Phyllotaxis and patterns on plants

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Energy FunctionalE{A m , A m ∗, ...} = − ∑ σ m A m A ∗ m−∑ τ jpq (A ∗ j A pA q + (∗))+ γ ∑ ( 1 2 |A m| 4 + ∑ m≠sδ ms |A m | 2 |A s | 2 )Adiabatic responseSolve stationary amplitude equationsChoose l m , l n to minimize EA j (l m , l n , r)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants
OutcomeTheorem: The amplitude equations for ANY Fibonacci sequence areinvariant under R → Rφ, φ = √ 5+12, the golden number;A s j(R) → A−sj+1 (Rφ), l j(R) → −l j+1 (Rφ), m j → m j+1 (≃ m j φ)1,2,3,5,8,13,...Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants
Page 1 and 2: Phyllotaxis <stron
Page 4 and 5: More Examples of Patterns on Plants
Page 6: Diagrams of phylla formationAlan C.
Page 9 and 10: Diagrams of phylla formationAlan C.
Page 11 and 12: Mechanisms and mod
Page 13 and 14: Biochemical factorsReinhardt (2000)
Page 15 and 16: Biochemical factorsReinhardt (2000)
Page 17 and 18: Mechanical FactorsFvKD Equation(Des
Page 19 and 20: Patterns arising from the governing
Page 21: Amplitude Equations 1. EquationsLin
Page 25 and 26: PDE SimulationFront propagating out
Page 45 and 46: PDE SimulationAlan C. Newell, Zhiyi
Page 47: TransitionsHow are transitions achi

Phyllotaxis and patterns on plants

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