Disk packing on logarithmic spiral lattices

Transcription

1 Disk packing on logarithmic spiral lattices Yoshikazu Yamagishi (Ryukoku Univ) (joint work with Takamichi Sushida (Hokkaido Univ)) 7-9 Jan 017, Tohoku University Disk packing Tohoku Univ 1 / 1

2 Phyllotaxis Phyllotaxis is the study of arrangement of leaves, florets, and other organs of plants. The main problem of mathematical phyllotaxis: why are the Fibonacci numbers so common in the spirals in living plants? Cylindrical model: Linear lattice L(z) = zz + Z, where Imz > 0. Centric model: logarithmic spiral lattice Λ(z) = { z j : j Z }, z = re 1θ D \ R. θ is called the divergence angle. r is called the plastochron ratio. { } Centric model: Archimedean spiral lattice je jθ 1 : j = 0, 1,.... { je Centric model: Parabolic spiral lattice jθ } 1 : j = 0, 1,.... Disk packing Tohoku Univ / 1

3 Voronoi tiling for the linear lattice Let L(z) = zz + Z be a linear lattice generated by z, where Imz > 0. The Voronoi region of the site point mz a is defined as V (mz a) = {ζ C : mz a ζ nz b ζ, nz b mz a}. The family {V (mz a)} mz a L(z) is a tiling of the plane C. An integer m > 0 is called a Voronoi parastichy number if the tile V (mz a) is edge-adjacent to the tile V (0) Disk packing Tohoku Univ 3 / 1

4 The parallellogram (0, mz a, (mz a) + (nz b), nz b) is an rectangle 0, mz a, (mz a) + (nz b), nz b) lie on a same circle The diagonals are of the same length (mz a) (nz b) = (mz a) + (nz b) (mz a)/(nz b) 1R The bifurcation diagram of Voronoi tilings consists of the half-circles (mz a)/(nz b) 1R, Im(z) > 0, for mb na = 1. Disk packing Tohoku Univ 4 / 1

5 Lemma Suppose that the tile V (0) is a hexagon and it is adjacent to V (mz a), V (nz b), V (mz a + nz b). Then a m < b n are (principal or intermediate) convergents of Re(z). Proof. We have mb na = 1, and a m < x < b n Disk packing Tohoku Univ 5 / 1

6 Circle packing along the linear lattice Consider the lattice L(z) = zz + Z. Denote by D(z, R) = {w C : w z < R}. The circle packing along the lattice L(z) is the family of disks {D(mz a, R) : mz a L(z)}, where R = min { mz a : mz a L(z) \ 0}. An integer m > 0 is called a packing parastichy number if R = mz a for some a Z. If it is the case, a/m is a (principal) convergent of Re(z). The bifurcation diagram of circle packings consists of sub-arcs of the half-circles mz a / nz b = 1, Im(z) > 0, for mb na = 1. The bifurcation diagrams for Voronoi tilings and circle packings are dual graphs to each other Disk packing Tohoku Univ 6 /

7 Logarithmic spiral lattice Consider a logarithmic spiral lattice Λ(z) = { z j : j Z }, z D \ R. Let 0 < R < 1. Suppose that the disks { D(z j, R z j ) } do not have j Z overlaps. That is, int(d(z j, R z j )) int(d(z k, R z k )) = for j k. The maximal R is given as R = min { z j z k z j + z k : j k }. An integer m > 0 is called a parastichy number of Λ(z) if a disk D(z j, R z j ) is adjacent to the disk D(z j+m, R z j+m ). Disk packing Tohoku Univ 7 / 1

8 Consider the function d(z, w) := z w z + w. Some of its properties are as follows. Boundedness: 0 d(z, w) 1. d(z, 0) = 1 for z 0. Symmetry: d(z, w) = d(w, z). Homogeneity: d(cz, cw) = cz cw cz + cw = z w z + w = d(z, w) for c 0. In particular, d(z, w) = d( 1 z, 1 w ). d(zm, 1) = d(z j+m, z j ). R = min { d(z j, z k ) : j k } = min { d(z j, 1) : j > 0 }. m > 0 is a parastichy number if R = d(z m, 1). Disk packing Tohoku Univ 8 / 1

9 Theorem d(z, w) = z w z + w is a distance function in the plane C. Proof. We will show that d(z, 1) + d(w, 1) d(zw, 1). (d(zw, w) + d(w, 1) = d(z, 1) + d(w, 1) d(zw, 1)) Case 1. If 0 < z 1 w, z 1 w 1 + z + 1 w + 1 z 1 zw z z 1 + zw z zw 1 = + z + 1 zw + z zw + 1 zw + 1. Case. If 0 < w 1 z, exchange z, w in Case 1. Disk packing Tohoku Univ 9 / 1

10 Case 3. If z, w 1, so z 1 z 1 w 1 w 1, z 1 w 1 z 1 + zw z + z + 1 w + 1 zw + 1 zw z 1 z = z 1 + w 1 ( z + 1)( zw + 1) ( w + 1)( zw + 1) ( ) 1 z 1 ( w 1) w 1 ( z 1) 0, zw + 1 z + 1 w + 1 z 1 w 1 z 1 + zw z zw 1 + z + 1 w + 1 zw + 1 zw + 1. z 1 Case 4. If z, w 1, z 1 w 1, exchange z, w in Case 3. w 1 Case 5. If 0 < z, w 1, d(z, 1) + d(w, 1) = d(1, 1 z ) + d(1, 1 w ) d(1, 1 ) = d(zw, 1). zw Disk packing Tohoku Univ 10 / 1

11 Generalizations of the metric d : p-relative metric, Li(1998), Hästö(00) z w ( z p + w p ) 1/p, (1 p + ) Triangular ratio metric, Klén-Lindén-Vuorinen-Wang(014). sup ζ G z w z ζ + w ζ (z, w G) Disk packing Tohoku Univ 11 / 1

12 Golé (007) The equidistance curve { of d is a real algebraic } curve of degree 4, {z : d(z, 1) = c} = (1+ce it ) : 0 t < π, called Pascal s limaçon. 1 c Let ν(z) = z 1 z +1, z C. We have ν(z) = 1 if z R := {z R : z 0}, ν(z) < 1 if z C \ R, and ν(z) = d(z, 1). Let µ(w) = (1+w) 1 w, w D. Disk packing Tohoku Univ 1 / 1

13 Lemma The function µ = (1+w) is a homeomorphism of the unit disk D onto the 1 w region C \ R, with the inverse function ν = z 1 z +1. Proof. Let w = ν(z) = z 1 z +1, where z C \ R. We have µ(w) = (1 + w) ( z z 1) z z + z z + z = 1 w ( z + 1) = z 1 z + z + z = z, since z 1 = (z 1)( z 1). Let z = µ(w) = (1+w) 1 w, where w D. We have ν(z) = z 1 z + 1 = (1 + w) (1 w ) 1 + w + 1 w = w + w + w w + w + w since 1 + w = (1 + w)(1 + w). = w, Disk packing Tohoku Univ 13 / 1

14 van Iterson s function σ(z 1, z ) = r 1 r cos θ 1 θ, r 1 + r where z j = r j e iθ j, r j > 0, π < θ j π, j = 1,. We have σ(z 1, z ) = 1 d(z 1, z ). The bifurcation locus σ(z m, 1) = σ(z n, 1) is written as a variable-separated form cos nθ = φ m,n(r), φ m,n (r) := (1 + r n m n )r 1 + r m. cos mθ The function φ m,n (r) is monotone increasing on 0 r 1, with φ m,n (0) = 0, φ m,n (1) = 1. Disk packing Tohoku Univ 14 / 1

15 Rothen-Koch (1989) Let z = re 1θ. If mθ π 0 < r < 1. If mθ π nθ π, then d(zm, 1) > d(z n, 1) for any nθ < π, then there exists a (unique) 0 < r 0 < 1 such that d(z m, 1) = d(z n, 1) if r = r 0, d(z m, 1) > d(z n, 1) if 0 < r < r 0, and d(z m, 1) < d(z n, 1) if r 0 < r 1. Proof. If mθ nθ π π, we have cos nθ mθ / cos 1 > φ m,n (r) for any 0 < r < 1, which implies that σ(z m, 1) < σ(z n, 1), and hence d(z m, 1) > d(z n, 1) for any 0 < r < 1. If mθ nθ π < π, we have 0 < cos nθ mθ / cos < 1. By the Intermediate Value Theorem, there exists 0 < r 0 < 1 such that φ m,n (r 0 ) = cos nθ mθ / cos, which implies that σ(z m, 1) = σ(z n, 1) for r = r 0. The uniqueness of r 0 follow from the monotonicity of φ m,n. Disk packing Tohoku Univ 15 / 1

16 Rothen-Koch (1989) Let z = re 1θ. Let m > 0 be a disk parastichy number for the spiral lattice Λ(z). Let a = [[ mθ mθ π ]] be the integer part of π. Then a m is a θ (principal) convergent of π. Proof. If a m is not a principal convergent of θ π, then there exists 0 < n < m such that nθ mθ π π. This implies that d(z n, 1) < d(z m, 1). m is not a disk parastichy number. G. van Iterson ( ). Bifurcation diagram of disk packing (1907). Disk packing Tohoku Univ 16 / 1

17 Bifurcation diagram of disk packing Disk packing Tohoku Univ 17 / 1

18 Voronoi tiling Logarithmic spiral lattice Λ(z) = { z j : j Z }. The Voronoi region of z j is defined as V j = { ζ C : ζ z j ζ z k, k } The family V = {V j } j is a tiling of the punctured plane C. m > 0 is called a Voronoi parastichy number if V 0, V m are (edge-)adjacent. Y-Sushida-Hizume, Nonlinearity 8 (015) Disk packing Tohoku Univ 18 / 1

19 Lemma (diagonals of the limaçon rectangle ) The points 1, z 1, z 1 z, z lie on a same circle in this order of vertices d(z 1, z ) = d(z 1 z, 1) If this is the case, we have d(z 1, z ) = d(z 1 z, 1) > d(z 1, 1), d(z, 1) z 1 z 1 z α 1 z Disk packing Tohoku Univ 19 / 1

20 Proof. The quadrilateral (1, z 1, z 1 z, z ) is inscribed in a circle in this order of vertices. Let α be the crossing point of the diagonals l(1, z 1 z ) and l(z 1, z ) (l(z, w) denotes the line segment with the endpoints z, w). Since the triangle (1, z 1, α) is similar to (z, z 1 z, α), we have z α 1 α = z 1z α z 1 α = z z 1 z 1 z 1 = z. The triangle (1, z, α) is similar to (z 1, z 1 z, α), so we have Thus we obtain z 1 α 1 α = z 1z α z α 1 α = z 1 α z 1 By the triangular inequality, we have = z 1 z 1 z 1 z = z α z = z 1. = z 1z 1 α, z 1 z d(z 1, 1) = z 1 1 z < z 1 α + α 1 z = z 1 α + α z = z 1 z z 1 + z z 1 + z = d(z 1, z ). Disk packing Tohoku Univ 0 / 1

21 Lemma The points 1, z m, z m+n, z n lie on a same circle in this order of vertices d(z m, z n ) = d(z m+n, 1) ψ m,n (θ) = ϕ m,n (r) ( cos mθ π ψ m,n (θ) := nθ π ) π cos ( mθ π + nθ π ) π, ϕ m,n(r) := r m + r n 1 + r m+n. If this is the case, d(z m, z n ) = d(z m+n, 1) > d(z m, 1), d(z n, 1) Disk packing Tohoku Univ 1 / 1

22 The bifurcation locus for Voronoi tilings g a m, b n r = g a m, b (θ) = ϕ 1 m,n (ψ m,n (θ)). n Y-Sushida-Hizume(015) : [ a m, b n ] [0, 1], Let z = re 1θ D \ R. Let ( a m, b n ) θ π, mb na = 1. If 0 < r = g a m, b (θ) < 1, V has Voronoi parastichy numbers m, n. n V has Voronoi parastichy numbers m, m + n, n if either a m < θ π < a+b m+n, g a m, (θ) < r < g b a (θ), n θ π = a+b m, a+b m+n m+n, g a m, (θ) < r < 1, or b n a+b m+n < θ π < b n, g a m, (θ) < r < g b a+b n m+n, b n (θ). Disk packing Tohoku Univ / 1

23 Bifurcation diagram of Voronoi tiling Disk packing Tohoku Univ 3 / 1

24 Voronoi spiral tiling Y-Sushida-Hizume, Nonlinearity 8 (015) Disk packing Tohoku Univ 4 / 1

25 Disk packing Tohoku Univ 5 / 1

26 Theorem A disk parastichy number is also a Voronoi parastichy number. Van Iterson s diagram of disk packing is a dual graph to the bifurcation diagram of Voronoi tiling. This shows that van Iterson s diagram has a tree structure, and it is connected and simply connected. Disk packing Tohoku Univ 6 / 1

27 Thank you! Disk packing Tohoku Univ 7 / 1