This paper reports on a Matlab program that represents asymmetric cell division and generates the nth row of the Fibonacci tree. Asymmetric cell division with a lag by newborn cells before continuous division and with lateral self-association in one dimension can be represented over unit cell-cycle time by classic Fibonacci trees.

1. FIBONACCI PHYLLOTAXIS BY ASYMMETRIC CELL DIVISION:
ZECKENDORF AND WYTHOFF TREES
Colin Paul Spears, Marjorie Bicknell-Johnson, and John J. Yan
Abstract. This paper reports on a Matlab program that represents asym-
metric cell division and generates the nth row of the Fibonacci tree. Asymmetric
cell division with a lag by newborn cells before continuous division and with lat-
eral self-association in one dimension can be represented over unit cell-cycle time
by classic Fibonacci trees. Both Wythoff and Zeckendorf forms of the classic
Fibonacci tree are explored for identifiers of Horizontal Para-Fibonacci (HPF,
cell Age), Zeckendorf (Z, cell generation), and Vertical Para-Fibonacci (VPF,
cousinship) sequences [17: A0335612, A007895, A003603] as well as Wythoff
pairs for modeling two- and three-dimensional displays. Routines were written
to evaluate displays up to F25 = 75,025 and higher.
Rectangular and helical displays of Fn populations parsed Fm demonstrate
regular Fibonacci phyllotaxis and floret spiral formation with uniform self-
association by Age. Generation Z clusters occur with the Age motif (1, 2,
3) as potential centers of nodal growth. Sequence VPF relates successive sets
of newborn cells by sister and first cousin relationships. The resulting pat-
terns may be mined for explanations of the appearance of Fibonacci numbers in
plant morphogenesis with broadening of patterns to include linear streaks and
symmetric groupings.
1. Introduction
Asymmetric binary cell division provides a rational basis for Fibonacci series
and other recursive phyllotaxic patterning in biologic structures [19]. Recogni-
tion in the 1990s that regular asymmetry in cell division is commonly observed
in biology led to discussion in 2002 that this may explain the helical spirals in
pine cones, with random occurrence of dexter vs. sinister arrangements [12].
This was questioned [3] with several active models for Fibonacci phyllotaxis
such as diffusion of growth factors [8, 18] and biophysical considerations of min-
imal energy surfaces [4, 5, 14]. This suggests a need for a unifying mechanism
for leaf/stem and floret/seedhead phyllotaxis.
Asymmetric division in plants is well known; for example, in Arabidopsis in
the polarization of the initial cell division of the zygote, and subsequently in the
progressive polarization of PIN1 auxin transport protein expression [16]. Models
of phyllotaxy address either the classic aspects of rotation angles of branch
appearance about a stem, or of spiral parastichies of end-organ structures. The
latter include a wide range of species and anatomy, such as the soreses, or
syncarps of fused fruits of pineapples, the bracts and inflorescences of conifers
in male and female pine cones, and the disc floret rays of the pseudoanthem
capitula of Compositae (Asteraceae) such as Helianthus species. However, our
present model addresses cylindrical structures more immediately applicable to
1

2. the former, classic phyllotaxy. The latter may relate more to conical shoot
apical meristem shapes.
We conjectured in 1998 [19] that identifiers of asymmetrical cell growth using
the most basic assignments of age and generation, and when evaluated modulo
a Fibonacci number, would lead to macroscopic structures with Fibonacci num-
bers apparent as the dominant theme. The only biological assumption to be
made is that cells can associate by age after production. Initial cylindrical
models were promising.
Therefore, a program for graphical display of this approach relevant to the
many thousands of cells present in early plant growth such as the subapical
procambial meristem was carried out using Matlab.
2. The Fibonacci Case
Consider the simplest case of asymmetric binary cell division, shown as a
classic (Wythoff-type) Fibonacci tree. The first generation parent cell, shown
as an open circle at the top of the tree, is non-dividing for one cell cycle, then
continuously produces second generation cells thereafter, shown as triangles.
After one cycle, mature cells ready for reproduction are shown filled. The second
and subsequent generations likewise show a one cell cycle lag prior to dividing
continuously as stem cells. The third generation newborn daughter cell is shown
as an open square; fourth generation, a diamond; and fifth generation, a star.
0 
1 
2  
3   
4     
5        
6             
7                    
8                                  ✩
w 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
Age (9) 1 2 3 1 4 1 2 5 1 2 3 1 6 1 2 3 1 4 1 2 7 1 2 3 1 4 1 2 5 1 2 3 1
GN 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0
A B A A B A B A A B A A B A B A A B A B A A B A A B A B A A B A A B
Figure 1: Classic Fibonacci Tree. Filled symbols represent mature cells; open,
immature cells. Generations tracked by , , , ✩, successively. Ancestral
lines trace cells. A(w) counts cells in each column.
If A represents an adult and B represents a newborn, one way to state
the rule of formation of the Fibonacci tree is to replace A by AB and B by
A, taking care to distinguish generations. The tree results in the Fibonacci
Golden Necklace (or Fibonacci Infinite Word) sequence [17]: A005614]: GN →
101101011011010110 . . . , when newborn and dividing (stem) cells are repre-
sented as 0 and 1 respectively. The age can be obtained by tracing an element
in the chart back along its branch to its newborn status. Successive ages A(w):
1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, . . . give the horizontal para-Fibonacci
sequence (HPF) [17: A035612]; note that row n ends with cell number Fn+1 ,
and the nth generation contains Fn+1 cells.
2

3. The cells in the classic Fibonacci tree above have been numbered to illustrate
Wythoff properties of the array [17]. In the chart, the newborns (open symbols),
have Age 1 and appear in cells w = 2, 5, 7, 10, 13, 15, . . . , 34, the Wythoff
numbers bk , k = 1, 2, . . . , 13; successive bk occur above the symbol B in the
ABAAB. . . row. Row r has Fr entries and Fr−2 newborns (the Fr−2 1s). The
Wythoff pairs (ak , bk ) include the two sequences A000201 and A001950 [17], √
ak : 1, 3, 4, 6, . . . , [kα] and bk : 2, 5, 7, 10, . . . , [kα2 ], α = (1 + 5)/2. Also,
ak + k = bk ; each ak is the first integer not yet listed. In the row (n – 1) of the
Fibonacci tree, the Fn entries contain Fn−2 1s in the bk positions with the other
Fn−1 entries in the ak positions. There are Fn−3 2s; Fn−4 3s; Fn−5 4s; . . . ; all
falling into patterned positions governed by ak numbers. For example, in the n
= 8 row, the F9 = 34 entries contain F6 = 8 twos, 5 threes, 3 fours, 2 fives, 1 six,
1 seven, and 1 nine, or 8 + 5 + 3 + 2+ 1 + 1 + 1 = 21 = F8 numbers falling
into cells numbered ak , and, of course, F7 = 13 ones corresponding to cells
numbered bk . Note that 1s appear below cells numbered by odd-subscripted
Fibonacci numbers. Figure 1 is numbered to show the properties of newborns;
the 1s always occur under a Wythoff-numbered cell bk . For reference, we list
the first ten Wythoff pairs (an , bn ):
(1, 2); (3, 5); (4, 7); (8, 13); (9, 15); (11, 18); (12, 20); (14, 23); (16, 26).
By renumbering Figure 1, Wythoff pairs can be read from the Fibonacci
tree. Since bk – 1 = an for n = ak from [6, 7], we let t = w – 1 in the classic
Fibonacci Tree and connect adjacent root branches, those given by the same
symbol in the tree and found one row farther back. Then, for example, adjacent
pairs (1, 2), (3, 5), (8, 13), (21, 34), . . . , are Wythoff pairs (ak , bk ) such that
1 + 2 = 3; 3+ 5 = 8; . . . ; am + bm = ak for k = bm . Take an open symbol in
the nth row at position t. Then the Wythoff pairs from adjacent root branches
are given by iterated subscripts as (at , bt ); (ap , bp ) for p = bt ; (ar , br ) for r =
bp and so on. Furthermore, the cells connected in Figure 1a are successive rows
of the Wythoff array [17: A035513] shown below for reference:
k-1 a(k)
0 1 1 2 3 5 8 13 21 34 55 89
1 3 4 7 11 18 29 47 76 123 199 322
2 4 6 10 16 26 42 68 110 178 288 466
3 6 9 15 24 39 63 102 165 267 432 699
4 8 12 20 32 52 84 136 220 356 576 932
5 9 14 23 37 60 97 157 254 411 665 1076
6 11 17 28 45 73 118 191 309 500 809 1309
7 12 19 31 50 81 131 212 343 555 898 1453
Wythoff Array
Each row k in the Wythoff array is a Fibonacci sequence with first two terms
(k - 1) and ak given in the two left columns. In each row, pairs of adjacent
columns give Wythoff pairs with ak in an odd numbered column and bk in the
adjacent even numbered column. Among many other properties [11, 15], each
column in the Wythoff array contains integers whose Zeckendorf representations
end in the same Fibonacci number. Compare the rows with the Fibonacci tree
of Figure 1a:
3

4. 0 
1 
2  
8,13
3   
4     
3,5 11,18 16,26
5        
6             
1,2 4,7 6,10 9,15 12,20 14,23
7                    
8                                  ✩
t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
HPF 1 2 3 1 4 1 2 5 1 2 3 1 6 1 2 3 1 4 1 2 7 1 2 3 1 4 1 2 5 1 2 3 1
VPF 1 1 1 2 1 3 2 1 4 3 2 5 1 6 4 3 7 2 8 5 1 9 6 4 10 3 11 7 2 12 8 5 13
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
Figure 1a: HPF gives Age. Wythoff pairs, i. e., 16,26, at root branches.
VPF counts newborns of successive generations.
The Age sequence HPF gives the column in the Wythoff array in which t
first appears. The vertical para-Fibonacci sequence VPF [17: A003603] connects
successive sets of newborn cells by sequential first-cousinship across generations
from the top down in the Fibonacci tree and gives the row in the Wythoff array
in which t appears. The first term in a new row in the Wythoff array is the
smallest integer that has not yet appeared in an earlier row. Here, VPF counts
first appearances of newborns as represented by open symbols in the nth row.
For example, in row 8, the first appearance of 4 is for t = 9 and that is under
the 4th open symbol (newborn) in row 8. If the symbol in row 8 is filled as in t
= 16, for example, the open symbol above 16 appears as the 3rd open symbol in
row 6, so VPF(16) = 3. The first open symbol above 20 is the 5th open symbol
in the 7th row; VPF(20) = 5. We note that VPF(Fk ) = 1. In the spirit of a
family tree structure, Wythoff pairs that are Fibonacci numbers denote (an , bn )
as sequential sisters. Other Wythoff pairs are all first cousins, distinguished by
sequence of birth order.
Earlier, we gave population sums of newborn and stem cells by generation
by a binomial c-column repeat spreadsheet [19]. The total number of cells of a
tree population cell cycle n is given below, where lag time c ≥ 2 and d is age
with d = 1 are newborns and up to d = c dividing (stem) cells:
n/c c
n−k(c−1)−d−1
Gn = Gn−1 + Gn−c = k
k=0 d=1
For the Fibonacci case c = 2, the inner summation gives the number of
newborn (d = 1) vs. stem cells (d = 2). If we write columns of Pascal’s triangle
in left-justified form with a drop of one for each successive column, the sums of
rows give successive Fibonacci numbers as the generational sums, and the c =
2 column-repeats break down each generation into ages:
1
11
111
4

5. 1121
11321
114331
1154631
1 1 6 5 10 6 4 1
...
For example, in row 7 with sum 21, the 6 first generation cells contain 1
newborn and 5 adults; the 10 second generation cells, 4 newborns and 6 adults;
the 4 third generation cells, 3 newborns and 1 adult. This study gives a nice
population count but cannot follow ancestral lines.
To track specific ages and generations at all cell-cycle times n, asymmetric
binary cell division is represented as a left-adjusted expansion of a cell popula-
tion as shown by the Fibonacci tree in Zeckendorf form in Figure 1. Newborns
are represented by empty symbols and are given age 1. Generations are denoted
by sequential symbols as in Figure 1. The count number of cell t is the subscript
of the largest Fibonacci number used in the Zeckendorf representation of t and
is obtained by counting back to the left edge of the tree from the symbol in cell
t along an ancestral line. Row 9 has 13 eights; 21, 22, . . . , 33 each need F8 = 21
in the Zeckendorf representation. Rows 3 through 8 illustrate the relationship
between cell number t and Age (HPF sequence):
3: 3, 1
4: 4, 1, 2
5: 5, 1, 2, 3, 1
6: 6, 1, 2, 3, 1, 4, 1, 2
7: 7, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1
8: 8, 1, 2, 3, 1, 4, 1, 2, 5, 1, 2, 3, 1, 6, 1, 2, 3, 1, 4, 1, 2
Notice that successive rows repeat the two preceding rows except for the
first term. Row n becomes: n, followed by row (n – 1) without its first term;
followed by row (n – 2). This information appears in the row labeled ‘count’
and gives the number of cells needed to trace the symbol in cell t in the nth row
back to the left edge along its ancestral line; in other words, the subscript of the
largest Fibonacci number used in the Zeckendorf representation of t. The 7s, for
example, are the eight numbers 13, 14, . . . , 20 whose Zeckendorf representations
begin with F7 . Note that t = Fk appears beneath a triangle.
The left-justified Fibonacci tree of Figure 2 is numbered to illustrate its
Zeckendorf properties. The generation sequence Z(t): 1, 1, 1, 2, 1, 2, 2, 2, 1, 2,
2, 2, 3, . . . is also the number of open symbols in the line tracing cell t back
to the left column along the most direct line; alternately, Z(t) is given from the
generation symbol appearing in cell t: triangle, 1; square, 2; diamond, 3; star, 4.
Sequence Z(t) is A007895 [17], the number of terms in the Zeckendorf represen-
tation of t; that is, the number of terms used when t is written as the (minimal)
sum of non-consecutive distinct Fibonacci numbers 1, 2, 3, 5, 8, 13, . . . The
VPF sequence lists successive founders’ ages per generation by tracking lineage
through top-branch newborns (open symbols) across generations. Also, VPF
becomes apparent by connecting successive sets of newborn cells by generation
5

6. 0 
1 
2  
3   
4     
5        
6             
7                    
8                                 ✩
t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
HPF (9) 1 2 3 1 4 1 2 5 1 2 3 1 6 1 2 3 1 4 1 2 7 1 2 3 1 4 1 2 5 1 2 3 1
Z(t) (0) 1 1 1 2 1 2 2 1 2 2 2 3 1 2 2 2 3 2 3 3 1 2 2 2 3 2 3 3 2 3 3 3 4
VPF (1) 1 1 1 2 1 3 2 1 4 3 2 5 1 6 4 3 7 2 8 5 1 9 6 4 10 3 11 7 2 12 8 5 13
count 1 2 3 4 4 5 5 5 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8
Figure 2: Fibonacci Tree in Zeckendorf form. HPF: Age of cell t.
Z(t): number of terms, Zeckendorf minimal representation of t
Z(t) labels generations. On chart: , 1; , 2; , 3; ✩, 4.
as in Figure 1a. Note that successive Wythoff pairs appear at root branches;
i.e., (16, 26).
Since each row in the Fibonacci tree contains Fk entries, the list of entries
soon becomes large. Arranging these large rows into Fm columns would make
the cell numbers in each column congruent modulo Fm and should line up
Wythoff positions of the 1s, for example, making several runs of 1s appear in
the columns. Return to numbering the columns 1, 2, . . . , Fn , as in the classic
Wythoff-form Fibonacci tree (Figure 1) and notice that, for n odd, Fn = bj
where j = Fn−2 so that the age 1 appears Fn−2 times; in particular, 1 appears
in the Fn cell. The numbers bk have the nice property that any number N
which can be represented by a sum of distinct Fibonacci numbers containing
2 equals some bi [6, 7] and thus will number a cell containing a one. The
column numbered Fn will contain cells which contain 1 and whose cell numbers
are multiples of Fn , making ones appear in the columns labeled 2 and 5, for
example. Consider odd subscripts:
F2k+1 – 1 = (F2k + F2k−2 + . . . + 21) + 8 + 3 + 1 = M + 8 + 3 + 1
F2k+1 = M + 8 + 3 + 2 = bj where j = F2k−1
F2k+1 + 1 = M + 8 + 3 + 2 + 1 = aw for some w
F2k+1 + 2 = M + 8 + 5 + 2 = bv for some v
F2k+1 + 5 = M + 13 + 3+ 2 = bv+1
A similar argument follows for multiples of F2k+1 :
2F2k+1 = F2k+1 + M + 8 + 3 + 2
3F2k+1 = 2F2k+1 + M + 8 + 3 + 2 = F2k+2 + F2k−1 + M + 8 + 3 + 2
4F2k+1 = F2k+1 + 3 F2k+1 = F2k+2 + F2k+1 + F2k−1 + M + 8 + 3 + 2
Of course, F2k+1 must be large enough to enjoy a good run of 1s. For 13,
ones remain in the right column through 11(F7 ) = 143, but 12(F7 ) = 156 ends
in a 1 so 156 = aw for some w. (See the table in the next section.) When Fn
cells are divided into Fm columns, the number of rows is either a Lucas number
or one less than a Lucas number with a Fibonacci number of terms left over in
6

7. the last row. When Fn /Fm , m < n, m, n positive, the quotient rounds off to a
Lucas number, and the remainder is a Fibonacci number or its negative. (See
[2]).
We hoped to capture runs of a given number to study the structure of asym-
metric binary cell division with a graphical representation. The powerful parse
command from the programming language Matlab answers all of these objec-
tives.
3. MATLAB Methods and Results
Matlab 7.1 was chosen for ease in programming the mathematical algorithm
and then conveying results in a graphics context. The numbers are generated
in the following process. An initial seed generation is created with parameters
of identity, age, and readiness to reproduce. The age is initialized as 1, as with
every newborn; with each generation, the age is increasingly incremented by 1.
The readiness to reproduce is initialized as 0; after one generation has elapsed,
the readiness to reproduce counter is changed to 1, signifying that the seed/cell
is ready to reproduce.
Results appear in matrix arrays and then in 3-D spiral graphs coded by
color, providing striking patterns in a helical presentation and in agreement
with physical models made previously up to F13 = 233 in which color-coded
beads were used to represent generation and age. Figure 3 shows a sample
spiral display. The mother cell is at the bottom, sequential ages and generations
wrapping clockwise and upward; generations are shown on the left, age on the
right, for row 13 with F13 = 233 entries, arranged in F7 = 13 columns using the
parse command. (Note: The row number need not equal the parse number.)
Tables 1a and 1b give data for Figure 4. In Table 1b, notice the grouping
of cells by generation into clusters, and, since the table represents an open
cylinder, values in the rightmost column can wrap around to join values in the
left column. Clusters of Generation Z values, the boxed 2 x 2, 2 x 3, and 3
x 3 areas, occur in phyllotaxic arrangement. Within these Z clusters, the Age
(HPF) motif (1, 2, 3) appears to be invariant in all samples examined.
7

8. 0 1 1 1 2 1 2 2 1 2 2 2 3
1 2 2 2 3 2 3 3 1 2 2 2 3
2 3 3 2 3 3 3 4 1 2 2 2 3
2 3 3 2 3 3 3 4 2 3 3 3 4
3 4 4 1 2 2 2 3 2 3 3 2 3
3 3 4 2 3 3 3 4 3 4 4 2 3
3 3 4 3 4 4 3 4 4 4 5 1 2
2 2 3 2 3 3 2 3 3 3 4 2 3
3 3 4 3 4 4 2 3 3 3 4 3 4
4 3 4 4 4 5 2 3 3 3 4 3 4
4 3 4 4 4 5 3 4 4 4 5 4 5
5 1 2 2 2 3 2 3 3 2 3 3 3
4 2 3 3 3 4 3 4 4 2 3 3 3
4 3 4 4 3 4 4 4 5 2 3 3 3
4 3 4 4 3 4 4 4 5 3 4 4 4
5 4 5 5 2 3 3 3 4 3 4 4 3
4 4 4 5 3 4 4 4 5 4 5 5 3
4 4 4 5 4 5 5 4 5 5 5 6 1
Table 1a: Pattern ID
Generation (Z) numbers associated with F13 = 233
Zeckendorf (Z) generation groupings shown
The structure of the Age array in Table 1b is characteristic of all our age
arrays, Fn (parsed Fm ). Cells self-associate by age, with the longest runs for
newborn 1s. The oldest (mother cell), age n = 13 in the upper left-hand corner,
is followed by age n – 2 = 11 as the next oldest. There is one cell 13; one cell
11; one cell 10; 2 cells 9; and 3 cells 8. The newborn cells, the 1s listed in bold
type, are clustered into several long runs, as are the 2s and the 3s, for example.
13 1 2 3 1 4 1 2 5 1 2 3 1
6 1 2 3 1 4 1 2 7 1 2 3 1
4 1 2 5 1 2 3 1 8 1 2 3 1
4 1 2 5 1 2 3 1 6 1 2 3 1
4 1 2 9 1 2 3 1 4 1 2 5 1
2 3 1 6 1 2 3 1 4 1 2 7 1
2 3 1 4 1 2 5 1 2 3 1 10 1
2 3 1 4 1 2 5 1 2 3 1 6 1
2 3 1 4 1 2 7 1 2 3 1 4 1
2 5 1 2 3 1 8 1 2 3 1 4 1
2 5 1 2 3 1 6 1 2 3 1 4 1
2 11 1 2 3 1 4 1 2 5 1 2 3
1 6 1 2 3 1 4 1 2 7 1 2 3
1 4 1 2 5 1 2 3 1 8 1 2 3
1 4 1 2 5 1 2 3 1 6 1 2 3
1 4 1 2 9 1 2 3 1 4 1 2 5
1 2 3 1 6 1 2 3 1 4 1 2 7
1 2 3 1 4 1 2 5 1 2 3 1 0
Table 1b: Pattern Age
Age sequence (HPF) for F13 = 233. Boxes are generation (Z)
groupings from Pattern ID given in Table 1a
Figure 4 shows a rectangular age display in gray tones of F20 (parsed F10 ),
or a population of 6765 cells parsed into 55 columns. The oldest or mother cell
is at (0,0). Cells are wrapped sequentially clockwise from above.
One can visualize the cylinder represented in Figure 3 as circular closure
of the row abscissas. The floret phyllotaxy number is given by the number of
diagonals intersecting the ordinate, with the appearance of 3, 5, and 8 diagonals
in alternate directions. Figure 5 shows the same data as Figure 4 but compressed
and rotated counterclockwise 90 degrees. The nodes are foreshortened; the
evident phyllotaxis now manifests as 8, 13, and 21 floret rays. (The rotate and
foreshorten 3-D rotate function of Mathlab caused ordinate row number display
8

9. changes.)
Analysis of larger trees produces floret rays with increased Fibonacci num-
bers, the maximum floret ray number being Fm−1 . The greatest floret ray num-
ber is associated with diagonals through the newborn cells. This phenomenon
may contribute to explaining the observation that the calathid, or specialized
capitulum, of Asteraceae as in Helianthus, typically shows increasing phyllotaxic
spiral ray number with growth, or rising phyllotaxis.
4. Discussion
Our Matlab 7.1 routine has provided us with a tool for exploring phyllotaxis
patterns that result from an asymmetrically dividing tree Fn modulo Fm to
answer the question of whether self-association of cells by age and generation
leads to Fibonacci spirals. Experimentally, cells are arrayed into age-nodes, ar-
ranged in alternate Fibonacci spirals with maximum size Fm−1 , the appearance
of parastichy dependent on Fn because of aspects of optical angles. Increase in
9

10. parastichy number with increased population size is well appreciated in work
going back to Adler and Turing [12].
The Matlab program allows us to explore results of parsing Fn (mod N) for
N a natural number and a non-Fibonacci number, with no apparent examples
that have the degree of age organization and phyllotaxic patterning as those
using Fm (or Lm ). In general, the further that the N used for a parsing number
is from a Fibonacci number, the greater the variance from Fibonacci phyllotaxis.
While the Fibonacci-like sequences {Hn }, Hn = Hn−1 +Hn−2 , have the prop-
erty that {Hn } is congruent to a sequence made of the original sequence and
negatives of those values: Hn ≡± Hr (mod Hk ), those subsequences are actu-
ally remainders of the divisor when Hn /Hk for only the Fibonacci and Lucas
sequences [20]. A simple rhythm of asymmetric binary cell division and age-
related self-association of cells during growth may be the only variables needed
to explain the occurrence of these classic number sequences in plant morphol-
ogy. The clustering of cells by generation Z occurring in symmetric blocks with
Age motif (1, 2, 3) shown in Table 1b is an attractive model for explanation of
macroscopic sites of leaf budding and branch formation. Parsing Fn by VPF
numbers leads to linear Fibonacci folding planes, and blocks of the Age (HPF)
sequence containing Wythoff pairs as well (as shown in Table 2). Age-dependent
mechanisms, which thus include Generational identifiers for regular asymmet-
ric cell division, are recognized as factors for regulation of plant growth from
vegetative to reproductive development [16, p.43].
0 1 1 1 2 1 3 2 1 4 3 2 5
1 6 4 3 7 2 8 5 1 9 6 4 10
3 11 7 2 12 8 5 13 1 14 9 6 15
4 16 10 3 17 11 7 18 2 19 12 8 20
5 21 13 1 22 14 9 23 6 24 15 11 25
16 10 26 3 27 17 11 28 7 29 18 2 30
19 12 31 8 32 20 5 33 21 13 34 1 35
22 14 36 9 37 23 6 38 24 15 39 4 40
25 16 41 10 42 26 3 43 27 17 44 11 45
28 7 46 29 18 47 2 48 30 19 49 12 50
31 8 51 32 20 52 5 53 33 21 54 13 55
34 1 56 35 22 57 14 58 36 9 59 37 23
60 6 61 38 24 62 15 63 39 4 64 40 25
65 16 66 41 10 67 42 26 68 3 69 43 27
70 17 71 44 11 72 45 28 73 7 74 46 29
75 18 76 47 2 77 48 30 78 19 79 49 12
80 50 31 81 8 82 51 32 83 20 84 52 5
85 53 33 86 21 87 54 13 88 55 34 89 1
Table 2: F13 = 233, parse F7 = 13
Vertical Para-Fibonacci Numbers (VPF) with Wythoff Pairs in Bold
Blocks of Wythoff pairs coincide with Age (HPF) grouping in Tables 1a, 1b
It was suggested recently that any phyllotaxis mechanism must include some
asymmetric component that cannot be explained by hypotheses of contact paras-
tiches, inhibitory fields, available space, pressure waves, and transport of key
growth hormones such as auxin [13]. Our model is a discrete, deterministic
approach, somewhat reminiscent of the cellular automata of Wolfram, but not
anticipated by L-systems (see [9] p.606). It may provide hypotheses to answer
the challenge of Meinhardt ([9], p. 730) that there has been no developmental
system capable of de novo [Fibonacci] pattern formation, and may provide ex-
planations for the occurrence of bijugate (i.e., double Fibonacci: 2, 4, 6, 10, 16,
10

11. . . . ), Lucas, and so-called exotic accessory phyllotaxic patterning (see Vakarelov
in [9] p. 213), that could occur by changing the conditions of initial growth and
cell cycle lag. While our rectangular arrays might evoke Coxeter-type lattice
models on a cylinder, our model does not include optimal squashing (see Dixon
in [9], p. 313) with addition of new growth from the perimeter, so that the
rising phyllotaxis of increasing contact parastichy pair numbers is not the result
of axial compression with mechanical buckling [5], [14]. Rather, our model is
one of growth from within, embedded in all regions at once. Likewise, our model
differs from packing and entropic models of spiral disk phyllotaxis (see Douady
and Couder, p. 539 and van der Linden, p. 487 in [9]).
Zeckendorf notation in asymmetric trees has been used by Kappraff [10], but
is not the generational identifier Z of this paper. Interesting comparisons of the
Wythoff Fibonacci form tree to Farey tree structures could be noted (see Jean
and Douady in [9]), but our rectangular (opened cylinder) arrays describe the
nth level of growth with cells linearly adjacent to one other, in keeping with the
fact that plant cells divide with a shared cell wall.
A limitation of the model, as presented, is a lack of mechanism for mainte-
nance of a fixed cylindrical diameter while accommodating side-to-side, pericli-
nal growth to achieve anticlinal, vertical growth. Cylindrical growth is central
to a vast variety of plant structures such as stems, stalks, scapes, corolla tubes,
and the stele. As known [16, p. 54], periclinal divisions in the vascular tissues
create the outer pericycle and the inner core of the xylem and phloem, in radial
axis formation.
Flexibility in cell wall structures occurs by symplastic growth with the Ex-
pansins class of proteins in which procambial elements can achieve extraordinary
length. A well-known example of plasticity relates to the fusiform initials of the
cambian meristem that undergo pseudotransverse, diagonal, and anticlinal cell
division followed by intrusive tip growth of daughter initials [1]. Development
of our Matlab programs by use of vector transforms for space-filling assignment
of such growth elements is a central challenge for future study.
Acknowledgement. The authors are indebted to the anonymous referee
whose review added significantly to the quality of the manuscript.
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AMS Classification Numbers: 92C15, 11B65, 11B39
4851 Oak Vista Drive, Carmichael, CA 95608
E-mail address: cpspears@aol.com
665 Fairlane Avenue, Santa Clara, CA 95051
E-mail address: marjohnson@mac.com
Dept. Engineering, UC Davis, Davis, CA 95616
E-mail address: jjyanca@gmail.com
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