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# Algorithm - Fibonacci Phyllotaxis by Asymmetric Cell Division

Patras 0707 2008 actual presentation + lindenmeyer

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### Algorithm - Fibonacci Phyllotaxis by Asymmetric Cell Division

1. 1. Colin Paul Spears Carmichael, California Marjorie Bicknell - Johnson Santa Clara, California John J. Yan Electrical and Computer Engineering University of California at Davis FIBONACCI PHYLLOTAXIS BY ASYMMETRIC CELL DIVISION: ZECKENDORF AND WYTHOFF TREES
2. 2. Cell Division is Asymmetric Parent & Daughter (Left) vs. Parent-Daughters-Granddaughters
3. 3. Asymmetric Cell Division: Binomial Identities for Age Analysis of Mortal vs. Immortal trees Applications of Fibonacci Numbers 7:377-391, 1998
4. 5. Age 7 1 2 3 1 4 1 2 5 1 2 3 1 Stem 1 0 1 1 0 1 0 1 1 0 1 1 0 Gen 0 1 1 1 2 1 2 2 1 2 2 2 3
5. 6. Fibonacci Age and Generation Necklaces F 13 = 233 Parse 21
6. 7. Fibonacci Age and Generation F 13 = 233 Parse F 11 = 89
7. 8. MATLAB Program <ul><li>Graphics display by processing number arrays in a loop for each cell parameter identifier, Age or Generation </li></ul><ul><li>Rectangular display by parse Number </li></ul><ul><li>Spiral display by 2 x Pi /parse number with assignment of x = sin(t), y = cos(t), z = t. </li></ul><ul><li>Sequential symbol and color assignments to Age and Generation by default Matab choices. </li></ul>
8. 9. MATLAB Program: Flow Diagram
9. 10. Matlab Cylindrical Wrap, F 14 = 377 Generation and Age Parse 13 (F 7 )
10. 11. Matlab Age F 20 = 6765 Parse F 10 (55)
11. 12. Age Array F 20 = 6765 Parse 55 Dominant Parastichies: 8, 13, 21
12. 13. Age (HPF) Array F 23 = 28657 Parse F 10 Dominant Parastichies: 13, 21, 34
13. 14. Structure of F n parse F m <ul><li>Number of Nodes = F n /L m Length of Nodes = L m = F m-1 + F m+1 </li></ul><ul><li>Oldest cell in a node alternates left-right in descending age </li></ul><ul><li>Newborn 1s occur in L m-2 runs </li></ul><ul><li>Cells associate by Age </li></ul><ul><li>Generations cluster in regular relationship to age, with the early Age 1,2,3,1 Age motif </li></ul>
14. 15. Left-adjusted Fibonacci Tree (Zeckendorf Form)
15. 16. Wythoff – Zeckendorf Array
16. 17. Classic Fibonacci Tree (Wythoff Form)
17. 18. Classic Fibonacci Tree (Wythoff Form) renumbered
18. 19. Wythoff Tree 1231 Motif with Wythoff Pairs N Age (HPF) Infinite Word Wythoff Pair VPF Age (HPF) WP N-1 Renumbered
19. 20. Classic Fibonacci Tree (Wythoff Form) renumbered with VPF Wythoff Pairs
20. 21. <ul><li>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 [5, 6] 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: </li></ul><ul><li>F2k+1 – 1 = (F2k + F2k-2 + … + 21) + 8 + 3 + 1 = M + 8 + 3 + 1 </li></ul><ul><li>F2k+1 = M + 8 + 3 + 2 = bj where j = F2k-1 </li></ul><ul><li>F2k+1 + 1 = M + 8 + 3 + 2 + 1 = aw for some w </li></ul><ul><li>F2k+1 + 2 = M + 8 + 5 + 2 = bv for some v </li></ul><ul><li>F2k+1 + 5 = M + 13 + 3+ 2 = bv+1 </li></ul><ul><li>A similar argument follows for multiples of F2k+1: </li></ul><ul><li>2F2k+1 = F2k+1 + M + 8 + 3 + 2 </li></ul><ul><li>3F2k+1 = 2F2k+1 + M + 8 + 3 + 2 = F2k+2 + F2k-1 + M + 8 + 3 + 2 </li></ul><ul><li>4F2k+1 = F2k+1 + 3 F2k+1 = F2k+2 + F2k+1 + F2k-1 + M + 8 + 3 + 2 </li></ul><ul><li>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. </li></ul>
21. 22. Matlab Age (HPF) F 13 = 233 Parse F 7 13-col Array
22. 23. Matlab Generation (Z) F 13 Parse F 7 13-col Array 2x3 and 3x3 Z-clusters
23. 24. Matlab Age (HPF) F 13 Parse F 7 13-col Array with 2x3 and 3x3 Z-clusters
24. 25. Matlab Age (HPF) F 13 Parse F 7 13-col Array with 2x3 and 3x3 Z-clusters and VPF Wythoff Pairs
25. 33. Green, Paul B. Calculus-based biophysical paradigms for patterning in plants. Am. J. Botany 86:1059, 1999
26. 34. AUXIN Transport by protein PIN Asymmetry: Science 312:383 and 858, May 12, 2006 Wisniewska et al; and Sieberer and Leyser