboth whorled and spiral phyllotaxes with the selec-tion between these alternative patterns depending onthe relative position of leaf initiation It follows thatthe appearance of Fibonacci relationships in phyllo-tactic patterning should not be considered as a generalrule for angiosperms but rather as a special casesolely applying to those plants capable of initiatingleaf primordia on their apical domes

Another unifying principle governing phyllotacticpatterns of seed plant shoots is that the positions ofmost if not all organs regardless of morphologicalidentity are probably specified by one common mech-anism or several related mechanisms associated withthe initiation of leaves and other homologous lateralorgans Such mechanisms may also control thearrangements of other prominent structures such asovuleriferous scales and disc flowers because thesestructures arise in the axils of subtending bracts



which are certainly homologous to leaves Howeveralthough related mechanisms for phyllotactic pattern-ing may be operating throughout the seed plants itmust be appreciated that leaves have independentlyevolved in several other lineages including leafy liv-erworts mosses and lycophytes (Cronk 2001 Fried-man Moore amp Purugganan 2004) Thus there is no apriori reason to believe that a universal mechanismfor positioning all types of analogous leaves is operat-ing in all land plant lineages

Lastly the common mechanism underlying phyllo-tactic patterns of seed plants as is the case with otherphysical and biological patterns (eg Goodwin 1994Ball 1999 Stewart 2001) appears to involve theinteraction of mathematical rules generating processand overall geometry In particular it seems quiteplausible that the mathematical rules for phyllotaxisarise from local inhibitory interactions among existingprimordia (Hofmeister 1868 Snow amp Snow 1952Douady amp Couder 1992) These interactions areapparently mediated by the expression of specificgenes whose products regulate growth hormones(Kuhlemeier amp Reinhardt 2001 Reinhardt et al2003) operating within the physical constraintsimposed by shoot apical geometry (Kelly amp Cooke2003) This interpretation will be evaluated in thenext paper in this series

NOTE ADDED IN PROOF

In larch somatic embryos the number of cotyledonsarranged in whorled phyllotaxis is directly correlatedwith apical diameter which is consistent with otherreports on whorled phyllotaxis presented in thispaper (Harrison LG von Aderkas P 2004 Spa-tially quantitative control of the number of cotyledonsin a clonal population of somatic embryos of hybridlarch Larix x leptoeuropaea Annals of Botany 93 423ndash433) Confocal imaging of green fluorescent proteinreporter genes is now being used to visualize the rela-tionships among auxin transport dynamics localizedgene expression and morphogenetic processes occur-ring in shoot apical meristems (Heisler MG Ohno CDas P Sieber P Reddy GV Long JA MeyerowitzEM 2005 Patterns of auxin transport and geneexpression during primordium development revealedby live imaging of the Arabidopsis inflorescence mer-istem Current Biology 15 1899ndash1911)

ACKNOWLEDGEMENTS

This paper is dedicated to my friend Don Kaplan onthe occasion of his retirement from the University ofCalifornia Berkeley I thank Wanda Kelly (Universityof Maryland) Donald Kaplan (University of Califor-nia Berkeley) Leor Weinberger (Princeton Univer-

sity) and Peter Endress (University of Zurich) for theirencouragement assistance and criticism

REFERENCES

Adler I 1974 A model of contact pressure in phyllotaxis Jour-nal of Theoretical Biology 45 1ndash79

Ball P 1999 The self-made tapestry pattern formation innature Oxford Oxford University Press

Bierhorst DW 1959 Symmetry in Equisetum AmericanJournal of Botany 46 170ndash179

Britton J 2003 Fibonacci numbers in nature httpbrittondistedcamosunbccafibslidejbfibslidehtm

Burtt BL 1978 Aspects of diversification in the capitulum InHeywood VH Harborne JB Turner BL eds The biology andchemistry of the Compositae Vol 1 London Academic Press41ndash59

Church AH 1920 On the interpretation of phenomena of phyl-lotaxis London Oxford University Press

Clark SE Running MP Meyerowitz EM 1993CLAVATA1 a regulator of meristem and flower developmentin Arabidopsis Development 119 397ndash418

Clark SE Running MP Meyerowitz EM 1995 CLAVATA3is a regulator of shoot and floral meristem developmentaffecting the same processes as CLAVATA1 Development121 2057ndash2067

Coen ES Meyerowitz EM 1991 The war of the whorlsgenetic interactions controlling flower development Nature353 31ndash37

Cook CDK 1999 The number and kinds of embryo-bearingplants which became aquatic a survey Perspectives in PlantEcology Evolution and Systematics 2 79ndash102

Coxeter HSM 1953 The golden section phyllotaxis andWythoff rsquos game Scripta Mathematica 19 135ndash143

Cronk QCB 2001 Plant evolution and development in a post-genomic context Nature Reviews Genetics 2 607ndash619

Douady S Couder Y 1992 Phyllotaxis as a physical self-organized growth process Physical Review Letters 68 2098ndash2101

Douady S Couder Y 1996a Phyllotaxis as a dynamical self-organizing process Part I The spiral modes resulting fromtime-periodic iterations Journal of Theoretical Biology 178255ndash274

Douady S Couder Y 1996b Phyllotaxis as a dynamical self-organizing process Part II The spontaneous formation of aperiodicity and the co-existence of spiral and whorled pat-terns Journal of Theoretical Biology 178 275ndash294

Douady S Couder Y 1996c Phyllotaxis as a dynamical self-organizing process Part III The simulation of the transientregimes of ontogeny Journal of Theoretical Biology 178295ndash312

Dunlap RA 1997 The golden ratio and Fibonacci numbersSingapore World Scientific

Endress PK 1987 Flora phyllotaxis and floral evolutionBotanische Jahrbucher fur Systematik 108 417ndash438

Erickson RO 1973 Tubular packing of spheres in biologicalfine structures Science 181 705ndash716

22 T J COOKE


Fleming AJ 2005 Formation of primordia and phyllotaxyCurrent Opinion in Plant Biology 8 53ndash58

Friedman WE Moore RC Purugganan MD 2004 Evolu-tion of plant development American Journal of Botany 911726ndash1741

Fujita T 1938 Statistische Untersuchung uumlber die Zahl kon-jugierten Parastichen bei den schraubigen OrganstellungenBotanical Magazine 52 425ndash433

Garland TH 1987 Fascinating Fibonaccis mystery and magicin numbers Palo Alto CA Dale Seymour Publications

Giulini A Wang J Jackson D 2004 Control of phyllotaxyby the cytokinin-inducible regulator homologue ABPHYL1Nature 430 1031ndash1034

Goethe JW von 1790 Versuch die Metamorphose der Pflan-zen Zu Erklaumlren Gotha Ettinger

Goodwin B 1994 How the leopard changed its spots the evo-lution of complexity New York Charles Scribnerrsquos Sons

Green PB 1999 Expression of pattern in plants combiningmolecular and calculus-based biophysical-based paradigmsAmerican Journal of Botany 86 1059ndash1076

Griffith ME da Silva Conceiccedilatildeo A Smyth DR 1999PETAL LOSS gene regulates initiation and orientation ofsecond whorl organs in the Arabidopsis flower Development126 5635ndash5644

Hofmeister W 1868 Allgemeine Morphologie des GewachseHandbuch der Physiologischen Botanik Leipzig Engelmann

Hoggatt VE Jr 1969 Fibonacci and Lucas numbers BostonHoughton Mifflin Company

Huntley HE 1970 The divine proportion a study in mathe-matical beauty New York Dover Publications

van Iterson G 1907 Mathematische und Mikroskopisch-Anatomische Studien Uumlber Blattstellungen Nebst Betra-schtungen Uumlber Den Schalenbau der Miliolinen JenaGustav-Fischer-Verlag

Jean RV 1984 Mathematical approach to pattern and form inplant growth New York Wiley-Interscience

Jean RV 1994 Phyllotaxis a systemic study in plant morpho-genesis Cambridge Cambridge University Press

Kappraff J 2002 Beyond measure a guided tour throughnature myth and number Singapore World Scientific

Kelly WJ Cooke TJ 2003 Geometrical relationships speci-fying the phyllotactic patterns of aquatic plants AmericanJournal of Botany 90 1131ndash1143

King S Beck F Luumlttge U 2004 On the mystery of the goldenangle in phyllotaxis Plant Cell and Environment 27 685ndash695

Knott R 2004 The Fibonacci numbers and the golden sectionin nature ndash 1 httpwwwmcssurreyacukPersonalRKnottFibonaccifibnathtml

Koshy T 2001 Fibonacci and Lucas numbers with applica-tions New York John Wiley amp Sons

Kuhlemeier C Reinhardt D 2001 Auxin and phyllotaxisTrends in Plant Science 6 187ndash189

Laux T Mayer KF Berger J Jurgens G 1996 TheWUSCHEL gene is required for shoot and floral meristemintegrity in Arabidopsis Development 122 87ndash96

Leigh EG Jr 1972 The golden section and spiral leaf-arrangement Transactions of the Connecticut Academy ofArts and Sciences 44 163ndash176

Leyser HMO Furner IJ 1992 Characterization of threeshoot apical meristem mutants of Arabidopsis thalianaDevelopment 116 397ndash403

Livio M 2002 The golden ratio the story of phi the worldrsquosmost astonishing number New York Broadway Books

Lyndon RF 1998 The shoot apical meristem its growth anddevelopment Cambridge Cambridge University Press

Mackay AL 1998 Prologue by a crystallographer phyllo-taxis In Jean RV Barabeacute D eds Symmetry in plants Sin-gapore World Scientific xxxvndashxxxix

Maksymowych R Erickson RO 1977 Phyllotactic changeinduced by gibberellic acid in Xanthium shoot apices Amer-ican Journal of Botany 64 33ndash44

McCully ME Dale HM 1961 Variations in leaf number inHippuris a study of whorled phyllotaxis Canadian Journalof Botany 39 611ndash625

Meicenheimer RD 1981 Changes in Epilobium phyl-lotaxy induced by N-1-naphthylphthalamic acid and α-4-chlorophenoxyisobutryic acid American Journal of Botany68 1139ndash1154

Meinhardt H 1984 Models of pattern formation and theirapplication to plant development In Barlow PW Carr DJeds Positional controls in plant development CambridgeCambridge University Press 1ndash32

Meinhardt H Koch A-J Bernasconi G 1998 Models ofpattern formation as applied to plant development In JeanRV Barabeacute D eds Symmetry in plants Singapore WorldScientific 723ndash758

Mitchison GH 1977 Phyllotaxis and the Fibonacci seriesScience 196 270ndash275

Niklas KJ 1988 The role of phyllotactic pattern as a lsquodevel-opment constraintrsquo on the interception of light by leaf sur-faces Evolution 42 1ndash16

Niklas KJ 1998 Light harvesting lsquofitness landscapesrsquo for ver-tical shoots with different phyllotactic patterns In Jean RVBarabeacute D eds Symmetry in plants Singapore World Scien-tific 759ndash773

Palmer JH 1998 The physiological basis of pattern genera-tion in the sunflower In Jean RV Barabeacute D eds Symmetryin plants Singapore World Scientific 145ndash169

Prusinkiewicz P Lindenmayer A 1990 The algorithmicbeauty of plants New York Springer

Reinhardt D Mandel T Kuhlemeier C 2000 Auxin regu-lates the initiation and radial position of plant lateralorgans Plant Cell 12 507ndash518

Reinhardt D Pesce E-R Stieger P Mandel T Balt-ensperger K Bennett M Traas J Friml J KuhlemeierC 2003 Regulation of phyllotaxis by polar auxin transportNature 426 255ndash260

Richards FJ 1948 The geometry of phyllotaxis and its ori-gin In Danielli JF Brown R eds Growth in relation to dif-ferentiation and morphogenesis Symposia of the Society forExperimental Biology Number 2 Cambridge CambridgeUniversity Press 217ndash245

Richards FJ 1951 Phyllotaxis its quantitative expressionand relation to growth in the apex Philosophical Transac-tions of the Royal Society of London Series B 235 509ndash564

Ridley JN 1982a Computer simulation of contact pressure incapitula Journal of Theoretical Biology 95 1ndash24



Ridley JN 1982b Packing efficiency in sunflower headsMathematical Biosciences 58 129ndash139

Rivier N Occelli R Pantaloni J Lissowski A 1984Structure of Benard convection cells phyllotaxis and crys-tallography in cylindrical symmetry Journal de Physique45 49ndash63

Running MP Meyerowitz EM 1996 Mutations in thePERIANTHIA gene of Arabidopsis specifically alter floralorgan number and initiation pattern Development 1221261ndash1269

Schooling W 1914 The Φ progression In Cook TA ed Thecurves of life London Constance 441ndash447

Schwabe WW 1971 Chemical modification of phyllotaxis andits implications In Davies DD Balls M eds Control mech-anisms of growth and differentiation Symposia of the Societyfor Experimental Biology Number 25 Cambridge Cam-bridge University Press 301ndash322

Selvan AM 1998 Quasicrystalline pattern formation in fluidsubstrates and phyllotaxis In Jean RV Barabeacute D eds Sym-metry in plants Singapore World Scientific pp 795ndash809

Sloane NJA 2004 The on-line encyclopedia of integersequences httpwwwresearchattcomsimnjassequences

Snow M Snow R 1952 Minimum areas and leaf determina-tion Proceedings of the Royal Society of London B 139 545ndash566

Snow M Snow R 1937 Auxin and leaf formation New Phy-tologist 36 1ndash18

Stewart I 2001 What shape is a snowflake New York WHFreeman

TAIR 2004 The Arabidopsis Information Resource databasehttpwwwarabidopsisorg

Thompson DrsquoAW 1942 On growth and form 2nd edn Cam-bridge Cambridge University Press

Thornley JHM 1975 Phyllotaxis I a mechanistic modelAnnals of Botany 39 491ndash507

Tucker SC 1961 Phyllotaxis and vascular organization of thecarpels in Michelia fuscata American Journal of Botany 4860ndash71

Vajda S 1989 Fibonacci and Lucas numbers and the goldensection theory and applications New York Halsted Press

Veen AH Lindenmayer A 1977 Diffusion mechanismfor phyllotaxis theoretical physico-chemical and computerstudy Plant Physiology 60 127ndash139

Vogel H 1979 A better way to construct a sunflower headMathematical Biosciences 44 179ndash189

Vorobyov NN 1963 The Fibonacci numbers Boston D CHeath

Williams RF 1975 The shoot apex and leaf growth a studyin quantitative biology Cambridge Cambridge UniversityPress

Wright C 1873 On the uses and origin of the arrangements ofleaves in plants Memoirs of the American Academy of Artsand Sciences 9 379ndash415

APPENDIX

The alert reader will have noticed that the reciprocal1φ (or φminus1) is numerically related to φ as

φ minus 1 = 06180339887 = φ minus 1

It turns out that φ2 is also numerically related to φ as

φ2 = 26180339887 = φ + 1

Indeed φ displays almost mystical numerical proper-ties For instance further calculations of the powers ofφ show that they exhibit a Fibonacci relationship toeach other (Schooling 1914 Huntley 1970 Dunlap1997 Kappraff 2002) In particular Table A1 showsthat the golden geometric progression of

φ1 φ2 φ3 φ4 φ5

corresponds to the additive sequence of

1φ + 0 1φ + 1 2φ + 1 3φ + 2 5φ + 3 respectively

This relationship can be generalized as a Fibonaccirule in the form of

φnminus2 + φnminus1 = φn

The negative golden geometric progression exhibitsthe same mathematical properties except that it is anoscillating sequence with the minus sign switchingback and forth between the two terms Thus this neg-ative geometric progression of

φminus1 φminus2 φminus3 φminus4 φminus5

is equivalent to the additive sequence of

1φ minus 1 minus 1φ + 2 2φ minus 3 minus 3φ + 5 5φ minus 8 respectively

Rearranging the above Fibonacci rule indicates thehigher negative power of φnminus2 is related to the previoustwo lower powers of φn and φnminus1 by

φn minus φnminus1 = φnminus2

Table A1 Numerical relationships between the geometricprogressions of φ (the so-called golden progressions) andthe equivalent values expressed in terms of additiveFibonacci sequences These relationships are generalizedas a Fibonacci rule in the form of φnminus2 + φnminus1 = φn

Positive geometricprogression

Negative geometric progression

Powers of φEquivalentvalues Powers of φ

Equivalentvalues

φ0 1 φ0 1φ1 1φ φminus1 1φ minus 1φ2 1φ + 1 φminus2 minus1φ + 2φ3 2φ + 1 φminus3 2φ minus 3φ4 3φ + 2 φminus4 minus3φ + 5φ5 5φ + 3 φminus5 5φ minus 8φ6 8φ + 5 φminus6 minus8φ + 13φ7 13φ + 8 φminus7 13φ minus 21

24 T J COOKE


Similar mathematical relationships are observedbetween primary fractional Fibonacci sequences andtheir limits (Table A2) For example the primary frac-tional sequence starting with 21 32 and 53 has alimit of φ It turns out that the fractional sequencestarting with 31 52 and 83 has a limit of φ2 the nextone starting with 51 82 and 133 has a limit of φ3etc An analogous pattern is observed with the corre-sponding reciprocal primary fractional sequences andtheir limits calculated as negative powers of φ Theserelationships can be summarized as the limit of anygiven primary fractional sequence is equal to φaminusbwhere a and b refer to the respective positions in theoriginal primary sequence of the two numbers (xaxb)used to start the fractional sequence under study

It should be clear from this brief discussion why φand related Fibonacci sequences are entrancing toeven those of us who are virtually untrained in formalmathematics

Table A2 Some characteristics of primary fractionalFibonacci sequences with an initial term of xaxb where xa

and xb are the ath and bth terms in the primary Fibonaccisequence (1 2 3 5 8 ) The limit of each fractionalsequence is calculated as φaminusb

Sequence Initial term Limit

12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

4

T J COOKE



2006

150

3ndash24

principles or underlying mechanisms What has thepotential to tie these disparate approaches together isthe widespread recognition that phyllotaxis displayssome truly remarkable and quite seductive mathemat-ical properties Moreover judging from the popular lit-erature the mathematics of phyllotaxis has allowedthis problem to transcend specialized scientific inter-ests so that it has also captured the imagination of theeducated public

It is often asserted that geometrical patterns in bio-logical structures are likely to result from simplephysical processes such as surface tension mechani-cal stress and fluid dynamics that are intrinsic tomatter itself (eg Thompson 1942 Ball 1999 Stew-art 2001) According to this perspective biologicalpattern is seen as the unavoidable consequence ofwhat might be called geometrical imperatives thatoperate on both inanimate and animate structures Noother phenomenon in plant morphology seems a morelikely candidate for arising from the action of a geo-metrical imperative than does phyllotaxis In seedplants despite an infinite number of conceivablearrangements the leaves are arranged in two basicpatterns spiral patterns composed of one leaf pernode and whorled patterns composed of two or moreleaves per node It is repeatedly claimed that thesepatterns can be described with reference to a simpleapparently universal and incredibly intriguing num-ber sequence known as the Fibonacci sequence

As the expression says fools rush in where angelsfear to tread and thus my colleague Wanda Kelly andI are proceeding in accordance with our naiumlve beliefthat the disciplines mentioned above have alreadymade the crucial discoveries for establishing the con-ceptual framework needed to solve phyllotaxis as ascientific problem In our opinion it has been the fail-ure of each discipline to take the discoveries from theother disciplines into account that has prevented thebotanical community as a whole from recognizing thisgreat achievement We are currently making selectedobservations designed to integrate those discoveriesfrom various disciplines into a coherent framework(for our first contribution see Kelly amp Cooke 2003)The present paper attempts to perform a clear-sightedanalysis of the Fibonacci sequence and its relationshipto plant phyllotaxis in an effort to separate botanicalessence from the Pythagorean mysticism plaguingmany scientific and popular expositions In particularthe objectives of this paper are (1) to describe the fun-damental properties of number sequences (2) to inter-pret the Fibonacci number sequence with respect tothese properties in order to illustrate its inherent lim-itations for describing certain phyllotactic arrange-ments and (3) to evaluate whether the phyllotaxesexhibiting genuine Fibonacci-based patterns arisefrom the universal geometrical imperative of optimal

packing or whether they are generated as the con-sequence of the underlying biological interactionsspecifying leaf position

FUNDAMENTAL PROPERTIES OF FIBONACCI SEQUENCES

A

PRIMER

ON

NUMBER

SEQUENCES

This section provides an elementary description ofthe critical features of number sequences A numbersequence is defined as any set of numbers that arearranged in a prescribed order Of particular interestare certain sequences known as recursive sequenceswhere each term is defined as a function of the pre-ceding term(s) A class of related recursive numbersequences is fundamentally defined by the mathemat-ical

formula

or rule used to generate each sequence inthat class For instance the number sequence

1 2 4 8 16 32 64 128 256 512

is generated by doubling the preceding term to pro-duce the succeeding term Its formula can be symbol-ized as

2

x

n

minus

1

=

x

n

where

x

n

minus

1

and

x

n

represent the values of the preced-ing term (

n

minus

1) and succeeding term (

n

) respectivelyUsing the same formula it is possible to generateanother number sequence in this class as

3 6 12 24 48 96 192 384 768

It is seen from these two examples is that one featurecapable of distinguishing between two particularsequences within a class is the

initial term

(or initialterms in those classes where the formula acts on morethan one preceding term to generate the succeedingterm) Indeed the formula and the initial term(s) areentirely sufficient together to define a unique recur-sive sequence (Vorobyov 1963 Koshy 2001)

Another feature that can help to characterize anumber sequence is its

limit

which represents thenumber that is ultimately approached by a sequencecomposed of many numbers Many sequences such asthe two listed above diverge without any finite limit sothat they can be said to reach a limit of infinity Farmore interesting are those sequences that converge ona specific number such as


x

n

minus

1

=

2

x

n

Of course the limit for this sequence is zero It isworth noting for future reference that the limit of aconverging sequence may have unique properties thatare not shared by the actual numbers in thatsequence In this particular example all the numbersno matter how infinitesimally small in this sequence

1 12

14

18

116

132

164


5



2006

150

3ndash24




small set


1 2 4 7 11 16 22 29 37 46


x

n

minus

1

+

p

n

minus

1

=

x

n

where

p

n

minus

1



S

ALIENT

FEATURES

OF

PRIMARY

F

IBONACCI

NUMBER

SEQUENCES


x

n

minus

2

+

x

n

minus

1

=

x

n


1 2 3 5 8 13 21 34 55 89 144 233 377


1 3 4 7 11 18 29 47 76 123 199 322 521 1 4 5 9 14 23 37 60 97 157 254 411 665

1 5 6 11 17 28 45 73 118 191 309 500 809 etc


1 1 1 1 1 1 112

34

78

1516

3132

6364





π


φ


φ

It turns out that

φ



φ


xx

xx

xx

n

n

n

n

n

n

-

-

-

- -+ =2

3

1

2 1

21

32

53

85

138

2113

3421

5534

8955

12

23

35

58

813

1321

2134

3455

5589

f = =ABAC

ACCB

Figure 1



13 21216

1 8 9 17

1 6 11 1751 4 9 145

1 6 13 207

1 3 4 11 187

11 17 197

1 9 1910

2 13 17 195 10

1 87 15

4 9 12 14 16 18 2010 15

6 8 9 12 14 16 18 2015 21





6

T J COOKE



2006

150

3ndash24


φ



φ


φ



ad infinitum


φ


f = =ABBD

BDBC



where

θ

l

and

θ

s


and

θ

s


φ

minus

2


U

NDER

-

APPRECIATED

MATHEMATICAL

CONSTRAINTS

ON

THE

APPLICATION

OF

F

IBONACCI

SEQUENCES

TO

BIOLOGICAL

PHENOMENA


hypothetical



fq

qq

= =360

1

1infins

qfl = =360

222 492infin infin

q qfs

l= = 137 507 infin

q fs = ( ) =360 137 507infin infin-2






1 4 5 9 14 2 4 6 10 16






1 1 1 3 112

12

12 n n n n-( ) +( ) +( )

2 2 2 3 212

12

12 n n n n-( ) +( ) +( )






Query sequence

Total matches


1235 100 9 (9)12358 100 37 (37)1235813 79 41 (52)123581321 40 26 (65)12358132134 26 22 (85)

8 T J COOKE



and the reciprocal




874

9187

17891

269178

447269

716447

1163716

18791163

487

8791

91178

178269

269447

447716

7161163

11631879








PHYLLOTACTIC WHORLS











Floral organ number



10 T J COOKE




















12 T J COOKE










14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5


5



2006

150

3ndash24




small set


1 2 4 7 11 16 22 29 37 46


x

n

minus

1

+

p

n

minus

1

=

x

n

where

p

n

minus

1



S

ALIENT

FEATURES

OF

PRIMARY

F

IBONACCI

NUMBER

SEQUENCES


x

n

minus

2

+

x

n

minus

1

=

x

n


1 2 3 5 8 13 21 34 55 89 144 233 377


1 3 4 7 11 18 29 47 76 123 199 322 521 1 4 5 9 14 23 37 60 97 157 254 411 665

1 5 6 11 17 28 45 73 118 191 309 500 809 etc


1 1 1 1 1 1 112

34

78

1516

3132

6364





π


φ


φ

It turns out that

φ



φ


xx

xx

xx

n

n

n

n

n

n

-

-

-

- -+ =2

3

1

2 1

21

32

53

85

138

2113

3421

5534

8955

12

23

35

58

813

1321

2134

3455

5589

f = =ABAC

ACCB

Figure 1



13 21216

1 8 9 17

1 6 11 1751 4 9 145

1 6 13 207

1 3 4 11 187

11 17 197

1 9 1910

2 13 17 195 10

1 87 15

4 9 12 14 16 18 2010 15

6 8 9 12 14 16 18 2015 21





6

T J COOKE



2006

150

3ndash24


φ



φ


φ



ad infinitum


φ


f = =ABBD

BDBC



where

θ

l

and

θ

s


and

θ

s


φ

minus

2


U

NDER

-

APPRECIATED

MATHEMATICAL

CONSTRAINTS

ON

THE

APPLICATION

OF

F

IBONACCI

SEQUENCES

TO

BIOLOGICAL

PHENOMENA


hypothetical



fq

qq

= =360

1

1infins

qfl = =360

222 492infin infin

q qfs

l= = 137 507 infin

q fs = ( ) =360 137 507infin infin-2






1 4 5 9 14 2 4 6 10 16






1 1 1 3 112

12

12 n n n n-( ) +( ) +( )

2 2 2 3 212

12

12 n n n n-( ) +( ) +( )






Query sequence

Total matches


1235 100 9 (9)12358 100 37 (37)1235813 79 41 (52)123581321 40 26 (65)12358132134 26 22 (85)

8 T J COOKE



and the reciprocal




874

9187

17891

269178

447269

716447

1163716

18791163

487

8791

91178

178269

269447

447716

7161163

11631879








PHYLLOTACTIC WHORLS











Floral organ number



10 T J COOKE




















12 T J COOKE










14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

6

T J COOKE



2006

150

3ndash24


φ



φ


φ



ad infinitum


φ


f = =ABBD

BDBC



where

θ

l

and

θ

s


and

θ

s


φ

minus

2


U

NDER

-

APPRECIATED

MATHEMATICAL

CONSTRAINTS

ON

THE

APPLICATION

OF

F

IBONACCI

SEQUENCES

TO

BIOLOGICAL

PHENOMENA


hypothetical



fq

qq

= =360

1

1infins

qfl = =360

222 492infin infin

q qfs

l= = 137 507 infin

q fs = ( ) =360 137 507infin infin-2






1 4 5 9 14 2 4 6 10 16






1 1 1 3 112

12

12 n n n n-( ) +( ) +( )

2 2 2 3 212

12

12 n n n n-( ) +( ) +( )






Query sequence

Total matches


1235 100 9 (9)12358 100 37 (37)1235813 79 41 (52)123581321 40 26 (65)12358132134 26 22 (85)

8 T J COOKE



and the reciprocal




874

9187

17891

269178

447269

716447

1163716

18791163

487

8791

91178

178269

269447

447716

7161163

11631879








PHYLLOTACTIC WHORLS











Floral organ number



10 T J COOKE




















12 T J COOKE










14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5





1 4 5 9 14 2 4 6 10 16






1 1 1 3 112

12

12 n n n n-( ) +( ) +( )

2 2 2 3 212

12

12 n n n n-( ) +( ) +( )






Query sequence

Total matches


1235 100 9 (9)12358 100 37 (37)1235813 79 41 (52)123581321 40 26 (65)12358132134 26 22 (85)

8 T J COOKE



and the reciprocal




874

9187

17891

269178

447269

716447

1163716

18791163

487

8791

91178

178269

269447

447716

7161163

11631879








PHYLLOTACTIC WHORLS











Floral organ number



10 T J COOKE




















12 T J COOKE










14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

8 T J COOKE



and the reciprocal




874

9187

17891

269178

447269

716447

1163716

18791163

487

8791

91178

178269

269447

447716

7161163

11631879








PHYLLOTACTIC WHORLS











Floral organ number



10 T J COOKE




















12 T J COOKE










14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5










Floral organ number



10 T J COOKE




















12 T J COOKE










14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

10 T J COOKE




















12 T J COOKE










14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5












12 T J COOKE










14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

12 T J COOKE










14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5




14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

14 T J COOKE



A SIMPLE MODEL












16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5









16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

16 T J COOKE








Total freespace ()

1φ (golden) 000 2146 214611 6667 715 738212 3333 1431 476423 1667 1788 345535 625 2012 263758 250 2092 23423455 005 2145 2150









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5









18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

18 T J COOKE



















20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5









20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

20 T J COOKE




CONCLUDING REMARKS









NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5





NOTE ADDED IN PROOF


ACKNOWLEDGEMENTS



REFERENCES




















22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

22 T J COOKE


































































APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5























APPENDIX


φ minus 1 = 06180339887 = φ minus 1


φ2 = 26180339887 = φ + 1


φ1 φ2 φ3 φ4 φ5















Equivalentvalues


24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

24 T J COOKE







12 23 35 x1x2 φminus1

13 25 38 x1x3 φminus2

15 28 313 x1x4 φminus3

18 213 321 x1x5 φminus4

113 221 334 x1x6 φminus5

21 32 53 x2x1 φ31 52 83 x3x1 φ2

51 82 133 x4x1 φ3

81 132 213 x5x1 φ4

131 212 343 x6x1 φ5

Date post:	27-Jun-2020
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