Book contents
- Frontmatter
- Contents
- Acknowledgments
- Prologue
- Part I Pattern recognition
- Part II Pattern generation: a key to the puzzles
- Part III Origins of phyllotactic patterns
- Part IV Complements
- Appendixes
- 1 Glossary
- 2 Answers to problems
- 3 Questions
- 4 General properties of phyllotactic lattices
- 5 The Williams–Brittain model
- 6 Interpretation of Fujita's frequency diagrams in phyllotaxis
- 7 L-systems, Perron–Frobenius theory, and the growth of filamentous organisms
- 8 The Meinhardt–Gierer theory of pre-pattern formation
- 9 Hyperbolic transformations of the cylindrical lattice
- Bibliography
- Author index
- Subject index
2 - Answers to problems
Published online by Cambridge University Press: 27 April 2010
- Frontmatter
- Contents
- Acknowledgments
- Prologue
- Part I Pattern recognition
- Part II Pattern generation: a key to the puzzles
- Part III Origins of phyllotactic patterns
- Part IV Complements
- Appendixes
- 1 Glossary
- 2 Answers to problems
- 3 Questions
- 4 General properties of phyllotactic lattices
- 5 The Williams–Brittain model
- 6 Interpretation of Fujita's frequency diagrams in phyllotaxis
- 7 L-systems, Perron–Frobenius theory, and the growth of filamentous organisms
- 8 The Meinhardt–Gierer theory of pre-pattern formation
- 9 Hyperbolic transformations of the cylindrical lattice
- Bibliography
- Author index
- Subject index
Summary
Chapter 1
Problem 1.1
In pineapples we may find families of 3, 5, 8, 13, and 21 spirals, so that the visible opposed pairs are (3,5), (8,5), (8,13), (21,13). Similar observations can be made on cones, except that the consecutive Fibonacci numbers generally obtained are smaller, as we have seen in the text.
Problem 1.2
1. Figure 1.3(1) shows the visible opposed spiral pair (7, 11).
2. Given a system (7,11), in the family containing 7 spirals, the numbers on adjacent primordia on a spiral must differ by 7, while the numbers on two consecutive primordia on any of the 11 spirals must differ by 11. So the thing to do is to choose a primordium near the rim of the specimen, let us say the one we marked with a dot in its center, and to put the number 1 on it. This primordium is at the junction of two opposed contact spirals, one in each family. The primordium with two dots in it is primordium #8 and the one with three dots is #12. Then we have everything we need to put a number on every other primordium, by applying the Bravais–Bravais theorem. For example, the primordium adjacent to #1, #8, and #12 is #19; the one on the right of #12 is #5; the one above #5 is #16; the one on the left of #16 is #23; and on the right of #16 is #9; on the right of #9 is #2, etc.
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- PhyllotaxisA Systemic Study in Plant Morphogenesis, pp. 290 - 298Publisher: Cambridge University PressPrint publication year: 1994