Phyllotaxis or the properties of spiral lattices. III. An algebraic model of morphogenesis M. Kunz, F. Rothen

To cite this version:

M. Kunz, F. Rothen. Phyllotaxis or the properties of spiral lattices. III. An algebraic model of mor- phogenesis. Journal de Physique I, EDP Sciences, 1992, 2 (11), pp.2131-2172. ￿10.1051/jp1:1992273￿. ￿jpa-00246692￿

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213 1992,

(1992) 2131-2172 Phys. France 2 I J. NOVEMBER PAGE

Classification

Physics Abstracts

61.00 87,10

spiral algebraic

Phyllotaxis properties of lattices. III. An the or

(*) morphogenesis model of

M. Kunz and F. Rothen

CH-l0l5Lausanne-Dorigny, Exp6rimentale, Physique Lausanne, Universit£ Institut de de

Switzerland

July 1992, 1991, accepted 24July (Received 1992) May 23 revised 22

phyllotactique

d'une Rdsumd. Une les celles constituent dcailles

structure que pomme comme

pin

prend apical qui naissance dtroit de

croissance. local On de

entoure centre sur un anneau un

simple

morphogenbse £tudie modble d'une £tablissant la de telle formation des relations

un en

algdbriques

premiers chaque permutations correspondant

h naturels, des nombres nombre entre n

d'apparition

l'ordre 6caille d'une d'une feuille. Ce modble conduit h la d6finition naturelle ou

divergence divergence. prendre s'ordonnent d'une valeurs Les

peut

cette structure que en une

hidrarchique

parentd qui

6vidence des 6vidente l'arbre le r61e nombres nobles. Il de

met y en a avec

Farey long spirales alignds obtient classant de l'on

les r6seaux cercles le de tangents que en

6quiangulaires.

phyllotactical

originates

such of Abstract. the scales of fir-cone A the pattem

arrangement as a

ring along surrounding thin study simple growth.

local of of model the We

center a a a

considering morphogenesis algebraic

by botanical relations between of such

structure permu- a

given

tations first corresponding

the birth of the natural each numbers, number order of scale

to a n

divergence angIe possible leaf. model allows define in This The values of the natural

to way. a or a

according divergence

places which

be classified hierarchical the noble numbers

to structure can a

prominent position. study of

Farey-tree in

in the This construction is similar the obtained to a

equiangular aligned along spirals. lattices of circles tangent

Morphogenesis phyllotads. 1. and

PHYSIOLOGISTS.-Phyllotaxis study spiral 1-1GEOMETERS the of

helicoidal is AND or

frequently

objects spiral plants. instance the Its for

pattems encounters among one are

daisy florets sunflower,

of the helicoidal distribution of in the scales of

arrangement

a or a or a

pineapple. pine

Fibonacci The has of the in these been

pattems sequence emergence cone or a a

(I).

long Among speculation people

of for time this

those who interested in

matter very were a

(*) Panially supported by Swiss the (FNIRS). National Foundation

(I)

history phyuotaxis short of of A be found

the [4]. in reference

general For account may a

phyllotaxis, [5-10]. references [I] introduction and

to e-g- see

PHYSIQUE JOURNAL N° DE I I I 2132

physicists,

mathematicians, problem, crystallographers, notices naturalists,

one numerous

truly,

who, Goethe biologists considered himself scientist. and such

poets even as as a

Facing biology, people essentially intrusion of arithmetic into this it

see mere some as a

following. They

will Other of be called in the geometry. geometers consequence space

geometrical phyllotactic plant scientists look for evidence of the in metabolism the of structure

They

growth plants.

hope they during find hormones secreted there the of how

pattems to

are

physiologists. morphogenesis. They deny during do But will called the role of be geometry not

during

emphasize inhibitors of they influence activators this and

the want to stress process, or

taking development chemical of individual contribution of in the the substance the part every

plant.

phyllotaxis point

scientists different of

Recently, third looked from still of category

at a a

physical physicists they consider which might because One call them systems view. pure

point being

far-off, self-organization starting [1-3]. similar

botanical Our exhibit

to very ones a

longer

this their work in discuss

shall

paper. we no

opposition genetics

attribute view, point between of the those who From

to common a our

morphogenesis, for that essential mechanisms determining in those who think and the role

independent

in

the of selection of be discussed needs

to nature, pattem not case are an

phyllotaxis.

splitting

by [I I] 1935 Snow that the by observed indeed in Snow and in

It parts two apex was

having

plant naturally phyllotaxis), often obtains Epilobium (a of hirsutium decussate two one

phyllotaxis. requires clearly spiral approach experiment apices developing theoretical this A to

independent genetic growth which information. of is of the model the a

1.2 APICAL emphasized geometrical Geometers have the RING AND MORPHOGENESIS.-

properties phyllotactic

highly

symmetric of the thought these

pattem structures to emerge : are

during growth particular point

without call for adopted (2). mechanism view

This of in any was

by

[lo]

Koch and of found; where references be

two

papers many one us can as a

morphogenesis

problem

of phyllotactic only the side

of issue in

structures consequence, was a

[lo]. However, references references [12-14] in

idea which take Marzec of

and up an

[15], Kappraff morphogenesis considered is point the physiologists

of from the of view the :

moulding primordia

growth apical of (as the and the called) shoots the the

young new on are

ring modelled. is

fact,

producing differentiation As of cellular place

shoot

new-bom does take

matter not at a a

ring-shaped tip (apex),

surrounding

within of the the stalk but

region [7]. the the

stem or apex a

apical

ring This remains linked the during growth.

For the observer the

to apex apex, on new an

(Fig.

radially I). shoots to appear go away

interval Let that time growth successive and the of the between

shoots two rate suppose us

phyllotaxis problem

settled down stable value. We follows then the of

have

to state may a as :

apical ring completely explain why

successive (the the shoots in ordered the

to appear way a on

becoming angle

approximately

after between successive shoots transient short constant, two a

period) why, generically, angular golden ~137.5°...).

and this angle distance is the

angular divided explaining by divergence. is This 2

the In called

distance aim

at

we ar « »

giving growth

apical ring, dynamical is, deterministic model of

define the the that

to on a a

physical interpretation generically which chemical

does has reasonable

and

system

some or « »

reproduce phyllotactic observed. the pattem

phyllotaxis

is of mechanism that be We here that the modelled in result the

assume a can

plant. As introduction,

in discussed basic in

the in

every same manner was assume we

particular independent genetics. of it is that

exist, special Or,

interest if mechanism does is (2) such it hidden and of

geometers. to no a

2133

MORPHOGENESIS

MODEL OF ALGEBRAIC PHYLLOTAXIS N° II AN

Apex

~~~~~~~~

°

Q Q

growth

~ QO

O

Q Q

growing

surrounding of ring-shaped region the

bom inside Fig.

shoots I. The stem.

apex a are a new

ring

radially where

from the

shoots moving the newbom

with the For observer

to go away appear apex, an

appeared. they first

ofmorphogenesis. dynamical

chemical the model

1.2. A

circle In order I

system to on as a

apical [12-15] illustrate growth, existence the of of references authors advocate the the process

interpreted acting

growth inhibitory of hydra of factor substance in the

a as an as case

development [16]. following. Such

called substance will be inhibitor in the The inhibitor

a » «

specify Turing

morphogen by [17] is of

word coined chemical

substance type

to

a a one », «

playing

growth morphogenetic during the role

process. some

hydra, high neighbourhood

inhibitor concentration is In the of the in the the of

case a

sprouting

long value,

and this concentration exceeds head threshold wherever

as, as some no

develop. morphogen is head elapsing, by

is diffusion able As time this fades

to away new or

description through

has also Thomley by [16]. chemical used been reaction This

model the to a

growth phyllotactic

[18]. of pattem a

profile

apical ring, already placed inhibitor of Consider the leaves the the result

; on a new

leaf its minimum. Because of this concentration of inhibitor will the

sprout, at appear new

profile locally (through reaction) increases. fades diffusion chemical Then the whole

away or

until the leaf next appears.

Kapraff

[15] [14] Sadik Koch, Bemasconi Guerreiro, and have studied Marzec and and a

dynamical Douady [3]

proposed which illustrates and Couderc idea this have

system

a

physical experiment dynamical analogous

exhibits behaviour. shall this that We refer to an

dynamic general of construction in the model. scheme our

algorithm The growth permutation model and the describe 1.2.2 A. We the to want process

defining Thomley's simple algorithm apical ring, following by deterministic the

a on

[18]. scheme

analytical study dynamics [12-15] proposed of references The of in the the is systems

simplify

why extremely difficult this is model further. the shall

we

(o,

integers permutation P~(0)

0, I, P~ of Consider the and I such that

a n n

=

,

write

:

~~'

(i) Pn

(~

~j~

"

n-I I

defining P~ dynamical

of time We think of time the discrete

system

state at

our as can « »

imagine

points integer

distribution point of circle and associate each

to C~ an n : a a so n on

given successively that,

integers 0, when the circle is orientation, the

meets an one

C~_ Ci, C~, Fig. (see 2). around the circle

i ,

PHYSIQUE JOURNAL 2134 DE I N° 11

Co

Ci

C~_1 ~

~2

Fig. 0, Co integers 2. distribution

Cj, C~, C~_

The of circle. around the

=

i ,

apical

ring geometrical simple representation

of of the picture of P is the

this

Now state at

a

~

point birth of

by indicates where I-th

is bom time labelled I the the

sprout

moreover,

n :

provides

for

just P~

with

occurred time thus the n-th has Permutation

sprout

state at one n. us

reproducing growth evolution sketched in give law, the

the Let

system. process an us now

1.2. I.

permutation n=2; given

by time trivial its is the Consider the system at state

arbitrarily which (Fig. the P~ decide 3). At the is broken have

symmetry

step, to next

on we

point (Fig. 4). place 2 point of 0 side

we

o o

2

Fig. 4. Fig. 3.

Fig. disregards 3. If angles, the of value possible only the ordering

there is points along of

one two one

the cwcle.

Fig.

4. broken The is (here point point 2)

thkd symmetry

is located the number

circle. as a soon as on

following points placed

the be principle

will according algorithm All that takes the to an

simply

looking for possible

1.2. each in I into

absolute the stated

account step, at as as : we are

of

inhibitor is, the

concentration, place point possible minimum that far from

try

to as we as n

the latest ones.

placed (Fig. 5).

Point therefore 3 between 0 Suppose placed is that 3 has been and I now

(because is point). close 0 0 the oldest place should then 4 between We

the oldest

to

very « »

point (namely neighbour I) its again placed (namely 2). after 0 and older that We is 4 argue

neighbour, (Fig. 6).

close between and its I, that 5 has be namely 2 older 0

put

to

to

very so » «

2, given neighbour replacing repeated (5),

rule be 0 has been that the for thus

new so can a

following points. the

algorithm

by neighbour denote P~. Suppose the older Let of

in that I

A)~~ up : us sum our

placed

P~

time 0 in Then time I, has been

between and I I

next to at

at set + + n. n n n

+~~

~~

k,

k time between k

+ and

A)~

put A)~ at + + ; n n

2135 MORPHOGENESIS PHYLLOTAXIS ALGEBRAIC MODEL OF AN

N° 11

o o

3

5

3

2

~

4

Fig. 5. Fig. 6.

Algorithm point Fig. points demands A 3 be 5. 0 and located between I. to

Fig. according algorithm points 6. Distribution six A. of to

A)~+~~#0

n+j

between Vk

that j such exists set Now there ;

+J~

and

and +

I begin

again j Put between 0, I

and A)~+J j

and

on. so ~~,

+ + n A)~ :

=

),

obtain time15 permutation (~

algorithm P~ Applying

this

at

to

we =

2 3 4 9 5 8 lo 12 13 14 6 7 11

p (0

~~

8 3 614 9 412 13 0 11 7 lo 5 2

just algorithm the evolution Let A the have We don't claim that law call constructed.

us we

following perfectly dynamical

by Thomley's defined mimics the behaviour of A real system a

algorithm forgets

scheme. Our it into the older does take the

sprouts account exact never nor

apical

gives insight ring. Nevertheless, good

into location their it of birth the what

may on a

beginning Stationary happen growth. regimes growth should described the of the of also be at

perrnutations by point. this shall later

retum to ; we

dynamical Anyway algorithm

exhibits A the from main features would two expect

we a

phyllotaxis

modelling

circle the

system on :

organize

angular

produce shoots distance the themselves between

constant to any as so a

immediately

and older shoot the

one

angular divergence. by this distance divided 2 golden is the

ar

of mathematically these The aim this is assertions and them. A to to paper express prove

description general of the construction properties permutations displaying of first and the

given. Algorithm

bijugate phyllotactic feature also be will produce trijugate A also

can or

giving proof, this without We will discuss last for mathematical the sake property pattems. any

brevity. of

Angular permutations, 2.

2,I DEFINITIONS.

of divergences.

phyllotac-

2. I. The main The characterizes idealized I that parameter set an

divergence,

[0,

is its is, divergence angular (recall number tic that

in that the

is pattem

an a

by ar). divided 2 distance

PHYSIQUE N° I1 JOURNAL I DE 2136

divergences throughout belonging will consider without

We the whole loss to paper, (0,

generality they divergences by in l related of inversion of orientation

the

to a mere : are

,

circle. of the

phyllotaxis generically

dynamical unique modelling should within select value A system a

divergences

possible divergence, golden of the Fibonacci

this the is is number that

set or : ;

T~~

0.381966 9~

= =

help with It golden the defined of is the

mean :

/

+

~

2

which being has the of well irrational known the

in words, property other is worst it

the

; « »

slowly approximated by irrational the of

rational [19, fractions

20]. most

sequences « »

Any real

between number represented

and o unique

in I be continued

can way a a as a

fraction

[al,

(2)

~3,

~2,

"

" ~

j

~~ ~

'

a~

+ a3

representation integer. positive (2) unique, each is where The is

if is

rational. In

except

a; a a a

the latter ambiguity

continued fraction is and the the requires

if finite is removed that case,

one

development last of

(2) exceeds the As belonging the

I.

term

consequence, to every a a

development

has the form of jo, a

I_

irrational

[at

a

=

(3)

la l integers

at, a~, a~ ~

rational

[a~ a~]

; ; a a =

simple development particularly 9~ has

: a

ii

[2, [2, 1, 1, 1, 9~ (4)

= =

periodic period

represented by its On is hand, where overlined. Lucas the other the

sequence a

divergence (3) divergence is anomalous given

9~, observed which the main by in is

nature, :

[3, 1]. (5) 9~

= =

3 r~ +

infinite development numbers, consists of of

after Those irrational whose

sequence an ones

simplest (~). 9~

notices that the 9~ noble numbers One noble

and

stage,

are some are « »

palled

~'

j'

~~~~~~~

~~

2

angles expressed belonging

Biologists prefer divergence degrees interval in and the

(3) to to use

speak angle, equal golden angle,

According they ]0°,180°]. this of Fibonacci rule, the

to to or

by spirit, ' divergence angle replace they Lucas 37°, the the Lucas

360° ( ) 507 the I In l

same

r

=

'

(3 99°, equal 360°. 501

r~

to ~)~ +

=

~

~~

integers

where d such

b, and that

Any be written

(4) noble number

c

a,

can are v as v =

+ cr

[ad -bc[ =1.

ALGEBRAIC

N° MODEL MORPHOGENESIS PHYLLOTAXIS AN 2137 11 OF :

(relation (I)) Definition. P~ be written 2.1.2 Each

as can

°

~

(6) P~

= Co Cn-i Cl

(o, )

N~_i P~ permutation of I, is subjected

I

to

a

n

= ,

(7) co=o.

through C~ define convenience, For

(8) C~=Co=0.

Co, C~_i integers circle, P~ Ci,.., above, discussed of

As

represents sequence a a on

predefined (Fig. 7). integer

according disregard

orientation location of each If the

to exact a we

permutations

circle, correspondence have of and these between the the

set one-to-one on we a

geometrical pictures.

lo, that

such angular I permutation P~ exists if there

We shall call e an

(C~_i (Co (Ci

(9)

0 <1

@) @) @)

< < < =

(x)

divergence fractional of (where x). denotes We shall that the is the number

part say a

(P~) compatible P~. P~ divergences compatible with write div of with It is and the to set easy

compatible

permutation figure o.35 with the check that is shown in 7.

=

picture

geometrical points P~ by angular permutation placing We of the

construct a an can

(C )

Co, C, angular

(P circle distance 2

from where div

the In this

~).

at

e case, on an ar

;

(Fig. 8).

P~ T~(@) realization of have

a we

ci o co=o

Ci=3

_~ ~~ ~

~

C~=I

C~

C4

C~

C4=2

_~ ~

3-

Fig. Fig. 8. 7.

permutation Fig. representation 7. Circular of

5) (0 3

2 4

p

~ Ci C~ Cs C~ C~ C4

) (0.35 (0.35 Co)

divergence Cl P~ compatible 0.35

number is with because 0 The o

a

~ ~

=

=

(0.35

C4) (0.35 (0.35 Cs). (0.35 P~ (9)). 2) 3) (See is relation C As

C

consequence, a

~ ~ ~

angular.

(0.35) angular of Fig. figure

8. Realization the angular permutation of 7. The P distance between 0 T~

~

(C, o).

equals

C, points C,

order 2 The relative figure and the is the of in 7.

same as ar.

2138

JOURNAL PHYSIQUE DE I N° I1

equivalent

Notice that permutation

is it

circle the represent

to

segment

a on or on a

[o, particular, definition the extension of In

I

the of

the is immediate. The latter

[. to T~ case

merely

circle phyllotaxis. allows closer connection

to a

study

equivalent 2.2 THE angular The perrnutations is of cuT-AND-PROJECTION METHOD. to

(9),

following

problem: given lo, number the I[, how @e numbers the

a are

(2 (no 9), ) [o,

I] ordered ? the

segment

on

,

simplified geometrical quasicrystals cut-and- analysis by The of has the well-known been

(@), (2 (n Similarly, ordering

@), the projection problem [21-25]. of

method the

@)

on

...,

simplified by

by [o, lift

the greatly

in dimensions and circle the is I

segment

two on a or

)-lattice) following periodic

section) (W~~(@ (in of two-dimensional lattice introduction the a

fk, strip

this defined 2.2.4. lattice, in by choice subset of of the be section and the to a

W~~(@)-lattice. W~~(@)-lattice by We the define 2.2.I The

n

~

lo) integers

(9

(

)

+

W~~~

n, m m

=

0

of

lo, divergence

is called simplify,

the lattice. where I the In order

to e we assume

points

irrational distinct of lattice the

be the have

two to consequence as a ; never same x-

(5). point coordinate will be lattice written A of the

-m(~)

W~~=n(~) (ll)

sign integers pair (k, I) convenience). (notice minus for For of the chosen have

any we

~'~n'm' ~'~nm ~kn

(12)

~

In', ~ lm' km

+ +

W(~ W(~ that, W~~ 0, interest of construction if o and The this is is such that l then,

w

~ ~

l~~l

(13) W~~

=

unambiguously point W~. Such written W~~ be

a can =

W~~(9)-lattice. envelopes of (13) directly Relation the 2.2.2 Convex the connects

W~~(9)-lattice T~(9) permutation angular

of P~. future with the realization For

an conve-

envelopes simplifies of study nience, the construction which

the recall the of

convex we

[26]. T~(9 of

structure

periodic plane P~ lattice P_

broken lines which the In and

construct two any can one are

half-plane

envelopes by lattice restricted of the of defined the the

parts to two upper convex

Envelopes neighbourhood (Fig. P~

) (x ) 9). o o describe the o, o,

P_ (x and and

y y

~ ~ ~ ~

origin) periodicity (the therefore, lattice, point because of the of the and, of o lattice of any

important proposition envelopes well known but point. In order of the

to state very convex a on

W~~(9)-lattice, properties

section main of

in recall the the the of

convergent next an we

lo,

[at,

1[. irrational number b

e

a~, a~,.

= ,

(~)

We shall y-coordinate

the (5) call x-coordinate and rather of

than components two vector a

b

abscissa ordinate. and

2139 ALGEBRAIC

MORPHOGENESIS PHYLLOTAXIS MODEL OF N° AN 11 :

y

~-

p

m

+ m m

m

m m

m m .

m m

m m

m .

m m

m m m

m

.

m m

m m

.

m

.

m m

m

.

m m

X

o

m

, .

m m m

m

.

m .

m m

m m

m

W~~(o

).

envelopes Fig. plane point The P_ 9. and lattice the of P~

two upper convex on a

representation of fraction of fraction. the continued Convergents continued From

2.2.3

an a

fractions infinite rational [at, of ], defines

irrational 9

sequence

an a~, a~, one

= ,

~~

~~

a~]

[at, [a,

= =

= =

l

~2 ~i al

~

~

a~

~~

a~] [at,

a~,

=

~3

(14)

~~

a~] [a,,

a~,

=

, qn

~~

by defined truncating

development the

of after n-th the 9 The fractions known

term. are as

qn

principal the the of irrational satisfy They number 9. the convergents relations

with

°

l

PO Pi

" ~

l (16)

qo qi aj

~ ~

properties Further found be 20]. [19, in references particular In can

~~

~'~

~~ ~~

~ ~ ~

9

(s

)

(17)

<

~

r

~ ~

~2 ~2

~r ~s

I

+ n n

(n, o).

s r, ~

PHYSIQUE N° 11 I JOURNAL DE 2140

following intermediate of The defined

in the for If l 9

convergents

a~ way.

are some ~ ~~

o,

km the rational fractions constructs one

Pk~~Pk+I

Pk+~k+2Pk+1 Pk~Pk+I Pk+2 Pk

~~~~

~ ~ ~ ~ ~

~k ~k ~k ~k + + + ~k ~n ~k ~k ~k

+1 +1 2 2

I + + +

~~ ~~

~

~

(17), Because of parity larger according both either

and smaller than the 9

to are or

~k ~k

2 +

~ ~~ ~~ ~~~

'

~ ~

k.

fractions of The intermediate for o called of

convergents

a~

are

r

<

~

=

~,

, ~

+ ~k, ~k ~~k

I

+ r

principal

specified, Together they in which has the

with the be

9. convergents, to way are, a

approximations geometrical

[19,

20]. irrational This of the last has best rational 9 property a

expressed

through

which be counterpart, can

Proposition I :

W~~(9)-lattice, P_(P~) envelope through Wq~,~p~,~. In points

the Here

goes a

(even). 0 odd and is

a~ w w r n ~~

Proposition Appendix (18), that, is demonstrated in A. Notice because of the I of segment

Wq~~~ envelope joining straight Wq~p~ line, the with is points Wq~,~p~,~ with

p~~~ a

a~~~) dividing (o equal into this vertices

As the of the

segment parts.

consequence, r a

~ ~

principal envelope correspond Wq~ convergents to p~.

neighbouring strips points. 2.2.4 and W~~(9) need lattice of Lattice We subset

now a

9) (C,

yields

projection points x-axis

the whose the the of set segment onto on

[o, following (Fig. through lo). of horizontal This is done

best the axis. definition I the

y

m

,

m

"

m

,

m

"

"

m

.

m .

m m

b

m m

, L

m m

m m

m m

m .

m .

m m

m

.

m

.

m

Fig. strip

defines 10. extended projection point ). subset the lattice

The lattice of W~~ b~ The of the o a

(no)

points of

lo,1[ the x-axis allows the fractional the

order b~ pans segment onto to on (0~nwL).

ALGEBRAIC OF MORPHOGENESIS PHYLLOTAXIS AN MODEL 2141 N° I I

i~

extended strip lattice We call the set

i~=W~~(o)nRx

(W~~;neN~,meZ)

(19) [0,L]

=

N. where L e

strip lattice In the call the

b~

set way, same we

W~) (Wo, [0, W~~(o ) 1[ Wi, b~ (20) [o, L] n

x

= = ,

(13), projection directly Because of b~ of yields x-axis the the the realization onto

) of ( corresponding angular the 9 permutation

T~ P

i

~ i.

~ ~

Clearly, knowledge neighbourhood

the

of the W, of allows each build b~

to e one

T~~i(9).

periodicity W~~(9), Because of P_ of lattice the the the of and structure

P~ neighbourhood. determines this corresponding analysis, principle, The simple although in

cumbersome. is somewhat important

leads However, it proposition in

stated the to next an

(Prop. 2)

section angular which permutation allows

characterized. be to an

perrnutation

Given

2.3CHARACTERIzATION ANGULAR PERMUTATIONS.- any oF

~

), angular following proposition brings together (° ? P~

is it The result

a

=

Cn Co Cl

i

[27], conjecture reciprocal. by

Steinhaus its Swierczkowski followed and who of a

Proposition

2 :

differences if only and P~ angular if the permutation is A

,n-I) C;~,-C; (I =0,1,.

values only take two

positive and integers)

(a, fl fl a

possibly third value

and a

Y~"-fl

different Jkom only if 0 I.

which is

or n can occur

be found in proposition 2

demonstration of

for the material All the can necessary

latter permutations

and P In angular the Figures P

Appendix and I16 show B. I la

two case,

~. ~

2

in 5 also namely differences, fl

while only and 3

there

appears

two

y

a occur

= = =

angular. perrnutation Ps

which is hand, Figure

lc, the other shows former not I the

a on case.

perrnutation,

angular the P~ proposition If is obvious

Moreover, 2

has consequence. an an

C~_; and for all between I I replacing C; by by P( perrnutation obtained

new

orientation the circle the

has the choice of angular.

that

is still This -1 no on means q

permutation. corresponding

angularity of the the influence on

classijj angular permutations apply the of ? Our is How 2.3.I concept to to purpose

emphasize

permutation phyllotaxis.

will below,

it shall As be angular

to necessary to we see

classify angular

Moreover, geometrical various

shall need the this the of to concept. aspect we

phyllotactic

morphogenesis

permutations

toward the of To this road

structure.

a as access an

proposition useful classification. for

which such aim, will be

state a new a we

PHYSIQUE JOURNAL N° DE I I I 2142

o o o

~

~

Vi

~

aft

5 5 5

-~

~

q.-5

ja->-.2

~ ~

~

->~~/

~/

a=~

~

7 4 4

4 6

b)

a) c~

satisfy requirements

the C, which

three differences permutation C P~(r shows Fig. I. The ~) I

i

~

Pg(r~~)

third difference figure the lib,

of shown in lla). the Proposition (Fig. In of 2 case

differences).

(there four angular

figure P~ lc is in missing.

Permutation shown I is 2 not

are

y =

Proposition

3 :

of

triple n) integers satisfies (6) positive (a, fl, For

which the conditions

two any :

(",fl)=I

(21) ia, pj+lwnwa +p max

Ci C~_ angular permutation P~ unique

such that fl. there exists

a, a

= = i

Proposition Appendix propositions demonstrated

is in 2 Moreover, B. 3 of

consequence a as

3, triples correspondence

exists of described and between the in there set one-to-one a

proposition given angular permutations. by 3 the is relation of and The connection the set

c~ (22) c p

«

= = j i

C~_i, triple (Ci, angular permutation

Actually, P~, n) satisfies in the the

any con-

triple (21). fl)

permutations by Notice characterized (a, those

ditions fl, that the

+ a are a

C, only differences (I for I) which the I, take values o,

the

C~

two

n

= ~, ,

positive integers) and (a, fl fl

a

(Appendix B).

of

direct B.2 This is lemma property consequence a

Hierarchy permutations. angular 3. of

Proposition 3 allows classification

angular

hierarchy

permutations. The

set to up one among a

which in will be

in useful

make order connection with the tum to emerges very a

morphogenesis phyllotactic

of

manifest, this To make

connection shall structure. a we

geometrical underline interpretation permutation the

P~

of integers series of

a as a

Cj, C~

placed Co,

circle.

a on

i ,

(21)

Conditions positive integers highest fl.

and (",

divisor of fl) denotes the (~)

two common a

coprime. fl fact that and therefore the

expresses are a

ALGEBRAIC MODEL MORPHOGENESIS 2143 PHYLLOTAXIS AN OF N° II :

perrnutation shall

We

consider

P

Let I DESCENT 3.

note

AND ANGULAR DESCENT. us some

~.

Pj~~

~~

removing point by perrnutation P~ obtained from number

the

: n

k+ k- k I I I I

n-

(o p

~

Ck"n-I Cn-1 Ck-i Ck+I Cl- ~ °

~~ ~"~o

C~_jl'~~~~

C~_j C~+i Ck+2. Ct.

~~

introduce P~ orientation for circle. and P

We choose the the We

two can same on now )~

definitions.

of permutations for D(P~) P~ P~~i

descent

permutation call the of We shall the set a

(?) which

Pfl)1~ Pn (24)

particular,

D(P~) consisting of the subset perrnutations In D(P~) angular of all will be e

angular P~. descent called the of

descent is Notice

that the word

here used in the

lineage of this

The of

sense « » reason ». «

justified choice by will below be connection the angular between the of to concept appear

phyllotactic descent

and the of pattem. emergence a

of

angular

Angular descent 3.I.I P~ permutation. angular (a, n) p, Let be

an an

=

perrnutation.

question words,

angular other where arises does it have descent ? In

A natural

can now : an

perrnutation P~ resulting

(with

is still point P~

n)

number in the that further

put

we a so

i

~

angular ?

answering permutations. angular question, simple

If this notice of Before property

we a

through

permutation P~ angular, P~ is defined then

some i,

(25) P~e D(P~_1)

angular (9). definition obvious is this of the

is

consequence an :

therefore have We

Proposition 4 :

If angular P~_i, belongs of

then

P~

permutation descent the permutation

to an a

itsef angular. P~ is

i

is, consider P~ P~

Conversely, angular show that have be when

two to to can cases.

we

i ~

point neighbour of origin

o. The is the I.

n a new

(Fig.

12). definiteness, fl between and o let For

set us n

permutation P~~i clearly defined this fl. For, in

angular thus is The if

case, +

n a =

). C, C,, by only differences (see Sect. 2.3.

P~ namely

characterized and fl

I

two

was a

i

~

displays P~

fl itself and which

the three differences As

consequence,

n, a n a

a

= i

~

satisfy proposition it is

of 2. On other hand,

check that fl, conditions the if

to

easy

+ n a ~

differences. than three P~~ shows

more

i

conclusion holds between o of of the roles that

The for and because the

symmetry

same n a

permutation. play

p characterization of and in the

a a

(?) words, permutation

descent of P~ In the permutations the of

other is obtained all from set a

adjunction supplementary of P~ namely point by point,

a n.

PHYSIQUE I 11 DE N° JOURNAL 2144

a

~0

+a

~

n -n

n-J

fl

neighbour

origin. former of

0 and fl, the The point has been between Fig. 12.

The

put

a new new n

differences

by -fl three if permutation P~~j fl is characterized corresponding

n- -n; a =

differences exceeds three.

number of fl, the

If fl.

+

+ n n

a a ~ =

only if

angular if P~ and resulting permutation is

the summarize, To

j ~

(26) p.

+

n a =

of neighbour origin

the o. point

The 2. is

not a new n

unique angular is proposition

claims that there

One 3 which

construct to

way an a can use

provided

(when I) fl that I triple

(a, fl, by

P~~, permutation defined the

+ + + n w n a

D(P~), required.

proposition P~ 4, On impossible).

of

is Because fl, this

as e

+ +

n a ~ ~,

implies

fl.

In order reconcile the latter angular, which P~ hand, itself is

other the

to

+ n w a

require

inequalities, clearly

must one

(27) p

+ n a <

discussion table I. this in summarize We

can

Table I.

fl I.

+

n a =

(a, I)

+ n,

n

n)(

Bifurcation («, fl,

(n, fl, I)

+ n

fl

a+ n ~

+1)

(", fl, p,

(tY,

11) 11 ~

of angular Apart 3.1.2 permutations. Descent from detail of the the transition tree

algorithm ofangular (a,

n) (a, ), fl, the descent fl, I table I defines which

allows

+ n us ~

lineage permutation

P~ angular (a, fl, ) the between and its

member of state to

any some n

permutation angular P~ angular descent. Since is permutation the descent of

any an

'~

(Prop. angular permutations by

4), P application it is clear that obtain all the of

the

)~ we can

algorithm 3) (2, 2, angular angular (1, 3). of 1, permutations descent and The whole of to set

inverting

therefore by constitutes be obtained from the orientation other the

two trees

one can

equivalently, substituting

of by compatible divergence circle the the 9

to or, a one new

9.

ALGEBRAIC MORPHOGENESIS MODEL

2145 OF PHYLLOTAXIS AN N° I I :

divergences

2.I.I, ourselves section restrict According

made convention in

the to to we

together by 3) 2, with whole

(1,

constituted its therefore,

belonging o, the and,

tree to to

2

angular (8). descent

general of angular described of the Algorithm descent. have above the 3.1.3 We structure

interesting

explicitly transition

algorithm descent, how the angular is of but it to see

+1) (a, (28)

p, (a, n)~ p, n

27)). subjected (28) condition place (recall takes that is to

points B~_j (A~_i, figure13. point A~_i

located between and Consider Let I be n

points point points

P~).

Let and be that such B~_ A~ and successive in B~ I

two

are n

i

angular according

P~;

the P~~j, descent of the located will between them in be to n

possibilities.

successively

orientation,

B~. There A~, and

two meets are one n

positive

only difference -fl~o.

the is ~fl,

that As

I.

consequence, so a a a

directly

located 8( A~_ Therefore, be after I 8j and

A~

must

+

: n a,

a. a

= = = j

angular.

P~ for

be A~

to

n

a

= j ~

directly

similarly located before for fl be B~

2. has fl. One show that

to

n

can n a

~ =

P~~ angular.

be

to

j

+I B~

B

B~_

= j i

~

~n

I n

~2

b~ b~

~n-

An

An- +I

" I I

symmetrical correspond

number I and angular Points Fig.13.- Algorithm descent: of

to n- n

(see text). situations

already

3.2 have introduced We THE the OF DIVERGENCES. Set SET COMPATIBLE

(P~) compatible permutation n) divergences (a, fl, div div of the that the with 9

are =

P~

n). (a, fl, aim this Our determine is

Set.

to =

question

first To this have

state to answer we

Proposition 5 :

of If

point by

position

located the k points given number 3 and is the circle,

are on n m a

(k9), satisfies angle integer

and the k km irrational ow then where is I, 9 n-

kj neighbours of satisjj 0), origin point (I.e. number and the the k~, two

(29) kl k2

qm qm,

~ ~ +1 r

for

~~). r(I

and

a~ some m r w w

(8) Here,

this lineage P~ incIudes the whole offsprings. only of and

direct its

not « »

PHYSIQUE I N° I JOURNAL DE I 2146

~~'~ ~~~~

of Integers

denominators and and of

convergents two

q~ q~~j are

~ qm,r qm+i

equal

Moreover, integer other denominator

of of

@.

to convergent q~,~ any some a

such that

II

(30)

lam, qs, qm,

qm

~~~

~

~

t + r r

larger than is n.

Proposition (Appendix B). Moreover,

5 direct of B. it

is Lemma I shows that consequence a

(a, n) is div of for fl, the those which

set

fl if (31) fl

qm qm,

" "

~

~ ~

+ r

or

fl if (32) fl

q~,

q~

a a ~

= = i

~ ~

proposition

irrational for Notice that If needs be 5 be the numbers 9 not to true. to

(k9)

proposition applies,

o all 5 I,

is distinct for km also if rational.

even are n w

compatible divergences.

have of Computing -We the the 3.2.I compute set to set

n) (a, p, condition div with the

(33) (a, n)

p, div

(0, ~

determining equivalent (a, proposition p, n) div the 2.I,I). that is Now 5 shows (see to

satisfy integers which p

and

fraction from known

construction continued of the to

two are a

unknown) priori (32) and

(31) (a for

m r. Some or

briefly. [10]). Sketch it (See Ref. We here is Such construction well

known a

relation with the Let start us

(34)

a~ rq~ q~

+ q~ r

~

~

~ =

j

~ ~ ~

implies (34) (see (18)). generalization (15) for intermediate is which of

convergents a

0

(35)

qm qm, qm ~qm

~

~

" +1

+1 r

or

m

~

l~~

(36)

~m

~

~

2 ~ +

~~'

~

Similarly, (s) with

divisor remainder (34) Euclidean [s is the

From where

q~.

r s

=

qm

i +

~

(37)

~m

~ l~~

+

~m

~~~~

divisor

Now, is with remainder for the couple and principal

of

q~_i, any on. so

qm

convergents,

(q~ ) o if I (38)

q~

m m

= i, ~

2147 MORPHOGENESIS MODEL OF PHYLLOTAXIS ALGEBRAIC AN

N° 11 :

~~

~

implies (33) only

if that that of the remainder the division vanishes if and 1.

q~

so

=

qm

(39)

l I

qo q, at

~ = =

algorithm that Euclidean division o the of when stops.

so m =

summarize results follows. We

now our can as

unique angular (a, permutation

in first of the fl, n) determines An the

part way a

development fraction,

Moreover, namely of continued it

sets

a~,. a~~i. as at, a

,

elements words, if

the of the undetermined. In other and leaves

rest

a~ a~ a~ r

w

~ ~,

~,

~ ~ ~

(4°)

[~l, [~l, +21

~

~m ~m

~2, ~ ~2>

~ ~ +2,

+1,

, , m

with

l~am+2+~

[~m+2,~m+3; (41)

~m+2~

m+3

simple has the condition one

(42)

r~ r w ~~.

symbolize satisfy by fractions We shall the of continued this which property set

[aj, (a, (43) n) fl, ]. div

a~ +

r

= j,

~ ,

above,

As provided divergences, rational noted the result obtained extended here be to can

through (42) restrict

that we

(44)

rm

°l

r

~ ~

2 +

~

rational, points which insures that, if

is then all realization

that of the of the 9

q so

n

~ =

q

permutation (a, n) distinct. fl, are

lengthy

through the results of this We discussion

up can sum

Proposition 6 :

j.

o, determine

(a, ) that div fl, fl

Then and permutation (a, ) be

fl, such Let

a n a n

~ 2

for following hold of

which relation the integers

series

at,

a~, a~ r a

i,

~ ,

(45) 9~i (a,

n) p, div 19~,

=

if and,

is

even, m

(46) r] [aj, II [aj,

92 RI

a2> am

am

a2,

~ =

I, ~

+ + , ,

if odd,

while, is m

(47) r] [al, II [al, 92

RI

a2, am am

a2,

<

=

~ I>

+ + > ,

that

write so we

(48)

[at, (a, div n) ]. p,

a~

+ r

= i,

, ~

PHYSIQUE 2148 JOURNAL DE N° 11 I

example, successively (3, 9). 8, We div For have compute

2 remainder 2

a3 » r

= - =

~

remainder

l

a~

= = = 2

~

remainder o

2

aj

= = = l

figure14. there (3, 9) explicit that 8, is in One check The form of shown can

according 2) the chosen points,

8,9)~ (0, 1, ordered div(3,

because

to are (0,

2

according proposition 6 orientation. Moreover, to

~

~,

(.

(3, 8,

9) (49) div

=

obvious is It that

(,

(, (8, 3,

9) div

(50)

=

o

~

3

~6 5

2

7

Fig. Angular permutation characterized 14. triple by (3, 9). 8, the

modifying algorithm by generally, division

of the

in the

More where I, obtain

case

at we =

Proposition 7

:

l [al, ]91, °2[ if

n)

(£k, fl,

dlv

[°,

+

~m ~ ~2,

~

+1, ~ = ,

then

I,

[1, ]. 91[

]1-

(fl, n)

9~, div

a~, + at a~

r

a,

j, = =

~ ,

fact requires that the I. point

which only

The

comment at

now concems a "

relation Then I is for and the

Hence I.

true qo not

qi q~ q~

m

_1~ = = =

~~~~

(51) =1). (m

# a~~,

qm

2149 MORPHOGENESIS ALGEBRAIC MODEL OF PHYLLOTAXIS AN N° I I

vanishes remainder

algorithm the when the We then : stop

~~

l(remainder

o)

a(

=

=

qi

and put

a(

(52) I I

a~ q~

m

= =

Proposition

Regular depends only (a, ) 6 singular and fl, 3.2.2 shows that div descent. n

compatible

permutations, angular

fl. of the interval of

in the and As

tree consequence, on a a

bifurcation, divergences satisfying of points only is modified I-e- values of for the

the

n on

p (53)

+

n a =

permutation

(43) fl ) hold,

with and Let together (a,

with fl,

to start

+

us a assume a n =

fl.

a <

distinguish

between We

descents two :

Regular (9) (n, ). descent fl, I. n) (a, fl,

I

+ n ~

transition, Before

(see (32)). (I the

After fl transition, ~) the

q~ a~

q~

« « r

a

< = =

i

~ ~

~

possibility only

fl ).

(I the Actually, being

q~'~ is p, Smaller,

a~

q~

n s w

w

~ = =

,, ~

~ ~

~

principal

correspond Now the transition selects subset of convergent.

must to a a

divergences (a, n) [28] div

compatible of fl,

set a new : as

). (a, div (54) (n, 1) fl,

fl, div

+ n n ~

easily This computed, subset is since

~~~~

~m,

~m

" 2

r +

implies which

(56)

am r

2 = +

and

(~~)

fl

qm +

qm+2

qm+ +

n a

" i,1 i = = +

given by compatible divergences is

of The set new

(58)

].

[at,

I) (n, fl, div

+

a~~i, a~~2,

+

n

= ,

).

Singular I n) (a, 2. descent

(a, fl,

+ n, n ~

that Notice fl Now have

qm,~ qm,~~

+

qm+ q~

+ we

n

a

a

< = = = = i i.

~

(59)

a~

r

~ ~ ~

compatible divergences

(32) interval of Otherwise The

satisfied. would be reads not new

(r 1) ].

[at, (a, ) (60)

div

I

+

a~ +

+ n n,

= i,

~ ,

approximable Slowly regular and thus leads irrationals

I, Note that the descent

to

sets

r » « =

transitions regular by

leads noble rationals. infinite succession of An number. to a

singular

Interestingly, correspond regular and the of descents the of concepts concepts to

introduced in regular singular mentioned in the second which and transitions been have paper

(a, 1). fl, (a, fl, n) If then

fl.

corresponds

(9) The considered here

to

+ n,

case n

a a ~ ~ -

2150 JOURNAL PHYSIQUE DE I N° 11

[10]. reference, logarithmic

aligned spiral this self-similar In along of circles pattems tangent a

By varying steadily

studied. the of transform have been the parameters

pattem, can one a

neighbours close-packed

four lattice where each circle is with into lattice where each tangent a

neighbours.

is with six happens

of circle in Now, be the contact

parameters to one a

lattice, possibilities modify divergence close-packed

the lattice from there the

to two ; are

continuously.

development

Their divergence fraction effects the of the continued

on as a are

(58)

justifies exactly by equations

(60). the shown and This the chosen for the

names same as

by first Moreover,

by Levitov that described of descents. it shown in

types

system two was a a

potential

regular [29]. transitions favoured the

energy are

point.

Farey angular permutation Let bifurcation

3.2.3 consider

at structure. tree a us an

compatible divergences of written associate is The set

ibi, ) (61) 1. b~i («, P, P

div

ia<,

am + +

r

«

= = +<, ,

points according

) given parity

(46) (47) (I the interval The 2 of of

1, by end the

to

or b~ are =

~~

fraction,

is rational In

If write

any b~ case a

we m.

qi

[a,, (62) 1]

b'

+ a~+,, r

= ,

bifurcation, branch associated of the intervals after then, each will be with the one

easily b'[ 16', b'is from and 16,, b~[. b, and obtained Now 0~

=~'~~~ b'=~~'~~'=~~'~~~~~~

(63)

+~2 ~m,r+~m+1 ~m,r+< ~<

30]. exactly Figure [28, in 15 Farey construction This is the rule used the of the shows tree

angular Farey P~ together for 3 with «n«7, permutations all that the the such tree

compatible divergences. corresponding of intervals

uniquely perrautations

Accordingly ) (tx, index with the rational p, fl numbers

+ we can a

b'=~

particular q)

permutation

and obtain

fl. with in

I We the ~p,

+

q

can

a

= =

q

~

by building realization

) fl takes form the (a, fl, which the

+ T~

a q

lC,

0'l

(64)

=

whence

(65) =1. C~

result, intuitive, obtained This which is rather be from Lemma B,I Lemma B.2 and can

(Appendix B). will Appendix It used in section 5 and in D. be

complete

have We achieved

3.3 THE A

DIVERGENCE. THE GOLDEN ALGORITHM AND now a

divergence. of angular their permutations, of for value knowledge construction about the any

possible analyse 1.2.2

in algorithm A introduced is the have Within this frame it to to we

Thomley's dynamic scheme. simulate

regular only corresponds algorithm

angular with of descent

show that A the We to want to

golden divergence thus bifurcations, leads the 0~. and to

MORPHOGENESIS OF 2151 ALGEBRAIC MODEL

PHYLLOTAXIS AN N° 11

o 2~ (1,2,3) ~

~~° § ~

(1,3,4)

(2

4) 3

~

3 ~

/

~

~

~~~

(1,4,5) (3,2,5) ~f~~

(3,4,5) ~ ~

]1/3,1/2[ ]0,1/4[

4 2

3

~ , ~ ~ '

II

~

s

~~'~'~~

(5 6) (3 6) (3 6) (3 6) 4 5 4 ~ 2

4~2 ' ' '

~

~

o o

~

~ o ~

s 4

i

~

~

~

~~'~'~~ ~~~~'~~

~~'~'~~ ~~'~'~~ (~ ~) ~

~ fi

2

~

]l/6,1/5[ ]1/5,1/4[

~ ~ ~

'

~

~

j

] ~ ~ ° (5,2,7)

~

~ @

2 6 (3,5,8)

j~

i

~

]1/3,2/5[

4

maximal Only 3). 2, permutation (1, the of

angular descent Farey of Fig.15.- The the tree

label represented. Apart from the explicitely

p)) been through (a, have permutations (defined

p,

+ a

(singular) transition is regular b~[ A divergences ]bj, is shown. compatible n), the interval of (a, p,

angular

descent. line indicates (single)

broken A line.

labelled double with an a

regular permutation (a, ) Consider

fl fl, fl. with Its descent includes the

+ a

n

a a ~ =

permutation :

Pn>= («'=«+P,P,n'=«'+p)

Fig, 16). (see

(1, )

Figure

neighbourhood point shows

2, 17 the of k construction fl I P its

e a ; on

~,

...,

directly proposition (namely only from follows 2 from fact differences and the that two

fl) P~,, and a' a'+ since fl. in n'

can occur =

point placed k(k

2,. a'+ I) fl thus been =1, Each between k its has

and older

,

neighbour, that k is

+ a.

locating exactly by

A, is this defined Now the if consider 0 rule between and a' we

application

initial of the in

A. the

step as a

neighbour Moreover A a'+

fl fl will locate between and its older which is 0 this :

algorithm regular that corresponds descent. A the of demonstrates to

PHYSIQUE JOURNAL I N° 11 2152 DE

k+>

o k

a+k a

Fig. 17. 16. Fig.

Fig,

16. P~, belonging

the -Permutation regular perrnutation

(a,

descent of p)

p, to

+ n see a =

section 3.3.

3.3).

P~.

point (see Neighbourhood Sect. of

Fig. 17. on a

permutations. Stationary 4.

permutations for the

that does allow

evolution for far law considered

have not We

so an

time represented ring indeed

apical is regime

the steady

description of the at of state

any : a

appeared during the have

that permutation into the P~

takes n-sprouts that by

account

a n

growth.

permutation

language.

regime stationary the

in describe also

But

a can one

define section I,

notation

in 3, Using following definitions. similar

introduce the We we as a

Co. convenience, further

point by For P~ removing 0 from obtained permutation

P

we

)°~

a =

finally them

integers

I and C~(I remaining Sk permute

of the

each subtract

to « n one

through 0. remains image P)°~ of 0 circularly the that

so

n-lj

0 2., k-I k k+I,

p

~

C~ C~_, C~ Co C, ~ C~+,

C~ 0 I

= = _,

~ ~ ~

~

Pj°~

(66)

=

Ck+< Ck+2 Ck-< ° -1 I I

~'~,

P)~

P(°~ negative integers

permutation

is for As is still I the of the first

case a non n

leaving

unchanged.

0 able definition give further We

to now : are a

P~ stationary permutation there D(P~) if P~+, permutation is exists such that

e a a

~'f~

~'n.

~

Clearly stationary permutation

describes

(the finite with

oldest

sprouts system memory a a a

forgotten),

invariant whose is the evolution. under state are

following proposition

The permutations necessarily phyl- that correspond such

states

to «

which

lotactic is, divergence

(there that is intuitive exhibit

systems systems,

to

a an reason »

apical

for this concentration field the ring by if, the permutation is described the and

: on same

only if, undergoes

it rotation

growth each of angle the the

step at mere process; a

characterizing nothing this rotation is but divergence). the

Proposition

8 :

angular. if, if, and P~ only permutation stationary A is it is

figure given

18. proof Appendix example C, in is in complete be found and The

an can

interesting

angular permutations which proposition gives Moreover, characterization of 8 an

proposition

involves arithmetics in 2. contrast to no

N° ALGEBRAIC MORPHOGENESIS PHYLLOTAXIS MODEL I I AN 2153 OF

:

° ° ~

8 7

~ ~

3

5

5 4

5 6

6 8

2

9

2

6

7 7

~ ~

4

(al (c) (b)

8,

permutation angular a) 9), b) (3, Fig.18. angular permutation stationary, The The An is new

giving algorithm angular

9)

descent,

according point (3, birth 8, the of the been 9 has added

to to to new

points C, replaced by (3, 10). and have been 8, angular permutation c) Point removed all other 0 has been

8, 9). P~ (3, identifies

resulting permutation

(I 8). The with I

C~ w w

Group 5. structure.

section, 5.I MAXIMAL analyse properties In this of the

PERMUTATIONS. group some we

angular permutations introduced above.

According

3, proposition given (a, fl), for the maximal value

to q can assume

permutation

[28, angular given by is

(a, ) fl. fl,

P We call maximal such remain

to

+ q

a

a

=

~

30].

permutation

permutations, corresponds angular always maximal On the of

tree to a a

define bifurcation. Let us

q)) (H~ fl~

(a, (67) p, fl

+

a

= = =

fl~

H~ permutations q) elements. Given (a, fl, fl the of of maximal is all

set

q +

tx

= =

satisfy

only depends H~

which

must q, on a

q) (", (68)

=

here). be (a is assumed fl

not to ~

permutation only

by maximal characterized differences the of Moreover, is

two occurrence a

C,

giving

I), Before mentioned section 0,1,.., (I 2.3.I. in has been

C~

as an n

= +,

description

explicit

introduce notations. of fl~,

of shall useful Note the that structure two we

follows. integer considered fixed in is what

q as a

P

fl~,

associate rational 0' H~ the with

section 3.2.3 First,

recall

to

e one can any

we :

=

, q

p(a). belong

Notice both

that is fixed, write and q) Since I.

~p,

to

p p

q we a

can

= =

coprime with positive than and it. 4~(q), integers of smaller the set q

integer through following R~(I)

of division The I by of the shall the We denote rest q.

straightforward

properties )

of R~(I

are :

)) ) (I j (I )

R~

Rq ~j R~

(R~ + +

= ~~~~

Rq(iR~j ). ))

R~(

=

They integers. (I, j) couple of valid for

any are

We able

state to now are

N° 11 PHYSIQUE JOURNAL I DE 2154

Proposition 9 :

together fl~,

with The of usual the composition permutations, law

is

set group. a

following H~, fl~ identities The H~, hold for

any e

Rq(I« Ha

(I ) (i (70)

)

I

q « « =

H~.H~, H~ (71)

= ~~~,~

' Hi (72)

H~~~ =

proposition of Appendix fl~ 9 be A demonstration found in nothing D. Note is that else can

ring

integers than the of the units of the [31]. of modulo Z~, group q

Angular

permutations

which maximal. One extend the

cannot structure to not group are

permutations angular

permutation

(5, 7). maximal. 4, which Consider for instance not are

According algorithm angular descent, of the

it has be written

to to as

2 3 4 5 '

6j (°

(5

7) 4 (73) ~

"

5.

' ' 0 5 6 3 4 2

Now

satisfy requirements proposition of angular. and, therefore, 2 does the be not cannot

Bijugate phyllotaxis. 6.

There 6. I BUUGATE

two PERMUTATIONS. SYMMETRIC PATTERN AND excep- are common

parastichies. phyllotactic of Fibonacci One finds numbers for whose tions the rule pattems to

parastichy

(see I) either consecutive numbers of Lucas series Sect. 2, I,

numbers the

two or are

Fibonacci built with the double numbers

of the

sequence up :

26.. 10 16

Trijugate

bijugate. generally m-jugate) (or called also

Such

pattems

pattems may are more

encountered. be

by juxtaposition m-jugate

of such is obtained of geometrical model simplest Tile pattem a

golden

[10]. lattice, W~~(0

is strips of where the

0 b~ identical

mean a m

m-jugate

quite improbable produced by simultaneous of is that It

pattems

appearance are m

ring, long-ranged

apical growth of the Such correlation in the each

step sprouts at process. a on

yield quite in fact mechanism different from the sketched in 1.2. I in would the

system

a one

I'°).

m-jugate exceptional

of would be this the pattems

not case, occurrence

imposes slight geometrical

modification for in ideal model This

remark

may : we a our

strips

example b~, associate different each of the that each time

to at age so a m

unique corresponds

birth of the sprout

a n. n

permutation application

of cut-and- describing by The the is obtained such pattem a

projection picture. geometrical discussed

method in

2.2 have

to we our

plants systematically m-jugate phyllotaxis. exhibit that (1°) don't consider here We

2155

MORPHOGENESIS PHYLLOTAXIS ALGEBRAIC MODEL N° 11 AN OF

apical ring would of by the permutation The time following the thus be modelled at state n

bijugate

(we consider Fibonacci simplify)

and in order

pattem

to a n even :

l~~,

)~[ «

Ii

[)~

)j[1~

~~ ~~~~ Ill

cl ci,~- ci

C)

I, is for all where even

c) cl

for

i all

+

i =

and where

~

0 l

~ Pi

(76)

=

, ,

Cn<2-

C> ~

~

2 2

permutation. angular is an

2-symmetric. P~ We will is that say

(P~),

words div in other pattem), 0~

(bijugate

Fibonacci have

In

e we : our case

~' ~'

~'~~' '

and

2 2

Fibonacci numbers.

successive two are

Surprisingly, algorithm

SYMMETRIC 6.2 the A PERTURBED PERMUTATIONS AND DYNAMICS.

permutations

initial Consider such for abnormal when in

generates step. an occurs an error

example irregular permutation (Fig, 19). P5 the

correctly applied, placed been have If A between and 2. had 4 would Now been I suppose

apply

placed 0, that and A P As has been 5 is I its 4 and older between

put

to to next

go we on 5.

neighbour, permutation

that obtain 2, circular

is and We time19 whose

at on. a so

representation figure is in 20.

0

lo

~~

4

~

14 ~

2

9

2 3

13 18

~

8 3

17

~~

5 11

Fig, Fig. 19. 20.

Fig.

19. An in 4. occurred step error

Fig.

permutation order 20. with of A 2 is obtained the result of initial symmetry

a as one error.

PHYSIQUE I N° I I JOURNAL DE 2156

It is check P,~ properties 2-symmetric satisfies

stated of

that above all the to easy a

permutation.

applied generally, (with

More if

is convenient choice of step)

the initial A

to a an m-

symmetric permutation, permutation m-symmetric divergence the remains its then and tends

number. towards noble a

transcription simple that This result algorithm regular

is than of shows the of A

more a

descent, whose purely considerations. construction relies mathematical on

equivalent Thomley's

considered of In this dynamic A be

for scheme

may sense as an

permutations.

Conclusions. 7.

Following

phyllotaxis.

[18], reference tried in this introduce temporal into factor

to paper a we

equation [14], In

kinetic describes reference the evolution of the of concentration inhibitor

a an

apical phyllotactic

ring the where the shoots of

Inhibitor is the

structure

appear. on new a « »

given

morphogen, hypothetical chemical preclude

substance assumed the

to to name some an

immediate Each shoot by supposed

of new-bom is be

followed

sprout.

to emergence new a a

inhibitor concentration fades local increase through of which diffusion and

away soon

degradation.

profile concentration As of overall inhibitor various the

shows consequence, a

peaks sharp according shoot, of of less the each birth

assumed order in

to to appear more or

succession.

algebraic analyze

distribution Our points

finite aim of of number the of

structure to was a a

points

together temporal

the circle with their evolution. The circle assumed

to a on are on

correspond peaks apical ring.

distribution We of forward the the have been able

to put to on a

singular regular permutations.

transitions of between

This builds and system system a

similar hierarchical classification of the transitions observed between the of lattices structure to

10].

[32, helices equiangular spirals the along circles alined As

tangent consequence, a or

emphasized.

divergences

of be special noble numbers role the

as can

algebraic general enough The necessarily of this model it is is

related that structure not to so

biologists

fact, chemical for phyllotactic between model. In look relation the

many a a

morphogen. that, the existence of shows and could

Now

expect,

structure paper our a as one

divergences unlikely general give

information of certain the noble is the

to occurrence any over

precisely, if of

of this shoots. More kind ruling the chemical

appearance processes new

phyllotactic

in

ought it be looked for information be from could drawn structure, to a

divergence,

by characterized Lucas in the for in anomalous instance

pattems, structures a or

phyllotaxis. m-jugate of case

Farey-tree)

(I,e, hierarchical both from The

fact that the

structure emerges pure a a same

morphogenesis astonishing, is after analysis

of the of static and from model

not process so a

during growth, scaffolding, its builds its It of the fact the that all. is the pattem, consequence

nothing which is else its than structure. proper

large majority

study. botanical this from of A Another conclusion be drawn

pattems one can

phyllotaxis divergence golden

(their is mean), close the while belong golden

the to to meets a

divergence.

Moreover, proportion spiral there still less of show Lucas small the pattems are

together complicated divergence, with for frequent of observed values the pattems, more

spiralled Observing

rotation lattice. and the

of forth which both axis instance those

put an a

possible morphogenetic study

that the

which this Farey-tree from suggests now emerges

golden (I,e. different from responsible the random fluctuation for anomalous structure any

morphogenetic

early during

of

the absence In one), if is the there

must process. very any, occur

correspond

important possible transitions less only fluctuation, such the

to to seem a

MORPHOGENESIS

2157 ALGEBRAIC MODEL OF PHYLLOTAXIS AN N° 11

divergence unchanged. analyzing fact, leaving of the these almost

the value In

bifurcations

possible of lead into the chemical could then

fluctuations assumed

process. core a

Acknowledgments.

deep gratitude

of work Koch, and A.-J. We the initiator this

to constant want to express our a

problems. G.Bemasconi, together help, various scientific technical with for and We

acknowledge precious information J. Bo£chat Guerreiro from and fruitful and and J. comments

Bestgen, Grossmann, Kl£man, Rivier discussions W. S. M. L. S. Levitov. N. deserves with

special interesting

acknowledgments suggestions for of and the he

comments many was source

enough manuscript. Finally,

because,

least, and read last but he kind

not to was our over we

typing manuscript.

the

indebted Ghelfi Rothen and E. for B.

to are

Appendix A.

proposition following In first order I,

demonstrate lemmas. the

two to prove we

Lemma A,I.

R~.

(na

If

slopes of be lattice the and have V Let

in basis

mb Z b

vectors +

n, e m a a =

belong envelopes they P+ signs, of opposite the lattice.

then the and P_

to convex

Proof.

graphically.

it We by

that defined contains lattice First observe unit cell the and b

prove no a

point,

respective Fig. Al). slopes (see whatever their follows From this that the set open

by (Fig. A2). kb bounded lines

point fl R, the kb k contains If and

lattice and

b

+ e a, no a

opposite

slopes signs, R+ R+ have (Fig. A3). define of fl' fl n

x we =

P+

envelope contain

envelop (to the does it Now if the the

not vector must cut convex a, a

point a), (if line, imply angle

broken be it the the in fl' would of existence segment

a was a a

point fl'). impossible,

the this is lattice in Moreover it in fl'.

cannot y-axes cross or x-

consequently contains P+

similar. The of is b

vector a. case

W~~(0)-lattice,

(For 2.2,I). definition of Sect. Lemma A.2

see a a

If ofopposite

)-lattice.

W~~ W~~ slopes

basis Consider signs,

and with 0 W~,

vectors

are a ~,

if

n'~0, belongs

point W~+~,,~+~, and then ~0, there P+

P_ is

to no ; or moreover n

P+

[n, n'] with P_ n'. i

+

on or max n

W~~ ~ ~

y

n

I

I

I

o x

Fig. Fig. Al. A2.

PHYSIQUE JOURNAL 2158 DE I N° 11

y y

$ ~

- b

p o' o

x

x

o

o

Fig. A3.

Proof.

diagram point, figure

lattice Simply consider hatched the in A4. The contains since

no zone

by

basis W~~ W~,~, W~~ W~,~, defined and is the unit cell Hence and vectors, empty.

are as

necessarily belongs Lemma A. W~ W~ and I then shows that well W~,

as

~,,

~,. ~, ~,,

~,

+ + + + ~

~

P+.

P_ to or

implies Fig. A4). the P+ (we also of It W~~ P_ W~, and consider that Lemma A, I

e e case ~,

point n']

envelope belongs [n, for is which i n'

then clear that

to

W;~ + convex max a

a n ~ ~

impossible. then the hatched this of We is have only is

if it in

: course zone

y

Wn+n',m+m'

j~~~,~,

Wnm

x

o

Fig. A4.

Proposition 2.2.3). (Sect. I

W~~(0)-lattice,

(resp. In P+) through P_ points

a goes

2)

~~ ~n ~ ~

~

~~'qn,

rPn,

+

r

being even). (resp. odd

n

Proof.

that First notice

0

if 0

is

qn,

p~,

even n ~

r ~

0 if

0

is odd

qn, p

n

~

r ~, ~

Hence

R+

if is R+

e even x n W~~

,~~

,

(Al)

R+

if is odd R_

W~ e x n

~

proposition by induction. the Now demonstrate we

MORPHOGENESIS ALGEBRAIC MODEL AN OF PHYLLOTAXIS 2159 N° 11

P+

fulfil

points and the ~~), ~(l

P_ constitute

Suppose

for the that I that

q~, W~j a~ r w « «

proposition the of further and and requirements the that generate W~~,~~,

suppose W~~~

~~~~

lattice.

a~+~) by if is

(or A.2 know then Lemma that We

W~~~~,~~~,

W~~,~~~~,~~

on r

=

point

W; P+ i P+, that there is with i P_ P_

and

(q~,

q~ q~,

q~,

or

or no on

~ ~

~ ~

i, j j i

+ + ~ ~ ~

+~). if

a~ r =

(or proof and of A.2 shown in been It has also the Lemma that W~~,~~~~,~, W~~~,~~~~

hypothesis

induction is if the thus

and lattice the

generate

W~~~, a~ ~)

r W~~~~~~~~

=

~~~~, +

(or if

verified for I

I

~).

q~, q~ a~

w « r

~,, =

, ,

+ ~ ~

simply demonstration, and the

close have

To the that

generate to prove W~~~, we W~~~~

satisfied). P~ condition is

by points (we Lemma this if P_ lattice know I that these and

on are

Vn,

write Z, But

e we can m

(n

mwaj

W

Wnm

+

ma

=

~~~~

(n )

mwq<

+ ma,

Wq~

= Pi p~

Appendix B.

have In idea cut-and-projection the outlined of the (chapter the 2.2), method which text we

proposition

give allows 2 be Here

the details the method. of to proven. we

Definition: neighbouring A(~ points B(~ the point strip W~~ of and the in the

points only in projection the f~

whose x-axis the li~ projection

is the of next are to on

A(~ projection

B(~. projection of W~~ simply the will of of be taken left write the the We

; on

B( A(

neighbours point

and for the of W~. a

Figure illustrates this definition. Bl

y

L

m

_

bL

~L

~~ .

nm

x

0

.

Fig.

Bl.

PHYSIQUE 2160 JOURNAL DE I N° 11

B,I. Lemma

be (q~,~; W~~(0)-lattice,

Let strip lattice q~,~+l;. b~ with Le in

a a

)

I 0 and

I

q~, q~

+

a~ n m r « «

,

~.

+ , ~ ~

have We

Ai

B~ and

~~

W~~ ~~~~ ~

~'~qn.,Pn., '~

# "

p~

~, ~,

( At

f

and B (B1) odd is i W~

W~

n

= =

p ~

,

Al

B(

of and Moreover W~~(0 )-lattice. basis the

vectors are

Proof.

Al

B(

neighbouring

origin, If and slopes points they of then the opposite signs of have are

they

define and lattice, (Fig. they 82). the since cell

generate empty an

y

-L

Bo

ij

x

o

B(,

Fig. point. By definition hatched lattice 82. and of the contains

A~ zone no

apply therefore Since L write

We Lemma A,I.

q,, we can can ~

At Bi

~~~~ and

w~~,~~,

" "~~>.kP>.k =

k

with I and I

a~ a~ « « « «

~. ~

+ +

neighbouring points

Since and the consider We the of

are

now even case n.

W~~

W~,

~~,

~~

~

, ,

j of with which of 0 denominator in is is less than

the b~, convergent greatest qj,~ even

L.

Now

q~,~~L

(83)

L

qn, qn, +

qn

~

"

+ r + r

Similarly, is denominator

the of since is that greatest

convergent

q~,

qj, q~,

even.

so n

= ~

~,

odd less h with which is than L.

Now

+qn=qn,<~L qn+<~qn+<

+~n,r~1~~ (~4)

+~n+2~~n+< "~n+I ~n+>,<

since I is odd. that

+ q~ so n q~_

= ; +,,

ALGEBRAIC MORPHOGENESIS AN 2161 PHYLLOTAXIS MODEL OF

N° I I :

Lemma B.2.

l~.

W~~(0 strip )-lattice, and let Let be in have W~~ point be Then in b~

any a we a

),

A(~

If

necessarily

equal W~~ then

L l

I is

(q~, to q~ + + q~, q~, ; e ; ; . one

+,

~ ~

~

At

B(

(Al B()

of

the

vectors ; ;

only if equal if third and

and the is

to one never

L~qk,r+qk+I i

W,o).

If

(W~,

A(~ W~~ necessarily of

equal is L

the

to vectors q,, . one

~

Proof.

following

define the L FiRST Let

sets cAsE q,. us : m

l~~qk,r~qk+< "Wnm(°)nRX i'~+qk,r+qk+l~ ii n +

[n,n+q~,~+q~+, n~ =W~~(o)nRx (85) -Ii

=W~~(o)nRx [n-q~~-q~~,+I,n]. n_

also define We

Al

A±

Wnm

± =

Bi

B±

Wnm

± =

Bi) (Ai

C± (86)

Wnm

± ~

(because similarly

It is that fl f~ I). and One W~~ b~

L

to

easy q~;~ e q~~, + see « ~

C± Finally, B±, situation fl figure A±, is illustrated 83. notice the in checks that

that

e ; we

B(~.

simply A(~

B$J substitute

A$J

and defined

in the define

and

we way same as we can

definition. in the 12± to bL

Fig. 83. belong points

the white The hatched and

J2. do not to zone

PHYSIQUE JOURNAL 2162 11 N° DE I

J2+ translating by by

that be obtained

the

observe Now vector

can

b~~ ~_, +~~

implies

B,I also W~~ Lemma

as

'

~qk,, ~L

qk +

+

o o

I ~qi,, ~L

qk+

+ (~~)

o o

obtain we

Al A$r

A+

Wnm

+

= =

Bl B$r

(88)

B+

Wnm

+

= =

translating

by it the origin by inverting and by with the is obtained fl_

respect to

b~~

_,

~~ +

W~~. Thus vector

A$j

B_

=

B$j

(89) A_

=

A(~.

We interested in considered

Two be are may cases

(Ail'

Z ti

n A$ni #

fl+,

£l_

In this since fl

have

U lj~

case, we

~ =

A$j

(A+,

A(~ (A$£, (B lo) B_

e =

proved. proposition is then The

A$I

Z tt (A$4, n

. =

Figure meaning Suppose using 84 of this relation. is odd illustrates the B.I k Lemma and

only

figure realized definition of is that if if situation of is and the it 84 the lj~, to easy see

(k 0.

odd) is and In other ~L

words

q~ qk+ +

n

n

~

~

0«L-q~,~~n~q~+,. (Bll)

k,

Similarly, for even

(B12) °~L~qk+<~~~qk,r.

A+

C+

;

[

h

h

~q

n m

B

Fig. The hatched of

84. is out EL- zone

MORPHOGENESIS 2163 PHYLLOTAXIS ALGEBRAIC MODEL OF AN 11 N°

inequalities They

only N* if also if I. These have solution and L

q~+, q~,~ e + a n ~

imply

0wL-q~+,~n+q~,~-q~+i~q~,~wL foroddk

0«L-q~~~n-q~~+q~+,~q~+,SL (B13) forevenk

C+ I-e- lj~. e

At

B(

A(~

figure (since lattice, C+

It is then 84 that and the cell in the clear generate

=

proposition by C+ point).

lattice defined of A+ B_ contains first The the is W~~ part no

proved. now

strip

L with SECOND the lattice basis It is the

lj~ construct vectors to cAsE Sqi. easy :

proposition (Fig. 85), Wio W,o (0, 1). Woj 0q, because The obvious in and is l and

~ =

periodicity W~~(0)-lattice. figure

of 85, of the the

consequence as a

y

"~qil

W~i

~~

~

~

~_~) ~l

~

~

~0

Fig. 85.

B.3.

Lemma

a~, Let

Let

. us 0, define in

~

the

nterval

~2 ~ ~

-

a

0

=

~ aj

4)

~2

~2

~ ~2

V (na and

an

uch

Then [

e

I Jo,

number

exists there

Proof.

Since

- na~ 1 . (B15)

mb~ =

PHYSIQUE 2164 JOURNAL DE N° I 11

b,

Moreover

I, by

b2 Z, construction. Vk

Then the

construct vector + a, a2

e one can =

(~).

kb~a+ka~b=k= (B16)

satisfy (B.15) n', Now m' satisfy Z and k'e let Z e

+m'b, bwn'aj lo, +k'e I[.

defined The

irrational, of number have b is since

we course

bi n'

~'

~~~~~ , , ~'~'~

~

b~

b, easily is irrational. One and checks

(~).

(n'+k'b~)a+ n'a+m'b+k'= (m'+k'a~)b= (B18)

'

~

words, proved

In other have W~~(b) that

I-e- V, and V

V.

we

e

e ~

0

The is also direct since calculation shows true, that converse

(~) =a~(~)

(a~k'+m') a

~

b~ n') k' b (b~

(B19)

+ + =

We

prove now

2.3). (Sect. Proposition 2

P~ angular if differences only if permutation and is A the

C~~i-C, I=0,1,. ,n-I

only values take two

and

( N

p p

*

e a a

,

possibly third value the and

Y=«-P

if from different 0 I. only which I is

occur or n can

Proof.

lo, Suppose angular divergence, compatible irrational and let be choose I

is P b

e we a as

~

[0, T~(b) P~ projecting strip

1[ simplify. by realization of the

We

construct to on can a

W~~(b)-lattice, associating integer projection b~_, by and the of with of the I the

proposition

directly follows B,I B.2. then from Lemmas and the W~

proposition. satisfying

permutation Conversely, conditions of P We

be the the let

want to a

~

constructing corresponding

W~~(b)-lattice. angular by consider To this

aim show that it is a

the vectors

~'

(820)

(~') b

~=

=

-fl a

2165 ALGEBRAIC MODEL OF MORPHOGENESIS I PHYLLOTAXIS AN N° I :

«b<

'

~~'d oPnnlllllltll~~~~

lo, o. i ~~~ '~'~~«a' ~~ i/« and

~°~~~~'

~~'

~~ a,

~~~ ~~ w

~ - ,

are

P ) (B21) («,

=

N~_,

since exists I, there that such that I

C~

e so =

ip

(822)

I ka

C~

+ + my

= =

I,

), (na

k, apply B.3 if V where Lemma then there

Z. We then mb Z

+ e ; n, e m can : m =

W~~(b). Jo,

exists irrational I such that V b

an e =

Co,

points integers

In relation establish the lattice and the order between to a

C,,

C~

the let

construct vector sequence us

_,, ,

~

Ci Ci

if

"

+

Ci+<~Ci (823) ~ if b

~~~'

Ci+<-Ci

if ~~~ "~'

0, By defining 1, where I I.

n

= ,

0 Co

=

C, (824)

C~ +

v~

= , +, +

0, 1,..., I, for in W~~(b). I V and also in This

vector constructs

sequence a n one =

following has the

property sequence :

).

(825)

(" C~ =

C~

example Fig. point 86).

for The is (see constitutes line in lattice It broken the

to now see a

plane, [0, this of is in fact R+. II is confined in

In the there that this broken line

part no x

2.2.I). (13) (Sect. of simply ambiguity point W~, because Hence W~~ if is written

a

l. ~~ ~~~

(826)

C~

= Ci

is clear that It then

jc~ ~j jc~~,j jc~ ~j jc, ~j

(827) o

~. ~ ~ ~ m

P~ angular. which that is proves

points C,, by Co, C~. Call S the broken defined lattice of As

line the the

consequence a

...,

properties points

when it x-axis. it is clear that S ends of the the Now show C~, cuts want to we

by [0, R+.

1]

S confined it reduction ad Let S that is in absurdum and that

prove suppose us x

[0,

1] ends the x-axis.

on over

[0,

R+ shifting by 1] of S

by obtained Let T be the broken line the restriction

to vector a x

W~~(b)-lattice 0). periodicity points. (1, lattice it follows that From the of the T connects

Figure by Since R+, it

[1, 87 shows that if S 2 S constituted T. and T ends in

cut must are x

connecting points, endway lattice S and lattice b, T and ordered

b

vectors +

cross a, a on a

Cj). R+ point, [0, II exists

(.., ET, since such that there Now Cj Cj

e C~ say x

=

(j

(828)

Cj (~

C~ +

= =

2166

JOURNAL PHYSIQUE DE I N° I1

a

y

,

,

, , -

-

", ~, a+b

~

~ ,

', ',

, , ,

, , ,

, , - ,

V2

', ', ',

q

q

_ 5 ',

', 3

', ~

, ,

,

', 4 ',

c

,

i ,

',

,

,

Ci

Vi

=

x

Co=0

Fig. 86.

y

~

tl

II

T

I l

~

I /) S

L/ I /

' "

f

/

",J II "

/ L j

If J

S

0

2

Fig. 87.

impossible Ci R+. j.

[1, which because if S therefore 2 is in the end But

I cannot # # C~ x

finally [2,

R+,

in

be used 3 and conclude that S

argument cuts ; same can x on we never so

x-axis, which is contradiction. the

a

Let

prove us

Proposition 2.3, (Sect. I). 3

N~

triple satisfies

the conditions

(a, p, n) For which e any

(829) ) (", P

=

la, (830) pi

p I

+ + max « n « «

angular permutation P~ that unique there

such exists a

C>=« C~_,=p.

Proof.

P~. define Let first existence of the We

prove us

ll'l Ill

~~~~~ ~

~ ~

2167

N° II MORPHOGENESIS PHYLLOTAXIS ALGEBRAIC OF MODEL AN

txb,

b, 0. We lo, in know interval irrational number taken the I and where is la

a,

an ~ = fl

(a;b)

I[ that by irrational Jo, B.3 such generate Lemma that there exists be an

A[~'

and b~_, W~~(b). strip this clear lattice, it is that of If consider the b

we =

B[~'

[0, P~, 1[, strip simply Fig. project 88). this the (see To obtain

segment

on a =

beginning proof Proposition

2. according described the of the method of the at to

W~~(b

)-lattice, unique, strip

of the numbers P~ notice that Now that is in

to b~

any see a _,

fully determine I, of

the which

b~ vectors one a~, n,

~'

~'

' ' '

A[ B[ ' A[ B[ (832)

= =

, ,

~2 ~2

'

A(~

equal difference is (to this, the

refer the of

demonstration to to

prove e b~ VW~~ W~~ ,

A+

B_ neither Lemma B.2 either depends only A+ C+ B_ and this but is in

b~

: or or nor

,, ,

the b~). I,

parameters a~, n, on

permutation projection angular strip From thit follows that by defined it the of the a

b~_,

fully

determined by Of and the x-axis is

numbers the b~.

course, on a~, a~ n,

C~_, C,

correspond p. b~

to

a,

= =

y

tL+fi

~b

i~_

,

I

~

0

b)

(a

point W~(b).

lattice Fig. 88. hatched contains since The

generates

no zone

Appendix C.

P)~j,

familiar operations definitions that the reader is with of the and We the assume

P)°j,

produces of P~ permutation given P~ permutation (cf. them from 3.I each

a new ; a +,

4). introduce for and Let convenience the notation us

f~ where P P P

)nj ~')~~

j°~

m = , ~

~~

i

'~ (°) '~

P

where P P ~'~°~

j~

m = , ~

easily One checks that

jnj p

~~~~ p

joy

j0) jn

~

C,I. Lemma

P)~j,

If

stationary P~ then permutation, is P~ also stationary. is

a

+, =

PHYSIQUE 2168 JOURNAL I N° DE

Proof.

P)°)~

P~ Since stationary, is P~+~ D(P~~,) there exists

P~ such that

e

i = +,.

+

simply properties consider P~+, Now of the

Pn+< D(Pn) ~

_p(n+<)(o)_p(o)(n)_p(n)

p(o)

_p ,

Lemma C.2.

P~ angular n) (a, D(P~) P~ permutation. If be p, Let stationary, is then

e an

= +,

angular. P is

,

+ ~

Proof.

integer

stationary. that is There We P P

distinct the three

put

want to

so n on cases are :

i

+ ~ ~

points

I) A~, B~#0,

B~, and We A~ between and and put

two assume n we

permutation

Cl). (Fig. P~ stationary, Since A~,

has I the be thus constructed B~ #

to

n +,

adding point removing

by subtracting 0, able transform the I, it and I be

must to to + one n

remaining again. integer

P~+, words, circle other obtain In the the

must every on : one

figure Actually,

P~+,.

of C2 be found of conditions because the

segment must on on

point

suppression point modify

0 the introduction of do and the of the and the I B~, A~

not + n

P~+,. point

neighbourhood of the the

on n

being figure

(since P~ The of C2 I) and it also I

B~

A~ segment

on n

n ~ ~

2); (Prop. differences satisfies rule of the the three

the

segment consequence, as a

B~ P~ angular. satisfies it and is A~

too,

n +,

Bn-

Bn I

n-I n

A~

A

Fig, C2, Cl. Fig.

(Fig.

stationary,

C3). possibly

2) 0 Since has

P We and I be

put

next to next to to

n n

,

+ ~

figure P~~,

only possible C4. This is of if find of the

segment

must course one on

triple I, P~ corresponds between 0 and and 0 I. Thus have the and A~ put

to B~

we n n

= =

applied algorithm angular P~ angular. descent, is n) I, of therefore (n I, the have

: we +,

(Fig. figure C5).

the C6 This time of 3) 0 I

We but put segment

not next to to next n n

satisfy P~,

This of

P~+, then has the rule three be found and the

segment to must on on

P~ again angular, by

differences, fl. conclude that is since therefore From this

+

we n

a = +,

applied

placing permutation P~

(a, ) Ao

fl in the the fl, between 0 and

+

we

n n

a a

= = =

angular algorithm of descent.

We

prove : can now

2169 MORPHOGENESIS ALGEBRAIC MODEL OF

PHYLLOTAXIS AN

II N°

n-2 n-I

n-I n

A~ n

A~~0 if I

~~ lA~

her I

B~

ot w s e

Fig. C3. Fig. C4,

Ao i

Ao

-i

-i n-

a ~ =

n

Fig. Fig, C6. C5.

Proposition 4). (Sect. 8

if stationary if permutation P~ and A only angular. is it is

Proof.

permutation angular is stationary T~(b) (consider

It clear that is realization of

any a

( )°j P~, (P~

) ) from permutation the is obtained P D b observe that P

and

construct T~

e

, , ,

+

+

~

tuming point

by ) +,(b angle of with wb). 2

T~ every an

stationary

Conversely, P~ permutation of be considered the

descent

any may as as a

permutation P~_, stationary (Lemma C,I), permutations stationary of whole is the the set

stationary P~ orientations). (with P~ angular, is descent of both

since conclude But

we » «

permutation stationary Lemma C.2 that from angular. is any

AppendixD.

obviously We proposition (Sect. 5.1). here equivalent three lemmas 9

that to prove are

D.I. Lemma

Hi'

Hi'

precisely,

angular. More

fl~, VH~ is

H~~~j.

e =

Proof.

write Let us

Ha (I ) (D ) C I

= ;

C; (D2) 0, 1, C if

2 I C~

+ q

= = j~~~ ,

C~ (D3) -1.

q =

2170 JOURNAL PHYSIQUE DE I N° 11

Hi'

following fl~, verify

belongs that the

angular show is To that and have to to we

2) only (Prop. differences take values two

i)-nj<(c,) nj<(c~

(D4) j(<)-<

+ =

nj<(o)-nj<(q-i)=-m. (D5)

I I

H~

angular,

Since exists is k, k know that there such that 0 and

N~

we + e

q ~ ~

ill

C; C I ktx

(D6)

= j~,~ =

fl where

q tx.

=

I

I

unique

k leads value of (D6) that 0

k,

of such solution

that

We to

an q + any

see ~ ~

I'

I.

satisfy (D6), k', also then if fact, k In +

(I I')

(D7) (k k') o P

« =

implies, (tx, exists fl I, that such which since there Z that

e c =

I'=I+c« k'=k+cp (D8)

I I'

[0, belong q] if

hence k' 0. k

to

cannot # cq +

+ + c =

that Now it clear is

I

j(I) (I j )

if o then k I I

+

~

= ~~~~

I

(I j (I j )

if 0 then

k I I

+ q

~ =

I

k*,

H~ is angular,

exists consider of Since the

there such that Let value *

N~ e us now m,

I*

k* satisfy equation they 0 and

the

q +

~ ~

-I*p

=o-c~= k*« i-q= (Dio) i-(« +p)

i,e.

(I*

(Dii)

i)P (k* i i)

+ « =

implies which

=k*+I*=k+I, (D12)

+I*-I k*+I

Moreover

-m=k*+I*-q=k+I-q, (D13)

Hi'

Hi'

H~,,

definitions from the write it follows fl~. If that shown

have Thus

e we we

=

Hi'(I

proof complete.

is

(a that the in 3.2.3

noticed that H~,(I

a'. But

that

p so

we

= =

D.2. Lemma

(D14) H~ (I ) )

fl~ VH~ (I

R~

e

a

=

,

Proof.

p/q)

(C~

where ilq 3.2.3 that H~(I) We have in usually C~.

write Let

seen

as

us = =

=P(«).

P

words In other

(C ) p/q1

j1I~ (I ) (I (1I~ p/q )

p/q I

q

q

= =

(D15)

(H~ (I ) )

R~ p ~

2171 ALGEBRAIC MODEL OF MORPHOGENESIS PHYLLOTAXIS N° I I AN

or

llk~(I)

Rq(ip) np(I).

# =

simply

identity Notice that used the we

i;/qi (D16) R~(i ) I

q =

Conversely

'(I

Hi

H~ (I (D17) )

R~(I

a

= =

Lemma D.3.

fl~,

of provided with law composition permutations, the usual is

group. a

Proof.

Lemma for existence of inverse element of D,I shows the each is clear the It that

group. an

H,

corresponds identity belongs fl~. and the to to

still We have that to prove

VH~

fl~ H~ fl~. H~, H~,

(D18)

e e

o ,

D.2 Now Lemma shows that

Rq(«Rq

Ha Ha ) «')) (« «'))

Rq(iRq (D19)

>(I

(<

o = =

q) q) (a', (tx (a, tx', Since q) I, and have I and then I

we

= = =

(R~(txa'),

q) (D20)

=

We have therefore that proven

(D21) fl~ H~ H~ H~,

e

o ~~~,~ =

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