DEVELOPMENT OF THE FUNDAMENTAL 
THEOREM OF PHYLLOTAXIS

DMITRY WEISE


 
 

Name:Dmitry Weise, Physician (b. Moscow, U.R S.S., 1956).

Address: Belovejskaya, Complex 39, Building 2, Suite 133, 121353 Moscow, Russia.

E-mail: dweise@gol.ru

Fields of interest: Phyllotaxis, cognitive graphics, periodicity in chemical and nuclear physics, theory of music.

Publications:

Harmony of phyllotaxis, In: International Conference "Mathematics and Arts" proceedings, Moscow , 1997, pp. 218-222 (in Russian)

Principle of minimax and rise phyllotaxis (Mechanistic phyllotaxis model), In: Fourth Interdisciplinary Symmetry Congress and Exhibition of the ISIS-Symmetry, Technion, Haifa, Israel, September 13-18, 1998.

The Pythagorean approach to the problems of periodicity in chemistry and nuclear physics, In: Fourth Congress of the International Society for Theoretical Physics (ICTCP-IV), July 9-16, 2002, INJEP, Marly-le-Roi, France, p. 59.

 
 

Abstract: In the concept of the Fundamental Theorem of the Phyllotaxis integrated and extended formulas are present for Fibonacci type series beginning with any integer U1. The formulas have empirical nature. Application of these formulas results on the certain step in dense packing with additive properties typical for phyllotaxis.

 

1 FUNDAMENTAL THEOREM OF THE PHYLLOTAXIS 

The Fundamental Theorem of the Phyllotaxis (called so by Roger V. Jean) is understood as a number of the statements concerning interrelation of a divergence angle with parastichy number on families of a spiral phyllotaxis pattern. Key items of these statements are formulas that describe normal (for additive Fibonacci type series, starting with U1=1) and anomalous (for the additive series starting only with U1=2) phyllotaxis. Cases when U1=>3 were been named aberrant and problematic patterns.

1.1 Glossary

Phyllotaxis: The arrangement of leaf or floret primordia at the shoot apex, or on the stem. In 92% of the observation the contact on conspicuous parastichy pair (m, n) in phyllotactic patterns is such that m and n are consecutive terms of the Fibonacci sequence, and the divergence angle d converges rapidly toward the Fibonacci angle. When m and n do not belong to the Fibonacci sequence, they are generally consecutive terms of a Fibonacci-type sequence. This constitutes a part of the challenge of phyllotaxis.

Figure 1: Daisy florets are arranged in a pattern of sets of parastichies, 
which number 8, 13, 21 in the outer part. Along each spiral from a family of n spirals differ by n. 

F1 = 1 ; F2 = 1; á 1, 1, 2, 3, 5, 8, 13, 21ñ ; D = 222.5°; d = 137.5°. |D| + |d| = 360°

In new terms: s = 1; t = 2; U1 = s; U2 = st 1. 

Fibonacci sequence: The sequence of integers: 1, 1, 2, 3, 5, 8, 13, The kth term is built using the rule Fk = Fk-2 + Fk-2 from the initial values F1=1; F2=1.

Fibonacci-type sequence: A sequence built from the same recurrence rule as the Fibonacci sequence but starting with other initial terms. For example, Lucas sequence:

1, 3, 4, 7, 11, 18, 29, 47, ; Uk = Uk-2 + Uk-1 ; U1 , U2 integers ; n = 1, 2, 3,  

Golden ratio: a =(1+Ö 5)/2=1.6180339; b =(1Ö 5)/2 = 0.6180339; t = a ; t1 = b .

Divergence angle d: Any of the two angles at the center of a transverse section of a growing shoot tip determined by consecutively initiated primordia. (This definition differs from those, given by Roger V. Jean, a little).

Fibonacci angle: The divergence angle d =360*t1 ? 137.5° or D=360*(1 t1 ) ? 222.5°

Parastichy: Any phyllotactic spiral seen on plants, or in transverse sections of plant apices. If along a parastichy the numbers differ by n then we say that it is a n-parastichy.

Family of parastichies: Given any parastichy in the centric representation, the corresponding family is the set of parastichies with the same pitch, winding around a common pole in the same direction and going through the centers of all the primordia.
 
 

2 RESULTS - PREDECESSORS
 
Normal phyllotaxis
Anomalous phyllotaxis
first members U1 and U2 of sequence 
U1 = 1
U1 = 2
U2 = t
U2 = 2t + 1
divergence angles d
d = Fk/(Fk t + Fk-1)
d = (Uk t + Uk-1)/(2Uk t + Uk+1+ Uk-1)
limit-divergence angles d
d = 360 (t + t  1) 1
d = 360 (2 + (t + t 1 ) 1) 1

where t ³ 2 is an integer By R. V. Jean (1994, pp. 37-38) 
 
 

3 NEW RESULTS
 
Variant 1
Variant 2
U1 = s 
U2 = st + 1
U2 = st 1
divergence angles
d = (Fn-1 Un)/(U1 Un) =

= (Fn-2 + Fn-1 t)/(s (F n-2 + F n-1 t)+F n-1)

d = (Fn-1 + Un)/(U1 Un) =

= (Fn-2 + Fn-1 t)/(s (F n-2 + F n-1 t) F n-1)

limit-divergence angles
d = (1 (U2 b U1)) / (U1(U2 b U1)) =
= (t b ) / (s (t b )+1)
d = (1 + (U2 b U1)) / (U1(U2 b U1)) =
= (t b ) / (s (t b )1)

where s and t are integers 

Notice that on figures in central area the circles are not evenly distributed, but in the outer part a packing become denser. By analogy, a ratio of two consecutive Fibonacci-type numbers Uk and Uk+1 approaches Golden ratio as k become large.


a                                                                       b

Figure 2: (a) s = 3; t= 4; U1 = s = 3 ; U2 = st 1 = 11; U3 = 14; d=129.34°; D=230.66°
(b) s = 5; t= 2; U1 = s = 5 ; U2 = st + 1 = 11; U3 = 16; d = 66.89°; D=293.11°

 

a                                                                         b

Figure 3: (a) s = 4; t= 4; U1 = s = 4 ; U2 = st + 1 = 17; U3 = 21; d = 85.38°; D=274.62°
(b) s = 5; t= 3; U1 = s = 5 ; U2 = st + 1 = 16; U3 = 21; d = 68.23°; D=291.77°

References Roger, V. Jean. Phyllotaxis: A systemic study in plant morphogenesis, Cambridge University Press 1994.