Spears, C.P., Bicknell-Johnson, M., and Yan, J.J. Fibonacci Phyllotaxis by Asymmetric Cell Division: Zeckendorf and Wythoff Trees.
Congressus Numerantium 201: 257-272, 2009.
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. Both Wythoff and Zeckendorf forms of the classic Fibonacci tree are explored for identifiers of Horizontal Para-Fibonacci (HPF, cell Age), Zeckendorf (Z, cell gener-ation), and Vertical Para-Fibonacci (VPF) cousinship sequences [15: A0335612, A007895, A003603] as well as Wythoff pairs 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 demonstrate regular Fibonacci phyllotaxis and floret formation with uniform self-association by Age. Generation Z clusters occur with the Age motif as potential centers of nodal growth. Sequence VPF relates successive sets of newborn cells by sister and first cousin relationships. The resulting patterns can be mined for explanations of the appearance of Fibonacci numbers in plant morphogenesis, with broadening of patterns to include linear streaks and symmetric groupings.