Book contents
- Frontmatter
- Contents
- Acknowledgments
- Prologue
- Part I Pattern recognition
- Introduction
- 1 The centric representation
- 2 The fundamental theorem and its applications
- 3 Hierarchical control in phyllotaxis
- 4 Allometry–type model in phyllotaxis
- 5 Practical pattern recognition
- Epilogue
- Part II Pattern generation: a key to the puzzles
- Part III Origins of phyllotactic patterns
- Part IV Complements
- Appendixes
- Bibliography
- Author index
- Subject index
3 - Hierarchical control in phyllotaxis
Published online by Cambridge University Press: 27 April 2010
- Frontmatter
- Contents
- Acknowledgments
- Prologue
- Part I Pattern recognition
- Introduction
- 1 The centric representation
- 2 The fundamental theorem and its applications
- 3 Hierarchical control in phyllotaxis
- 4 Allometry–type model in phyllotaxis
- 5 Practical pattern recognition
- Epilogue
- Part II Pattern generation: a key to the puzzles
- Part III Origins of phyllotactic patterns
- Part IV Complements
- Appendixes
- Bibliography
- Author index
- Subject index
Summary
Lestiboudois–Bolle theory of duplications
The fundamental theorem (Chapter 2), in particular its special form called the Adler theorem, can be linked to early results in phyllotaxis and illustrated in the following way. Let us consider again Figure 2.3(2). In this representation of a phyllotactic pattern we noticed that the Fibonacci numbers get closer to the y–axis as we move up in the diagram, because of the value of the divergence angle. (We observed a similar phenomenon in Figure 1.4 around the line PC.) The phyllotactic fractions that can be obtained from this system are for example, 2/5, 3/8, 5/13, 8/21, corresponding to cycles of 5, 8, 13, 21 leaves, respectively.
Now for each of the cycles, let us project the points of the lattice on the X–axis. For a cycle of 8 points for example, the ordered list of points is then, from right to left, 8–3–6–1–4–7–2–5, the same as in Figure 3.1 for that cycle. In both figures, following the points in their natural order along the genetic spiral, we must go almost 3 times to the right,starting at 1, to meet 8 leaves. We can thus learn from Figure 3.1 that 3/8 is a phyllotactic fraction of the system. The numbers m and n in two consecutive cycles constitute the visible opposed parastichy pair (m,n), and the corresponding fractions are the end points of the intervals given by the Adler theorem.
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- PhyllotaxisA Systemic Study in Plant Morphogenesis, pp. 48 - 75Publisher: Cambridge University PressPrint publication year: 1994