Phyllotaxis and patterns on plants
Phyllotaxis and patterns on plants
Phyllotaxis and patterns on plants
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<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>Alan C. Newell, Zhiying Sun, Patrick D. ShipmanUniversity of Ariz<strong>on</strong>aMarch 23, 2009Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Patterns Are EverywhereAlan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
More Examples of Patterns <strong>on</strong> PlantsAlan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
How does leaves/flowers form <strong>on</strong> a plant?Maturesunflower headApical meristem(AM)Schematic of AMAlan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Diagrams of phylla formati<strong>on</strong>Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Diagrams of phylla formati<strong>on</strong>Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Diagrams of phylla formati<strong>on</strong>Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Plant <str<strong>on</strong>g>patterns</str<strong>on</strong>g>Challenges:Why Fib<strong>on</strong>acci?C<strong>on</strong>necti<strong>on</strong> between phyllotactic c<strong>on</strong>figurati<strong>on</strong>s <str<strong>on</strong>g>and</str<strong>on</strong>g> surfacedeformati<strong>on</strong>sTransiti<strong>on</strong>sUniversality, self similarityAre plant <str<strong>on</strong>g>patterns</str<strong>on</strong>g> seen anywhere else in the physical world orthe laboratory?Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Mechanisms <str<strong>on</strong>g>and</str<strong>on</strong>g> modelsParadigms: Hofmeister (1868),Snow <str<strong>on</strong>g>and</str<strong>on</strong>g> Snow(1950’s)Douady <str<strong>on</strong>g>and</str<strong>on</strong>g> Couder (1990’s)Mechanical stresses:Green, Steele, Dumais (1990’s)Shipman <str<strong>on</strong>g>and</str<strong>on</strong>g> Newell (2005)Biochemical agents: Reinhardt et al (2000)Meyerowitz et al (2007)Traas et al (2007)Coupled model:Growth affects stress/strainStress promotes growth.How do we c<strong>on</strong>nect paradigms <str<strong>on</strong>g>and</str<strong>on</strong>g> mechanisms?Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Paradigms <str<strong>on</strong>g>and</str<strong>on</strong>g> Mechanisms: c<strong>on</strong>necti<strong>on</strong>s?Paradigms are rules for the placement of the newest primordiumgiven the positi<strong>on</strong>s of all previous primordia.cf. packing efficiencyMechanisms involve instabilities of uniform states, the decompositi<strong>on</strong>of the phase space into active A <str<strong>on</strong>g>and</str<strong>on</strong>g> passive P(A) modes, thecoordinatizati<strong>on</strong> of A with (amplitude) order parameters, <str<strong>on</strong>g>and</str<strong>on</strong>g> orderparameter equati<strong>on</strong>s which may be gradient flows with a free energy.cf. energy minimizati<strong>on</strong>How to c<strong>on</strong>nect these ideas?Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Biochemical factorsReinhardt (2000) showed that horm<strong>on</strong>e Auxin is instrumental ingrowth <str<strong>on</strong>g>and</str<strong>on</strong>g> uneven distributi<strong>on</strong> of Auxin is the reas<strong>on</strong> for theformati<strong>on</strong> of phyllotaxisAlan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Biochemical factorsReinhardt (2000) showed that horm<strong>on</strong>e Auxin is instrumental ingrowth <str<strong>on</strong>g>and</str<strong>on</strong>g> uneven distributi<strong>on</strong> of Auxin is the reas<strong>on</strong> for theformati<strong>on</strong> of phyllotaxisMeyerowitz, Traas, Reinhardt et al (2006): Auxin efflux proteinPIN1 <strong>on</strong> the cell wall can redistribute themselves so as to pumpAuxin form low c<strong>on</strong>centrati<strong>on</strong> to high c<strong>on</strong>centrati<strong>on</strong>dA(i)dt= D ∑ k(A(k) − A(i)) + T ∑ k(A(k)P(k, i) − A(i)P(i, k)) + c − dA(i)P(i, j) = P ij = PA(j)κ + ∑ (k)A(k) jA(i): Auxin c<strong>on</strong>centrati<strong>on</strong> in cell iP(i, k): PIN1 c<strong>on</strong>centrati<strong>on</strong> <strong>on</strong> cell wall (i,j) pointing to cell jAlan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Biochemical factorsReinhardt (2000) showed that horm<strong>on</strong>e Auxin is instrumental ingrowth <str<strong>on</strong>g>and</str<strong>on</strong>g> uneven distributi<strong>on</strong> of Auxin is the reas<strong>on</strong> for theformati<strong>on</strong> of phyllotaxisMeyerowitz, Traas, Reinhardt et al (2006): Auxin efflux proteinPIN1 <strong>on</strong> the cell wall can redistribute themselves so as to pumpAuxin form low c<strong>on</strong>centrati<strong>on</strong> to high c<strong>on</strong>centrati<strong>on</strong>dA(i)dt= D ∑ k(A(k) − A(i)) + T ∑ k(A(k)P(k, i) − A(i)P(i, k)) + c − dA(i)P(i, j) = P ij = PA(j)κ + ∑ (k)A(k) jA(i): Auxin c<strong>on</strong>centrati<strong>on</strong> in cell iP(i, k): PIN1 c<strong>on</strong>centrati<strong>on</strong> <strong>on</strong> cell wall (i,j) pointing to cell jThe c<strong>on</strong>tinuum limit PDE:g t = −Lg −H ▽ 2 g −▽ 4 g − ¯κ 1 ▽(g ▽g)− ¯κ 2 ▽(▽g ▽ 2 g)−δ ′ g 3g(r, α): auxin fluctuati<strong>on</strong> around stati<strong>on</strong>ary uniform c<strong>on</strong>centrati<strong>on</strong>∼ growth strainAlan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Mechanical FactorsFvKD Equati<strong>on</strong>(Describing deformati<strong>on</strong> of thin elastic shells):ζw t + D∆ 2 w + N∆ χw + 1R α∆ ρf − [f , w] + κw + γw 3 = 01Eh ∆2 f − 1R α∆ ρw + 1 [w, w] = 02w(r, α): out-of-surface deformati<strong>on</strong>. f (r, α): in-plane stress.Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Mechanical FactorsFvKD Equati<strong>on</strong>(Describing deformati<strong>on</strong> of thin elastic shells):ζw t + D∆ 2 w + N∆ χw + 1R α∆ ρf − [f , w] + κw + γw 3 = 01Eh ∆2 f − 1R α∆ ρw + 1 [w, w] = 02w(r, α): out-of-surface deformati<strong>on</strong>. f (r, α): in-plane stress.Combined ModelAlan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Equati<strong>on</strong>s for coupled modelw(r, α, t)f (r, α, t)g(r, α, t)Surface normal deformati<strong>on</strong>Airy Stress tensorGrowth/fluctuati<strong>on</strong> auxin c<strong>on</strong>centrati<strong>on</strong>ζw t + D∆ 2 w + N∆ χ w + 1R α∆ ρ f − [f , w] + κw + γw 3 = 01Eh ∆2 f − 1R α∆ ρ w + 1 2[w, w]+∆g = 0g t = −Lg − H ▽ 2 g − ▽ 4 g − ¯κ 1 ▽ (g ▽ g) − ¯κ 2 ▽ (▽g ▽ 2 g) −δg 3 + β∆fAlan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Patterns arising from the governing equati<strong>on</strong>sBoth mechanisms share the same structures in the equati<strong>on</strong>s.∂w∂t= −(∇ 2 + P) 2 w + ɛw + N.L..In 1-dimensi<strong>on</strong>al, The linear growth rate for e ikx+σt isσ(k) = (−k 2 + P) 2 + ɛ.In 2-dimensi<strong>on</strong>al, ∇ 2 = ∂2 + 1 ∂∂r 2 r ∂r + 1 ∂ 2The linear growth rate forr 2 ∂α 2e ilα+imx+σt is σ(l, m) = −(−l 2 − m2 + P) 2 + ɛ.r 2Triad: ⃗ k m = (l m , m), ⃗ k m + ⃗ k n = ⃗ k m+n ⇒ l m + l n = l m+n , m + n = (m + n)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Amplitude Equati<strong>on</strong>s 1. Representati<strong>on</strong>w(r, α, t) = ∑ A s m(r, t)e is ∫ l mdr−imα + (∗)where A m (r, t)e i ∫ l mdr is WKB approximati<strong>on</strong> for Hankel functi<strong>on</strong>sH s m(kr) in the r >> 1, m/r finite limit with l 2 m = k 2 − m 2 /r 2g(r, α, t) = ∑ B s m(r, t)e is ∫ l mdr−imα + (∗)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Amplitude Equati<strong>on</strong>s 1. Equati<strong>on</strong>sLinear∂A m∂t∂+ (2il m∂r + i ∂l m+ i l m∂r r )2 A m = (ɛ − (lm 2 + m2r 2 − P)2 )A mQuadratic n<strong>on</strong>linearitye i ∫ l mdr−imα e i ∫ l ndr−inα → e i ∫ (l m+l n)dr−i(m+n)α+τ(m, n, m + n)A ∗ nA m+nCubic Saturati<strong>on</strong> (e.g. elastic foundati<strong>on</strong> resp<strong>on</strong>se)−γA m (|A m | 2 + ∑ δ ms |A s | 2 )Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Energy Functi<strong>on</strong>alE{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 resp<strong>on</strong>seSolve stati<strong>on</strong>ary amplitude equati<strong>on</strong>sChoose l m , l n to minimize EA j (l m , l n , r)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
OutcomeTheorem: The amplitude equati<strong>on</strong>s for ANY Fib<strong>on</strong>acci 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<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>r = r 0 + Ut − r ′∇ 2 = 1 R∂∂r ′ R ∂∂r ′ + 1 R 2 ∂ 2∂α 2 R = r 0 + ( U V − 1)r ′Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Fr<strong>on</strong>t propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
PDE Simulati<strong>on</strong>Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
OutcomeTheorem: Under R → Rφ mn , φ mn = m+nφmφ+n = 1+φ22φ≃ φ − 1/2, theamplitude equati<strong>on</strong>s of two different Fib<strong>on</strong>acci sequences areisomorphic <str<strong>on</strong>g>and</str<strong>on</strong>g> in particular,E(1, 2, 3, 5, 8, ...; R) = E(1, 3, 4, 7, ...; 1.17R)1,3,4,7,...Alan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>
Transiti<strong>on</strong>sHow are transiti<strong>on</strong>s achieved between whorls <str<strong>on</strong>g>and</str<strong>on</strong>g> Fib<strong>on</strong>accisequences <str<strong>on</strong>g>and</str<strong>on</strong>g> between different Fib<strong>on</strong>acci sequences?e.g. 2,2,4 → 2,3,5e.g. 1,2,3,5, 8 → 1,3,4,7,..cf. Eckhaus, skew-varicose instabilitiesAlan C. Newell, Zhiying Sun, Patrick D. Shipman<str<strong>on</strong>g>Phyllotaxis</str<strong>on</strong>g> <str<strong>on</strong>g>and</str<strong>on</strong>g> <str<strong>on</strong>g>patterns</str<strong>on</strong>g> <strong>on</strong> <strong>plants</strong>