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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.

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- 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. Asymmetriccell 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 timeby classic Fibonacci trees. Both Wythoﬀ and Zeckendorf forms of the classicFibonacci 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 Wythoﬀpairs for modeling two- and three-dimensional displays. Routines were writtento evaluate displays up to F25 = 75,025 and higher. Rectangular and helical displays of Fn populations parsed Fm demonstrateregular 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 setsof newborn cells by sister and ﬁrst cousin relationships. The resulting pat-terns may be mined for explanations of the appearance of Fibonacci numbers inplant morphogenesis with broadening of patterns to include linear streaks andsymmetric groupings. 1. Introduction Asymmetric binary cell division provides a rational basis for Fibonacci seriesand other recursive phyllotaxic patterning in biologic structures [19]. Recogni-tion in the 1990s that regular asymmetry in cell division is commonly observedin biology led to discussion in 2002 that this may explain the helical spirals inpine cones, with random occurrence of dexter vs. sinister arrangements [12].This was questioned [3] with several active models for Fibonacci phyllotaxissuch 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 mechanismfor leaf/stem and ﬂoret/seedhead phyllotaxis. Asymmetric division in plants is well known; for example, in Arabidopsis inthe polarization of the initial cell division of the zygote, and subsequently in theprogressive polarization of PIN1 auxin transport protein expression [16]. Modelsof phyllotaxy address either the classic aspects of rotation angles of branchappearance about a stem, or of spiral parastichies of end-organ structures. Thelatter include a wide range of species and anatomy, such as the soreses, orsyncarps of fused fruits of pineapples, the bracts and inﬂorescences of conifersin male and female pine cones, and the disc ﬂoret rays of the pseudoanthemcapitula of Compositae (Asteraceae) such as Helianthus species. However, ourpresent model addresses cylindrical structures more immediately applicable to 1
- 2. the former, classic phyllotaxy. The latter may relate more to conical shootapical meristem shapes. We conjectured in 1998 [19] that identiﬁers of asymmetrical cell growth usingthe most basic assignments of age and generation, and when evaluated moduloa Fibonacci number, would lead to macroscopic structures with Fibonacci num-bers apparent as the dominant theme. The only biological assumption to bemade is that cells can associate by age after production. Initial cylindricalmodels were promising. Therefore, a program for graphical display of this approach relevant to themany thousands of cells present in early plant growth such as the subapicalprocambial meristem was carried out using Matlab. 2. The Fibonacci Case Consider the simplest case of asymmetric binary cell division, shown as aclassic (Wythoﬀ-type) Fibonacci tree. The ﬁrst generation parent cell, shownas an open circle at the top of the tree, is non-dividing for one cell cycle, thencontinuously produces second generation cells thereafter, shown as triangles.After one cycle, mature cells ready for reproduction are shown ﬁlled. The secondand subsequent generations likewise show a one cell cycle lag prior to dividingcontinuously as stem cells. The third generation newborn daughter cell is shownas 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 statethe rule of formation of the Fibonacci tree is to replace A by AB and B byA, taking care to distinguish generations. The tree results in the FibonacciGolden 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 elementin 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-Fibonaccisequence (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 illustrateWythoﬀ 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 Wythoﬀnumbers bk , k = 1, 2, . . . , 13; successive bk occur above the symbol B in theABAAB. . . row. Row r has Fr entries and Fr−2 newborns (the Fr−2 1s). TheWythoﬀ 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 theFibonacci tree, the Fn entries contain Fn−2 1s in the bk positions with the otherFn−1 entries in the ak positions. There are Fn−3 2s; Fn−4 3s; Fn−5 4s; . . . ; allfalling 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 fallinginto cells numbered ak , and, of course, F7 = 13 ones corresponding to cellsnumbered bk . Note that 1s appear below cells numbered by odd-subscriptedFibonacci numbers. Figure 1 is numbered to show the properties of newborns;the 1s always occur under a Wythoﬀ-numbered cell bk . For reference, we listthe 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 Fibonaccitree. Since bk – 1 = an for n = ak from [6, 7], we let t = w – 1 in the classicFibonacci Tree and connect adjacent root branches, those given by the samesymbol in the tree and found one row farther back. Then, for example, adjacentpairs (1, 2), (3, 5), (8, 13), (21, 34), . . . , are Wythoﬀ pairs (ak , bk ) such that1 + 2 = 3; 3+ 5 = 8; . . . ; am + bm = ak for k = bm . Take an open symbol inthe nth row at position t. Then the Wythoﬀ pairs from adjacent root branchesare 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 rowsof 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 adjacentcolumns give Wythoﬀ pairs with ak in an odd numbered column and bk in theadjacent even numbered column. Among many other properties [11, 15], eachcolumn in the Wythoﬀ array contains integers whose Zeckendorf representationsend in the same Fibonacci number. Compare the rows with the Fibonacci treeof 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 tﬁrst appears. The vertical para-Fibonacci sequence VPF [17: A003603] connectssuccessive sets of newborn cells by sequential ﬁrst-cousinship across generationsfrom the top down in the Fibonacci tree and gives the row in the Wythoﬀ arrayin which t appears. The ﬁrst term in a new row in the Wythoﬀ array is thesmallest integer that has not yet appeared in an earlier row. Here, VPF countsﬁrst 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 underthe 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 inrow 6, so VPF(16) = 3. The ﬁrst open symbol above 20 is the 5th open symbolin the 7th row; VPF(20) = 5. We note that VPF(Fk ) = 1. In the spirit of afamily tree structure, Wythoﬀ pairs that are Fibonacci numbers denote (an , bn )as sequential sisters. Other Wythoﬀ pairs are all ﬁrst cousins, distinguished bysequence of birth order. Earlier, we gave population sums of newborn and stem cells by generationby a binomial c-column repeat spreadsheet [19]. The total number of cells of atree population cell cycle n is given below, where lag time c ≥ 2 and d is agewith 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 ofnewborn (d = 1) vs. stem cells (d = 2). If we write columns of Pascal’s trianglein left-justiﬁed form with a drop of one for each successive column, the sums ofrows 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 1newborn 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 nicepopulation count but cannot follow ancestral lines. To track speciﬁc ages and generations at all cell-cycle times n, asymmetricbinary 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. Newbornsare represented by empty symbols and are given age 1. Generations are denotedby sequential symbols as in Figure 1. The count number of cell t is the subscriptof the largest Fibonacci number used in the Zeckendorf representation of t andis obtained by counting back to the left edge of the tree from the symbol in cellt along an ancestral line. Row 9 has 13 eights; 21, 22, . . . , 33 each need F8 = 21in the Zeckendorf representation. Rows 3 through 8 illustrate the relationshipbetween 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 theﬁrst 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 rowback to the left edge along its ancestral line; in other words, the subscript of thelargest Fibonacci number used in the Zeckendorf representation of t. The 7s, forexample, are the eight numbers 13, 14, . . . , 20 whose Zeckendorf representationsbegin with F7 . Note that t = Fk appears beneath a triangle. The left-justiﬁed Fibonacci tree of Figure 2 is numbered to illustrate itsZeckendorf 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 backto the left column along the most direct line; alternately, Z(t) is given from thegeneration 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, . . . TheVPF sequence lists successive founders’ ages per generation by tracking lineagethrough top-branch newborns (open symbols) across generations. Also, VPFbecomes 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 entriessoon becomes large. Arranging these large rows into Fm columns would makethe cell numbers in each column congruent modulo Fm and should line upWythoﬀ positions of the 1s, for example, making several runs of 1s appear inthe columns. Return to numbering the columns 1, 2, . . . , Fn , as in the classicWythoﬀ-form Fibonacci tree (Figure 1) and notice that, for n odd, Fn = bjwhere j = Fn−2 so that the age 1 appears Fn−2 times; in particular, 1 appearsin the Fn cell. The numbers bk have the nice property that any number Nwhich can be represented by a sum of distinct Fibonacci numbers containing2 equals some bi [6, 7] and thus will number a cell containing a one. Thecolumn numbered Fn will contain cells which contain 1 and whose cell numbersare multiples of Fn , making ones appear in the columns labeled 2 and 5, forexample. 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 endsin a 1 so 156 = aw for some w. (See the table in the next section.) When Fncells are divided into Fm columns, the number of rows is either a Lucas numberor 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 aLucas 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 parsecommand 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 algorithmand then conveying results in a graphics context. The numbers are generatedin the following process. An initial seed generation is created with parametersof identity, age, and readiness to reproduce. The age is initialized as 1, as withevery 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/cellis ready to reproduce. Results appear in matrix arrays and then in 3-D spiral graphs coded bycolor, providing striking patterns in a helical presentation and in agreementwith physical models made previously up to F13 = 233 in which color-codedbeads were used to represent generation and age. Figure 3 shows a samplespiral display. The mother cell is at the bottom, sequential ages and generationswrapping clockwise and upward; generations are shown on the left, age on theright, for row 13 with F13 = 233 entries, arranged in F7 = 13 columns using theparse 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 groupingof cells by generation into clusters, and, since the table represents an opencylinder, values in the rightmost column can wrap around to join values in theleft column. Clusters of Generation Z values, the boxed 2 x 2, 2 x 3, and 3x 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 agearrays, Fn (parsed Fm ). Cells self-associate by age, with the longest runs fornewborn 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 cell11; one cell 10; 2 cells 9; and 3 cells 8. The newborn cells, the 1s listed in boldtype, 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 cellis at (0,0). Cells are wrapped sequentially clockwise from above. One can visualize the cylinder represented in Figure 3 as circular closureof the row abscissas. The ﬂoret phyllotaxy number is given by the number ofdiagonals intersecting the ordinate, with the appearance of 3, 5, and 8 diagonalsin alternate directions. Figure 5 shows the same data as Figure 4 but compressedand rotated counterclockwise 90 degrees. The nodes are foreshortened; theevident phyllotaxis now manifests as 8, 13, and 21 ﬂoret rays. (The rotate andforeshorten 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 phenomenonmay contribute to explaining the observation that the calathid, or specializedcapitulum, of Asteraceae as in Helianthus, typically shows increasing phyllotaxicspiral ray number with growth, or rising phyllotaxis. 4. Discussion Our Matlab 7.1 routine has provided us with a tool for exploring phyllotaxispatterns that result from an asymmetrically dividing tree Fn modulo Fm toanswer the question of whether self-association of cells by age and generationleads to Fibonacci spirals. Experimentally, cells are arrayed into age-nodes, ar-ranged in alternate Fibonacci spirals with maximum size Fm−1 , the appearanceof parastichy dependent on Fn because of aspects of optical angles. Increase in 9
- 10. parastichy number with increased population size is well appreciated in workgoing back to Adler and Turing [12]. The Matlab program allows us to explore results of parsing Fn (mod N) forN a natural number and a non-Fibonacci number, with no apparent examplesthat have the degree of age organization and phyllotaxic patterning as thoseusing Fm (or Lm ). In general, the further that the N used for a parsing numberis 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 andnegatives of those values: Hn ≡± Hr (mod Hk ), those subsequences are actu-ally remainders of the divisor when Hn /Hk for only the Fibonacci and Lucassequences [20]. A simple rhythm of asymmetric binary cell division and age-related self-association of cells during growth may be the only variables neededto explain the occurrence of these classic number sequences in plant morphol-ogy. The clustering of cells by generation Z occurring in symmetric blocks withAge motif (1, 2, 3) shown in Table 1b is an attractive model for explanation ofmacroscopic sites of leaf budding and branch formation. Parsing Fn by VPFnumbers leads to linear Fibonacci folding planes, and blocks of the Age (HPF)sequence containing Wythoﬀ pairs as well (as shown in Table 2). Age-dependentmechanisms, which thus include Generational identiﬁers for regular asymmet-ric cell division, are recognized as factors for regulation of plant growth fromvegetative 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 someasymmetric component that cannot be explained by hypotheses of contact paras-tiches, inhibitory ﬁelds, available space, pressure waves, and transport of keygrowth hormones such as auxin [13]. Our model is a discrete, deterministicapproach, somewhat reminiscent of the cellular automata of Wolfram, but notanticipated by L-systems (see [9] p.606). It may provide hypotheses to answerthe challenge of Meinhardt ([9], p. 730) that there has been no developmentalsystem 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 Vakarelovin [9] p. 213), that could occur by changing the conditions of initial growth andcell cycle lag. While our rectangular arrays might evoke Coxeter-type latticemodels on a cylinder, our model does not include optimal squashing (see Dixonin [9], p. 313) with addition of new growth from the perimeter, so that therising phyllotaxis of increasing contact parastichy pair numbers is not the resultof axial compression with mechanical buckling [5], [14]. Rather, our model isone of growth from within, embedded in all regions at once. Likewise, our modeldiﬀers from packing and entropic models of spiral disk phyllotaxis (see Douadyand Couder, p. 539 and van der Linden, p. 487 in [9]). Zeckendorf notation in asymmetric trees has been used by Kappraﬀ [10], butis not the generational identiﬁer Z of this paper. Interesting comparisons of theWythoﬀ Fibonacci form tree to Farey tree structures could be noted (see Jeanand Douady in [9]), but our rectangular (opened cylinder) arrays describe thenth level of growth with cells linearly adjacent to one other, in keeping with thefact 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 centralto 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 tissuescreate the outer pericycle and the inner core of the xylem and phloem, in radialaxis formation. Flexibility in cell wall structures occurs by symplastic growth with the Ex-pansins class of proteins in which procambial elements can achieve extraordinarylength. A well-known example of plasticity relates to the fusiform initials of thecambian meristem that undergo pseudotransverse, diagonal, and anticlinal celldivision followed by intrusive tip growth of daughter initials [1]. Developmentof our Matlab programs by use of vector transforms for space-ﬁlling assignmentof such growth elements is a central challenge for future study. Acknowledgement. The authors are indebted to the anonymous refereewhose review added signiﬁcantly to the quality of the manuscript.References [1] C. B. Beck, An Introduction to Plant Structure and Development: Plant Anatomy for the Twenty-First Century, Cambridge University Press, New York, 2005, 168-170. [2] M. Bicknell-Johnson and C. P. Spears, Lucas Quotient Lemmas. Submit- ted to Thirteenth International Conference on Fibonacci Numbers and Their Applications. [3] A. J. Fleming, Plant Mathematics and Fibonacci’s Flowers, Nature 418 (2002), 723. 11
- 12. [4] P. B. Green, Expression of Pattern in Plants: Combining Molecular and Calculus-Based Biophysical Paradigms, American Journal of Botany 86 (1999), 1059-1076. [5] H. Hellwig, R. Engelmann, and O. Deussen, Contact Pressure Models for Spiral Phyllotaxis and their Computer Simulation, Journal of Theoretical Biology 240 (2006), 489-500. [6] V. E. Hoggatt, Jr., M. Bicknell-Johnson, and R. Sarsﬁeld, A Generaliza- tion of Wythoﬀ ’s Game, The Fibonacci Quarterly 17 (1979), 198-211. [7] A. F. Horadam, Wythoﬀ Pairs, The Fibonacci Quarterly 16 (1978), 147- 151. [8] H. J¨nsson, M. G. Heisler, B. E. Shapiro, E. M. Meyerowitz, and E. o Mjolsness, An Auxin-Driven Polarized Transport Model for Phyllotaxis, Proceedings of the National Academy of Sciences 103 (2006), 1633-1638. [9] R.V. Jean, and D. Barab´, Symmetry in Plants, Series Mathematical Bi- e ology and Medicine 4, World Scientiﬁc Publishing Co. Pte. Ltd., London, 1998.[10] J. Kappraﬀ, Growth in Plants: A Study in Numbers, Forma 19 (2004), 335-354.[11] C. Kimberling, The Zeckendorf Array Equals the Wythoﬀ Array, The Fi- bonacci Quarterly 33 (1995), 3-8.[12] A. J. S. Klar, Fibonacci’s Flowers, Nature 417 (2002), 595.[13] R. W. Korn, Anodic Asymmetry of Leaves and Flowers and its Relation- ship to Phyllotaxis, Annals of Botany (London) 97 (2006), 1011-1015.[14] D. Kwiatkowska and J. Dumais, Growth and Morphogenesis at the Vege- tative Shoot Apex of Anagallis Arvensis, Journal of Experimental Botany 54 (2003), 1585-1595.[15] W. Lang, The Wythoﬀ and the Zeckendorf Representations of Numbers are Equivalent, Applications of Fibonacci Numbers 6, Edited by G. E. Bergum et al, Kluwer Academic Publishers, Dordrecht, 1996, 321-337.[16] O. Leyser and S. Day, Chapter 4: Primary Axis Development, Mechanisms in Plant Development, Blackwell Science Ltd., Malden, MA, 2003, 48-73.[17] N. J. A. Sloane, On-line Encyclopedia of Integer Sequences, http://www.research.att.com/˜jas/sequences n[18] R. S. Smith, S. Guyomarc’h, T. Mandel, D. Reinhardt, C. Kuhlemeier, P. Prusinkiewicz, A Plausible Model of Phyllotaxis, Proceedings of The National Academy of Sciences 103 (2006), 1301-1306. 12
- 13. [19] C. P. Spears and M. Bicknell-Johnson, Asymmetric Cell Division: Bi- nomial Identities for Age Analysis of Mortal vs. Immortal Trees, Appli- cations of Fibonacci Numbers 7, Edited by G. E. Bergum et al, Kluwer Academic Publishers, Dordrecht, 1998, 377-391.[20] L. Taylor, Residues of Fibonacci-like Sequences, The Fibonacci Quarterly 5 (1967), 298-304.[21] J. Wisniewska, J. Xu, D. Seifertov´, P. B. Brewer, K. Ruzicka, I. Blilou, D. a Rouqui´, E. Benkov´, B. Scheres, J. Friml, Polar PIN Localization Directs e a Auxin Flow in Plants, Science 312 (2006), 883. 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 13

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