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.

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 F_k = F_k-2 + F_k-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:

Golden ratio: a =(1+Ö 5)/2=1.6180339…; b =(1–Ö 5)/2 = –0.6180339…; t = a ; t^–1 = –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*t^–1 ? 137.5° or D=360*(1– t^–1 ) ? 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 U_k and U_k+1 approaches Golden ratio as k become large.

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°