Spiral disk packings

Accepted Manuscript Spiral disk packings Yoshikazu Yamagishi, Takamichi Sushida PII: DOI: Reference:

S0167-2789(16)30200-7 http://dx.doi.org/10.1016/j.physd.2016.12.003 PHYSD 31872

To appear in:

Physica D

Received date: 3 May 2016 Revised date: 21 August 2016 Accepted date: 19 December 2016 Please cite this article as: Y. Yamagishi, T. Sushida, Spiral disk packings, Physica D (2016), http://dx.doi.org/10.1016/j.physd.2016.12.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Spiral disk packings Yoshikazu Yamagishia,∗, Takamichi Sushidab a Department

of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, 520-2194 Japan b Research Institute for Electronic Science, Hokkaido University, Sapporo, 060-0812 Japan

Abstract It is shown that van Iterson’s metric for disk packings, proposed in 1907 in the study of a centric model of spiral phyllotaxis, defines a bounded distance function in the plane. This metric is also related to the bifurcation of Voronoi tilings for logarithmic spiral lattices, through the continued fraction expansion of the divergence angle. The phase diagrams of disk packings and Voronoi tilings for logarithmic spirals are dual graphs to each other. This gives a rigorous proof that van Iterson’s diagram in the centric model is connected and simply connected. It is a nonlinear analog of the duality between the phase diagrams for disk packings and Voronoi tilings on the linear lattices, having the modular group symmetry. Keywords: spiral phyllotaxis, disk packing, Voronoi tiling, continued fraction 2000 MSC: 11A55, 52C15, 52C20, 92C80

1. Introduction Phyllotaxis is the study on the distribution of leaves, florets, and seeds around an axis. A historical overview is in [2], and an extensive review on recent developments is given in [13]. Among the classical works in geometric approach, van Iterson [17] studied, in 1907, the packing of disks and the shape called ‘folioids’, in the cylinder, the plane (called the centric model) and the cone. Since his work (see Figure 1), the phase diagrams of disk packings with the Farey tree structure have been reproduced by other researchers under various models and hypotheses of phyllotaxis [6, 11, 12]. The cylindrical model usually considers the linear lattices, and its geometry is well established. The parameter space has the modular group(P SL(2, Z)) symmetry [3]. In the cylindrical model, it is known that the phase diagram of disk packings is dual to the phase diagram of Voronoi tilings [9].

∗ Corresponding

author Email addresses: [email protected] (Yoshikazu Yamagishi), [email protected] (Takamichi Sushida)

Preprint submitted to Physica D

August 21, 2016

1 4

2 7

1 3

3 (4, 3) 2 8 (1, 4, 3) (3, 5) 3 5 (3, 5, 2) 7 (5, 2)

1 1 6 7 1 8

(1, 5)

(1, 4)

(1, 3)

(3, 2)

1 2

1 5

(1, 3, 2)

(1, 2) (2, 1) (2, 3, 1)

−

(2, 3) (3, 1) (2, 5) 3 (2, 5, 3) 7 (5, 3) (4, 1) (3, 4, 1) 2 1 (5, 1) − − 5 3 (3, 4) 1 8 − 8 1 1− 7 − − 2 1 −1 3 6 − − 5 7 4

Figure 1: The punctured disk D∗ =

{

z = re

√ −1θ

: 0 < r < 1, θ ∈ (−π, π]

}

is the parameter

space for the logarithmic spiral lattices Λ(z), and the corresponding spiral disk packings. The solid curves denote Van Iterson’s diagram F consisting of disk packings with two or three parastichy numbers. A curve with the label (m, n) denotes a set of parameters for disk packings with parastichy numbers m, n. A node point with the label (m, n, l) is a parameter for a disk packing with parastichy numbers√m, n, l. Dashed radial lines are for reference. A fraction a/m on the unit circle denotes e2π −1a/m .

2

1 3

3 2 8 3 5 7

1 5

(4, 3) (1, 4)

(3, 5)

(5, 2)

1 2

3 − 7 2 − 5 3 − 8

1 4

2 7

(2, 5)

1 1 6 7 1 8

(1, 5)

(1, 3)

(3, 2)

(2, 3) (3, 1)

(5, 3)

(3, 4)

(4, 1) (5, 1)

− −

1 3 −

2 7

1 1− 7 − 1 −1 6 − 5 4

1 8

Figure 2: Phase diagram of Voronoi tilings for logarithmic spiral lattices. A solid curve with the label (m, n) denotes a family of quadrilateral tilings with parastichies m, n. The complement of the solid curves corresponds to hexagonal tilings with three parastichies. See [20].

There are several choices for the spiral in the √ Iterson { centric }model. Van considered a logarithmic spiral lattice Λ(z) = z j : j ∈ Z , z = re −1θ , 0 < r < 1, θ ∈ (−π, π], having a symmetry of similarity transformations. For each site point z j ∈ Λ, j ∈ Z, he considered a disk D(z j , R|z|j ) centered at z j , with the radius R|z|j being proportional to the absolute value |z|j . In the phyllotaxis theory, 1/r is called the plastochrone ratio, and θ is called the divergence angle. See [19] for many illustrations. The GeoGebra script [7] draws spiral disk packings for any parameters z. In the previous paper [20], we studied Voronoi tilings for logarithmic spiral lattices, see Figure 2. The aim of this paper is to show that van Iterson’s diagram for disk packings in the centric model, which is not often mentioned in the recent literature, is dual to the diagram of Voronoi tilings, as is apparent in Figure 3. First we show in section 2 that the bounded function d(z, w) =

|z − w| , |z| + |w|

z, w ∈ C,

(1)

which determines the coefficient R in van Iterson’s disk packing, satisfies the axioms of distance function. This was conjectured by Christophe Gol´e in 2007, who also showed that the equidistance curve is a lima¸con of Pascal. We call d the packing distance, or the lima¸con distance, after Gol´e. Van Iterson used the 3

1 3

3 2 8 3 5 7

2 7

1 4

1 5

1 1 6 1 7 8

1 2

−

3 (a) 7 2 (b) − 5 3 − 8 −1 2 1 3− − 7 4 ●

●

1 − 1 8 1− 7 1− − 6 5

Figure 3: Phase diagrams of disk packings and Voronoi tilings, superimposed. Each partition for hexagonal Voronoi tilings (surrounded by dotted curves) is divided into three subregions, depending on the disk contact parastichy numbers. The parameters marked (a), (b) correspond to the disk packings in Figure 4.

4

function |σ(z1 , z2 )| =

√

1 − d(z1 , z2

)2 ,

√ 2 r1 r2 θ1 − θ2 cos , σ(z1 , z2 ) := r1 + r2 2

√

instead of d(z1 , z2 ), for zj = rj e −1θj , rj > 0, θj ∈ (−π, π], j = 1, 2. The function σ is variable-separated and easier to compute than d. An integer m > 0 is called a disk parastichy number for the spiral lattice Λ(z) if { } d(z m , 1) = min d(z j , 1) : j = 1, 2, . . . .

Section 3 reformulates some results by Rothen and Koch [15] on the relationship between the disk parastichy number and the continued fraction expansion of θ/2π. The Voronoi tiling for the spiral site set Λ(z) has its own notion of parastichy number, which is also related to the continued fraction expansion of θ/2π. The bifurcation of Voronoi tilings occurs when the quadrilateral (1, z m , z m+n , z n ) is inscribed in a circle [20]. Section 4 shows that the equation for the bifurcation of Voronoi tilings can also be written in terms of the packing distance function d. Moreover we show that the diagonals of the inscribed quadrilateral (1, z m , z m+n , z n ) are longer than its sides in the packing distance d. Section 5 reformulates the results in [20] that the phase diagram of Voronoi spiral tilings has a ‘semi-triangular’ decomposition with Farey tree structure. Section 6 shows that a disk parastichy number is also a Voronoi parastichy number. Finally we prove in Section 7 the binary tree structure of van Iterson’s graph F. 2. Distance function determined by van Iterson’s disk packing In this section, the function d is derived from van Iterson’s disk packing, and it is shown that d is a distance function in the plane. Denote the disk centered at z with the radius R|z| by D(z, R|z|). For distinct points z, w ∈ C∗ , the disks D(z, R|z|), D(w, R|w|) are tangent to each other if R = d(z, w), where d is defined by (1). { the function } Let Λ(z) = z j : j ∈ Z , z ∈ D∗ , be a logarithmic spiral lattice. Since the function d is invariant under complex multiplication, i.e., d(z, w) = d(cz, cw),

c ∈ C∗ ,

(2)

we have d(1, ), { z ) = d(z , z } d(1, z ) = d(1, z ), for any j, m ∈ Z. Let D(z, R) = D(z j , R|z|j ) : j ∈ Z be a family of disks. If { } R = R(z) := min d(z j , 1) : j = 1, 2, . . . , m

j

j+m

m

−m

D(z, R) is a disk packing, and D(z, R0 ) is not a packing for any R0 > R (with overlappings of disks). An integer m > 0 is called a contact parastichy number, or disk parastichy number, for the spiral lattice Λ(z) if { } d(z m , 1) = min d(z j , 1) : j = 1, 2, . . . . Figure 4 shows some examples of disk packings. 5

(a)

(b) √

Figure 4: Disk packings and Voronoi tilings for spiral lattices Λ(z), z = re −1θ . Dashed lines are the boundary of Voronoi tiles. (a) Three disk parastichy numbers 2, 3, 5, with r = 0.605237, θ = 2π · 0.730002. (b) Two disk parastichy numbers 3, 5 and quadrilateral Voronoi tiles, with r = 0.616980, θ = 2π · 0.816314. See also [7], a GeoGebra script that draws the spiral disk packing for any parameter z.

2.1. Plane distance function The following proposition was conjectured by Christophe Gol´e in 2007. Note that, the complex multiplicative symmetry (2) implies that d(z, w) = d(1/z, 1/w) for z, w ∈ C∗ . Proposition 1. The function d : C × C → R is a distance function of the plane C. Proof. It is straitforward to see that 0 ≤ d(z, w) ≤ 1, and

d(z, w) = 0 ⇔ z = w, d(z, w) = d(w, z).

We are going to show the inequality d(z, 1) + d(w, 1) ≥ d(zw, 1),

z, w ∈ C∗ ,

(3)

which implies the triangular inequality d(zw, w) + d(w, 1) = d(z, 1) + d(w, 1) ≥ d(zw, 1) for z, w 6= 0. Case 1. If 0 < |z| ≤ 1 ≤ |w|, we have |z − 1| |zw − z| |z − 1| + |zw − z| |zw − 1| |z − 1| |w − 1| + = + ≥ ≥ . |z| + 1 |w| + 1 |z| + 1 |zw| + |z| |zw| + 1 |zw| + 1 Case 2. If |z| ≥ 1 ≥ |w| > 0, we exchange z, w to apply Case 1. 6

Case 3. If |z|, |w| ≥ 1 and |z − 1|(|w|2 − 1) ≥ |w − 1|(|z|2 − 1), we have |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 and so

|z − 1| |w − 1| |z − 1| + |zw − z| |zw − 1| + ≥ ≥ . |z| + 1 |w| + 1 |zw| + 1 |zw| + 1

Case 4. If |z|, |w| ≥ 1 and |z − 1|(|w|2 − 1) ≤ |w − 1|(|z|2 − 1), we exchange z, w to apply Case 3. Case 5. If 0 < |z|, |w| ≤ 1, we use the results of Cases 3 and 4 to obtain d(z, 1) + d(w, 1) = d(1, 1/z) + d(1, 1/w) ≥ d(1, 1/zw) = d(zw, 1). The special cases of the triangular inequality d(0, z) + d(z, w) ≥ d(0, w),

and

d(z, 0) + d(0, w) ≥ d(z, w)

are obvious, as we have d(z, w) ≤ 1, and d(z, 0) = 1 whenever z 6= 0. 2.2. Pascal’s lima¸con Here we show that the equidistance curve {z ∈ C : d(z, 1) = s}, for 0 < s < 1, is a lima¸con of Pascal. This section is based on [8]. √ Let z = 1 + λe −1t , λ > 0, and suppose that 0 < s = d(z, 1) = |z−1| |z|+1 < 1. Cosine’s rule on the triangle 4(0, 1, z) is written as |z|2 = 1 + λ2 + 2λ cos t. This, together with s(1 + |z|) = λ, implies that λ= and

2s(1 + s cos t) , 1 − s2 √

(4)

(1 + se −1t )2 z = 1 + λe = . (5) 1 − s2 The parameterization (4), called √ a lima¸con of Pascal, was obtained in [8]. In the real coordinates z = x + −1y, it is an algebraic curve of degree 4 written as ((x − 1)2 + y 2 − s2 (1 + x2 + y 2 ))2 = 4s4 (x2 + y 2 ), √ −1t

or equivalently

((1 − s2 )((x − 1)2 + y 2 ) − 2s2 (x − 1))2 = 4s2 ((x − 1)2 + y 2 ). 7

2.3. Lima¸con foliation The parameterization (5) leads us to consider the complex function µ(w) =

(1 + w)2 , 1 − |w|2

Let ν(z) =

z−1 , |z| + 1

w ∈ D. z ∈ C.

It is easy to see that ν(z) = −1 if z ∈ R− := {z ∈ R : z ≤ 0}, |ν(z)| < 1 if z ∈ C \ R− , and |ν(z)| = d(z, 1) . The following two lemmas show that the function µ is a homeomorphism of the unit disk D onto the region C \ R− , with the inverse function ν. Lemma 2. µ(ν(z)) = z for z ∈ C \ R− , Proof. Let w =

z−1 |z|+1 ,

where z ∈ C \ R− . We have

(|z| + 1 + z − 1)2 z z¯ + 2z|z| + z 2 (1 + w)2 = = = z, 2 2 2 1 − |w| (|z| + 1) − |z − 1| 2|z| + z + z¯ since |z − 1|2 = (z − 1)(¯ z − 1). Lemma 3. ν(µ(w)) = w for w ∈ D. Proof. Let z =

(1+w)2 1−|w|2 ,

where w ∈ D. We have

z−1 (1 + w)2 − (1 − |w|2 ) 2w + w2 + ww ¯ = = = w, |z| + 1 |1 + w|2 + 1 − |w|2 2+w+w ¯ since |1 + w|2 = (1 + w)(1 + w). ¯ The family of lima¸cons {µ(w) : |w| = r}, 0 < r < 1, is a foliation of the region C \ R− with a singularity at µ(0) = 1, see Figure 5. 3. Disk parastichy and continued fractions In this section we reformulate some results by Rothen and Koch [15] that show the relationship between the parastichy numbers of disk packing and the continued fraction expansion of the divergence angle. For each integer m > 0, let S(m) be the set of z ∈ D∗ such that the spiral lattice Λ(z) has∪a disk parastichy number m. It is a closed set. We have a partition D∗ = m>0 S(m). Let F be the set of z ∈ D \ R such that the spiral lattice Λ(z) has at least two disk parastichy numbers m1 6= m2 . The set F, or F ∪ {0}, is called van Iterson’s diagram, see Figure 1. Let √ 2 r1 r2 θ1 − θ2 cos (6) σ(z1 , z2 ) = r1 + r2 2 8

y

x s = 0.6

s = 0.7

s = 0.8

s = 0.9

Figure 5: The equidistance curves of the distance function d. They are lima¸cons of Pascal.

√

for zj = rj e −1θj , rj > 0, θj ∈ (−π, π], j = 1, 2. Note that σ(z1 , z2 ) ≥ 0 if |θ1 − θ2 | ≤ π, and that √ |σ(z1 , z2 )| = 1 − d(z1 , z2 )2 .

Van Iterson [17] used |σ(z1 , z2 )| in his study of disk packings, instead of d(z1 , z2 ). Let m > n > 0 be integers. If z ∈ S(m) ∩ S(n), we have σ(z m , 1) = σ(z n , 1). This equation is rewritten in a variable-separated form cos nθ 2 = φm,n (r), cos mθ 2

where

φm,n (r) := and

m−n 2

(1 + rn )r 1 + rm

cos nθ = cosh nθ iπ > 0, 2 2π

,

(7)

cos mθ = cosh mθ iπ > 0. 2 2π

The function φm,n (r) is increasing on 0 ≤ r ≤ 1, with φm,n (0) = 0, φm,n (1) = 1. √

Lemma 4. Let z = re −1θ , 0 < r ≤ 1, θ ∈ (−π, π], and fix θ. Let m > n > 0 nθ m n be integers. If |h mθ 2π i| ≥ |h 2π i|, then d(z , 1) > d(z , 1) for any 0 < r < 1. nθ mθ If |h 2π i| < |h 2π i|, then there exists a (unique) 0 < r0 < 1 such that • d(z m , 1) = d(z n , 1) if r = r0 , • d(z m , 1) > d(z n , 1) if 0 < r < r0 , and 9

• d(z m , 1) < d(z n , 1) if r0 < r ≤ 1. nθ nθ mθ Proof. If |h mθ ≥ 1 > φm,n (r) for 0 < r < 1, 2π i| ≥ |h 2π i|, we have cos 2 / cos 2 which implies that σ(z m , 1) < σ(z n , 1), and hence d(z m , 1) > d(z n , 1) for 0 < r < 1. nθ nθ mθ If |h mθ < 1. By the 2π i| < |h 2π i|, we have 0 < cos 2 / cos 2 Intermediate mθ Value Theorem, there exists 0 < r0 < 1 such that φm,n (r0 ) = cos nθ 2 / cos 2 , which implies that σ(z m , 1) = σ(z n , 1) for r = r0 . The uniqueness of r0 and the rest statements follow from the monotonicity of the function φm,n . √

Proposition 5. Let z = re −1θ , θ ∈ (−π, π]. Let m > 0 be a disk parastichy mθ number for the spiral lattice Λ(z). Let a = [[ mθ 2π ]] be the integer part of 2π , see θ a Appendix. Then m is a (principal) convergent of 2π . a θ Proof. If m is not a principal convergent of 2π , there exists n > 0, such that nθ mθ n < m and |h 2π i| ≤ |h 2π i|. By Lemma 4, this implies that d(z n , 1) < d(z m , 1), and hence m is not a disk parastichy number.

Lemma 4 is a reformulation of [15, Proposition 1]. Lemma 5 is a reformulation of [15, Proposition 2]. 4. Voronoi tilings and the packing distance In this section, we extend the result of [20] to show that the Voronoi tiling for a logarithmic spiral lattice is related to the packing distance d. Let T = {Tj }j∈Z be the Voronoi tiling of C∗ for the spiral lattice Λ = √ { j} z j∈Z , z = re −1θ ∈ D \ R. An integer m > 0 is called a Voronoi parastichy number if T0 is adjacent to Tm . The bifurcation of Voronoi tilings occurs when the quadrilateral (1, z m , z m+n , z n ) is inscribed in a circle. We are going to show that the equation for the bifurcation of Voronoi tilings is written in terms of the packing distance function d. √ Let z1 , z2 ∈ D∗ . Denote by zj = rj e −1θj , 0 < rj < 1, θj ∈ (−π, π], j = 1, 2. Let (z1 − z1 z2 )(z2 − 1) z1 (z2 − 1)2 χ(z1 , z2 ) = = (z2 − z1 z2 )(z1 − 1) z2 (z1 − 1)2 be a cross-ratio of the four points 1, z1 , z2 , z1 z2 . Let √

√ √ √−1θ1 /2 (r2 e −1θ2 − 1) r1 e √ χ(z1 , z2 ) = √ √−1θ /2 2 r2 e (r1 e −1θ1 − 1)

be a square root of χ(z1 , z2 ). √

Lemma 6. Let zj = rj e −1θj , 0 < rj < 1, θj ∈ (−π, π], j = 1, 2. Suppose that the four points 1, z1 , z2 , z1 z2 are distinct and not collinear. Then the following conditions are mutually equivalent. 10

(i) (1, z1 , z1 z2 , z2 ) is a quadrilateral, in this order of vertices, inscribed in a circle. (ii) χ(z1 , z2 ) < 0. (iii) ξ(z1 , z2 ) = 0, where ξ(z1 , z2 ) = (1 + r1 r2 ) cos

θ1 − θ2 θ1 + θ2 − (r1 + r2 ) cos . 2 2

(iv) θ1 θ2 < 0, |θ1 − θ2 | < π and σ(z1 , z2 ) = σ(1, z1 z2 ).

(8)

(v) θ1 θ2 < 0, |θ1 − θ2 | < π and d(z1 , z2 ) = d(1, z1 z2 ).

Moreover, if one of the abve conditions holds, we have

d(z1 , z2 ) = d(1, z1 z2 ) > max(d(1, z1 ), d(1, z2 )).

(9)

Proof. (i) ⇔ (ii). Obvious. √ √ r (ii) ⇔ (iii). ξ = √r21 |z1 − 1|2 Re χ. (iii) ⇔ (iv). First we suppose that θ1 θ2 ≥ 0 and show that ξ > 0. We may assume, without loss of generality, that θ1 , θ2 ∈ [0, π]. Then we obtain 2 2 cos θ1 −θ ≥ cos θ1 +θ 2 2 , 1 + r1 r2 > r1 + r2 > 0, and hence ξ > 0. (A geometric proof can be obtained in a similar way as in [20, Lemma 3].) Next we suppose that θ1 θ2 < 0 and |θ1 − θ2 | ≥ π, and show that ξ < 0. We may assume, without loss of generality, that −π < θ2 < 0 < θ1 ≤ π and 2 2 θ1 ≥ θ2 + π. Then we obtain cos θ1 −θ < 0 < cos θ1 +θ 2 2 , and hence ξ < 0. If θ1 θ2 < 0 and |θ1 − θ2 | < π, we have θ1 + θ2 = Arg(z1 z2 ), and hence ξ(z1 , z2 ) =

(1 + r1 r2 )(r1 + r2 ) (σ(z1 , z2 ) − σ(1, z1 z2 )). √ 2 r1 r2

(iv) ⇒ (v). Obvious. (v) ⇒ (iv). If θ1 θ2 < 0 and |θ1 − θ2 | < π, we have σ(z1 , z2 ), σ(1, z1 z2 ) > 0, and so √ √ σ(z1 , z2 ) = 1 − d(z1 , z2 )2 = 1 − d(1, z1 z2 )2 = σ(1, z1 z2 ).

Finally, suppose that θ1 θ2 < 0, |θ1 − θ2 | < π and d(z1 , z2 ) = d(1, z1 z2 ), and show the inequality in (9). The quadrilateral (1, z1 , z1 z2 , z2 ) is inscribed in a circle in this order of vertices. Let α be the crossing point of the diagonals `(1, z1 z2 ) and `(z1 , z2 ) (`(z, w) denotes the line segment with the endpoints z, w). Since the triangle 4(1, z1 , α) is similar to 4(z2 , z1 z2 , α), we have |z2 − α| |z1 z2 − α| |z2 − z1 z2 | = = = |z2 |. |1 − α| |z1 − α| |1 − z1 |

The triangle 4(1, z2 , α) is similar to 4(z1 , z1 z2 , α), so we have |z1 z2 − α| |z1 − z1 z2 | |z1 − α| = = = |z1 |. |1 − α| |z2 − α| |1 − z2 | 11

z1

●

z 1 z2

α ●

●

●

1

●

z2

Figure 6: The sides of a quadrilateral (1, z1 , z1 z2 , z2 ) inscribed in a circle are shorter than the diagonals in the lima¸con metric.

Thus we obtain |1 − α| =

|z2 − α| |z1 z1 − α| |z1 − α| = = , |z1 | |z2 | |z1 z2 |

see Figure 6. By the triangular inequality, we have d(z1 , 1) =

|z1 − 1| |z1 − α| + |α − 1| < |z1 | + 1 |z1 | + 1 |z1 − α| + |α − z2 | |z1 − z2 | = = = d(z1 , z2 ). |z1 | + |z2 | |z1 | + |z2 |

A similar argument applies to d(z2 , 1). It can happen ξ(z1 , z2 ) < 0 and yet d(1, z1 z2 ) = d(z1 , z2 ), when −π < θ2 < 0 < θ1 < π and |θ1 − θ2 | > π. The following lemma is a counterpart to Lemma 6. It is not used in this paper. √

Lemma 7. Let zj = rj e −1θj , 0 < rj < 1, θj ∈ (−π, π], j = 1, 2. Suppose that the four points 1, z1 , z2 , z1 z2 are distinct and not collinear. Then the following conditions are mutually equivalent. (i) χ(z1 , z2 ) > 0. (ii) η(z1 , z2 ) = 0, where η(z1 , z2 ) := (1 − r1 r2 ) sin

θ1 − θ2 θ1 + θ2 + (r1 − r2 ) sin . 2 2

(iii) Either (1, z1 , z2 , z1 z2 ) or (1, z2 , z1 , z1 z2 ) is a quadrilateral inscribed in a circle, in this order of vertices. Moreover, if one of the above conditions holds, we have θ1 θ2 > 0. 12

Proof. (i) ⇔ (iii). Obvious. √ √ r (i) ⇔ (ii). η = √r21 |z1 − 1|2 Im χ. Finally, we are going to show that η > 0 if θ1 θ2 ≤ 0. We may assume, without loss of generality, that −π < θ2 ≤ 0 ≤ θ1 < π. Then we obtain 2 2 sin θ1 −θ ≥ | sin θ1 +θ 2 2 |, 1 − r1 r2 > |r1 − r2 |, and hence η > 0. 5. Farey tree structure of the Voronoi phase diagram In this section, we reformulate the result in [20] on the bifurcation of Voronoi √ tilings, in terms of the lima¸con distance function. Let z = re −1θ , r > 0, a b θ ∈ (−π, π]. Let ( m , n ) ⊂ (− 21 , 12 ) be a Farey interval, i.e., m, n > 0 and mb − na = 1. We are going to define a continuous function [ ] 2πa 2πb ga,b : , → [0, 1] m n m n such that the spiral lattice Λ(z) with the generator [ ] √ 2πa 2πb z = g a , b (θ)e −1θ , θ ∈ , , m n m n

(10)

gives a quadrilateral Voronoi tiling with parastichy numbers m, n if 0 < g a , b (θ) < m n 1. The curve parameterized by (10) is labelled (m, n) in Figure 2. See also Figure 7. The equation (8) for z1 = z m , z2 = z n is written in a variable-separated form ψm,n (θ) = ϕm,n (r), where ( ) )( ( ) )−1 mθ nθ mθ nθ cos h i−h i π cos h i+h i π , 2π 2π 2π 2π rm + rn ϕm,n (r) := . 1 + rm+n

ψm,n (θ) :=

Let

(

g a , b (θ) := ϕ−1 m,n (ψm,n (θ)) . m n

1 nθ Note that 0 < ψm,n (θ) < 1 if Arg(mθ)Arg(nθ) < 0 and h mθ 2π i − h 2π i < 2 . The function ϕm,n (r) is increasing on 0 ≤ r ≤ 1, with ϕm,n (0)[ = 0, ϕm,n ] (1) = 1. If a b 2bπ (m , n ) ⊂ (0, 12 ), the function g a , b is naturally defined on 2aπ , if m, n ≥ 3, m n [ ] [ ] m[ 0n 1 ] 2π 2π a b or on 2n−2 , n if m , n = 1 , n and n ≥ 3. We furthermore extend the a b 2π for n ≥ 2. For ( m , n ) ⊂ (− 21 , 0), the functions by g 01 , n1 (θ) = 0 on 0 ≤ θ ≤ 2n−2 function g a , b is defined by g a , b (θ) = g −b , −a (−θ). m n

m n

n

m

a b , n ] to emphasize the interval. At We also denote by g a , b (θ) = gθ [ m m n 2πb the endpoints of the Farey intervals, we have g a , b ( 2πa a b ( ,n m ) = gm n ) = 1, m n −1 m−2 2π a 1 (π) = ϕ g0, n1 ( n ) = 1, and g m ( ), for m, n ≥ 3. See Figure 2. ,2 m,2 m+2

13

Lemma 8 (Monotonicity). [ ] [ ] a b a a+b gθ , < gθ , , m n m m+n if

a m

<

θ 2π

<

a+b m+n

a a+b and gθ [ m , m+n ] > 0.

gθ if

a+b m+n

<

θ 2π

<

b n

[

] [ ] a b a+b b , < gθ , , m n m+n n

a+b b and gθ [ m+n , n ] > 0.

Proof. This is a so-called Fundamental Theorem of Phyllotaxis [20, Lemmas 11, 12]. 0

0

a b a b By applying Lemma 8 repeatedly, we see that if ( m , n ) 6= ( m 0 , n0 ) are Farey θ a b a0 b0 a b intervals such that 2π ∈ ( m , n ) ⊂ ( m0 , n0 ) ⊂ (0, 1) and gθ [ m , n ] > 0, then a b a0 b0 gθ [ m , n ] > gθ [ m 0 , n0 ]. a b a b Lemma 9. Let ( m , n ) ⊂ (− 12 , 12 ) be a Farey interval such that ( m , n ) 6= √ 1 1 θ a b 1 1 −1θ ∈ D \ R, and suppose that 2π ∈ ( m , n ). (− 2 , 2 ), (0, 2 ), (− 2 , 0). Let z = re

1. If

a b , ] < 1, m n the tiling T (z) is a quadrilateral tiling with (Voronoi) parastichy numbers m, n. 2. If either 0 < r = gθ [

θ a a+b a b a a+b ∈( , ) and gθ [ , ] < r < gθ [ , ], 2π m m+n m n m m+n θ a+b b a b a+b b ∈( , ) and gθ [ , ] < r < gθ [ , ], or 2π m+n n m n m+n n θ a+b a b = and gθ [ , ] < r < 1, 2π m+n m n

(11)

then the tiling T (z) is a hexagonal tiling with (Voronoi) parastichy numbers m, m + n, n. Proof. [20, Lemmas 11, 12]. √

a b Let V[ m , n ] be the set of z = re −1θ ∈ D \ R defined by (11). We have obtained a ‘semi-triangular’ decomposition of the parameter space D \ R into a b , n ], with the Farey binary-tree structure as is shown in Figure the regions V[ m 2.

14

6. Disk parastichy and Voronoi parastichy In this section, we show that a disk parastichy number is at the same time a Voronoi parastichy number. It is not a trivial result. The continued fraction expansion and the inequality (9) play key roles in the proof. √

a b , n ) ⊂ (− 21 , 12 ) be a Farey interval Lemma 10. Let z = re −1θ ∈ D \ R. Let ( m θ a b containing 2π . For r ≥ gθ [ m , n ], |m − n| is not a disk parastichy number of a b Λ(z). For r ≤ gθ [ m , n ], m + n is not a disk parastichy number of Λ(z). a b Proof. For r = r0 := gθ [ m , n ], we have

d(z m−n , 1) = d(z m+n , 1) > max {d(z m , 1), d(z n , 1)} by (9). Since ) ( (m − n)θ h > max h mθ i , h nθ i > h (m + n)θ i , i 2π 2π 2π 2π

we obtain

for r > r0 , and

d(z m−n , 1) > max {d(z m , 1), d(z n , 1)}

(12)

d(z m+n , 1) > max {d(z m , 1), d(z n , 1)}

(13)

for r < r0 , by Lemma 4. This completes the proof. √

Lemma 11. Let z = re −1θ , θ ∈ (−π, π]. Let cl be a (principal) convergent of θ 2π , and suppose that l is a disk parastichy number of the spiral lattice Λ(z). Let a b θ (m , n ) be a Farey interval containing 2π . a b (i) If 0 < gθ [ m , n ] ≤ r, then a b , n ], then (ii) If 0 < r ≤ gθ [ m

c l c l

a b ∈ [m , n ]. a b 6∈ ( m , n ).

a b a Proof. (i). Suppose that 0 < gθ [ m , n ] ≤ r and cl < m . Then there exists 0 0 b a b a0 b0 a 0 0 a Farey interval ( m0 , n0 ) ⊃ ( m , n ) such that l = m − n and gθ [ m 0 , n0 ] > 0. 0 0 a b a b By the remark after Lemma 8, we have g [ , ] < g [ , ] ≤ r. By (12), θ m 0 n0 θ m n { } 0

0

we have d(z l , 1) > max d(z m , 1), d(z n , 1) , which implies that l is not a disk parastichy number. a0 b0 a b If cl > nb , then there exists a Farey interval ( m 0 , n0 ) ⊃ ( m , n ) such that 0 0 a b l = n0 − m0 and gθ [ m 0 , n0 ] > 0. The remaining argument is similar. a b a , n ]. Then there exists a (ii). Suppose that m < cl < nb and 0 < r ≤ gθ [ m 0 0 b a b a 0 Farey interval ( m0 , n0 ) ⊂ ( m , n ) such that l = m + n0 . By the remark after a0 b0 a b l Lemma { 8, we have gθ [ m }0 , n0 ] > gθ [ m , n ] ≥ r. By (13), we have d(z , 1) > 0

0

max d(z m , 1), d(z n , 1) , which implies that l is not a disk parastichy number.

15

Theorem 12. Let z ∈ D \ R. A disk parastichy number for the spiral lattice Λ(z) is also a Voronoi parastichy number for T (z). √

Proof. Let z = re −1θ , θ ∈ (−π, π]. Let l be a disk parastichy number for Λ(z). θ By Proposition 5, it is a denominator of some (principal) convergent cl for 2π . a b a b θ Case 1. Suppose that r = gθ [ m , n ] for some Farey interval ( m , n ) 3 2π . By a b , n , and so l = m, n, either of which is a Voronoi Lemma 11, we have cl = m parastichy number by Lemma 9. a b a a+b a a+b Case 2. Suppose that gθ [ m , n ] < r < gθ [ m , m+n ], where ( m , m+n ) is a Farey θ c a b a a+b interval containing 2π . By Lemma 11, we have l ∈ [ m , n ] \ ( m , ). Since cl { m+n } θ a a+b a a+b b ∈ (m , m+n ), we have cl ∈ m , m+n , n , and is a (principal) convergent of 2π so l = m, m + n, n, either of which is a Voronoi parastichy number by Lemma 9. a b a+b b a+b b Case 3. The case gθ [ m , n ] < r < gθ [ m+n , n ] with a Farey interval ( m+n , n) 3 θ is similar to the Case 2. 2π a b a a+b θ Case 4. Suppose that gθ [ m , n ] < r < 1 for a Farey interval ( m , m+n ) 3 2π , θ a+b c a b and moreover that 2π = m+n . By Lemma 11, we have l ∈ [ m , n ]. This implies { } a a+b b that cl ∈ m , m+n , n , and so l = m, m + n, n, either of which is a Voronoi parastichy number by Lemma 9. a b Theorem 12 shows that V[ m , n ] ⊂ S(m) ∪ S(n) ∪ S(m + n), and that the curve in D \ R defined by the equation (10) is contained in S(m) ∪ S(n).

7. Duality of phase diagrams In this section we describe the relationship between van Iterson’s tree F and the phase diagram of Voronoi tilings. a b a b Let ( m , n ) ⊂ (− 12 , 12 ) be a Farey interval such that ( m , n ) 6= (− 21 , 12 ), (0, 12 ), (− 21 , 0). a b a b Denote the closure of V[ m , n ] in D \ R by cl(V[ m , n ]). Since cl(V[

a b , ]) ⊂ S(m) ∪ S(n) ∪ S(m + n), m n

we have cl(V[

a b a b , ]) ∩ F = (cl(V[ , ]) ∩ S(m) ∩ S(n)) m n m n a b ∪ (cl(V[ , ]) ∩ S(m) ∩ S(m + n)) m n a b ∪ (cl(V[ , ]) ∩ S(n) ∩ S(m + n)). m n

(14)

The arcs S(m)∩S(n), S(m)∩S(m+n), S(n)∩S(m+n) are written by the equations σ(z m , 1) = σ(z n , 1), σ(z m , 1) = σ(z m+n , 1), and σ(z n , 1) = σ(z m+n , 1), respectively.

16

Consider the function f a , b (θ) := m n

φ−1 m,n

) ) ( (√ cos nθ 1 + cos nθ −1 2 = φm,n cos mθ 1 + cos mθ 2

for m > n, or f a , b (θ) := φ−1 n,m m n

in (7). Note that 2π(a+b) m+n . 2π(a+b) m+n ,

(√

f a , b ( 2π(a+b) m+n ) m n

1+cos mθ 1+cos nθ

)

for m < n, where φm,n is defined

= 1, since we have cos nθ = cos mθ for θ =

The domain of definition of f a , b (θ) is an interval with an endpoint m n

satisfying −1 < cos nθ ≤ cos mθ for m > n, or −1 < cos mθ ≤ cos nθ a b for m < n. For ( m , n ) ⊂ (0, 12 ), the domain of definitions of f a , b are given as m n follows. ] [ 1 1 Dom(f 01 , 21 ) = , , 3 2 [ ] 1 1 Dom(f 10 , n1 ) = , , n≥3 n+1 n−1 [ ] (m − 3)/2 (m + 1)/2 Dom(f (m−1)/2 , 1 ) = , , m≥5 m 2 m−2 m+2 [ ] 1 2 1 ) = 1 , , n≥2 Dom(f n+1 ,n 2n 2n + 1 [ ] a−b a+b Dom(f a , b ) = , , m > n ≥ 3, m − n ≥ 3 m n m−n m+n [ ] a+b b−a , , n > m ≥ 3, n − m ≥ 3. Dom(f a , b ) = m n m+n n−m a b For ( m , n ) ⊂ (− 12 , 0), f a , b is defined by f a , b (θ) = f n−b , m−a (−θ). In either m n

m n

n

m

2π(a+b) case, the function f a , b is increasing in the (sub)interval [ 2πa m , m+n ] if m > n, m n

2πb or decreasing in the (sub)interval [ 2π(a+b) m+n , n ] if m < n.

Lemma 13. (i) If n ≥ 3, g 01 , n1 (θ) is an increasing function on the subinterval π [ n−1 , 2π n ]. , π]. (ii) If m ≥ 3, g (m−1)/2 , 1 (θ) is a decreasing function on the interval [ (m−1)π m m

2

(iii) If m > n ≥ 3 and a, b > 0, the function g a , b (θ) is decreasing on the m n

2π(a+b) subinterval [ 2πa m , m+n ]. If n > m ≥ 3 and a, b > 0, the function g a , b (θ) 2πb is increasing on the subinterval [ 2π(a+b) m+n , n ].

m n

Proof. (i) Suppose that n ≥ 3, and consider the function g 01 , n1 (θ) = ϕ−1 1,n (ψ1,n (θ)). π , 2π We are going to show that ψ1,n (θ) is an increasing function of θ ∈ [ n−1 n ]. In

17

fact, we have 0 ψ1,n (θ)

( )−2 sin nθ + n sin θ n+1 θ = cos 2 2   ( )−2 n−1 ∑ n+1 sin θ   n+ cos = cos(n − 1 − 2j)θ θ > 0. 2 2 j=0

(ii) Suppose that m ≥ 3. We are going to show that ψm,2 (θ) is a decreasing π function of θ ∈ [ (m−1)π , π]. Let ρ = π − θ ∈ [0, m ]. Since m ≥ 3 is odd, we have m 1 m+2 sin mθ sin 2θ sin mρ sin 2ρ 0 cos2 θ · ψm,2 (θ) = + = − m 2 m 2 m 2 ( ) ∫ m d sin tρ dt = dt t 2 ∫ m tρ cos tρ − sin tρ = dt < 0. t2 2 ( ) a θ a+b (iii) If m > n ≥ 3 and m < 2π < m+n , then cos h mθ i − h nθ i π is a 2π 2π ) ( nθ function of θ. decreasing function of θ, and cos h mθ 2π i + h 2π i ( π is an increasing ) a+b θ nθ If n > m ≥ 3 and m+n < 2π < nb , then cos h mθ i − h i π is an increasing 2π 2π ( ) nθ function of θ, and cos h mθ i + h i π is a decreasing function of θ. 2π 2π

a b a b Lemma 14. Let ( m , n ) ⊂ (− 12 , 12 ) be a Farey interval and suppose that ( m , n ) 6= 1 1 1 1 (− 2 , 2 ), (0, 2 ), (− 2 , 0). Then there exists a unique θ ∈ Dom(f a , b )∩Dom(g a , b ) m n m n such that f a , b (θ) = g a , b (θ). m n

m n

Proof. Case 1. Suppose that m > n. Then we have [ ] 2πa 2π(a + b) Dom(f a , b ) ∩ Dom(g a , b ) = , . m n m n m m+n 2π(a+b) The function g a , b is decreasing, and f a , b is increasing at [ 2πa m , m+n ], and m n m n 2πa we have f a , b (θ) < g a , b (θ) = 1 for θ = m . Moreover, we have m n

m n

g a , b (θ) < f a , b (θ) = 1, for θ = m n

m n

2π(a + b) . m+n

(15)

The Intermediate Value Theorem applies to see that there exists a unique θ ∈ 2π(a+b) [ 2πa a b (θ) = f a b (θ). ,n m , m+n ] such that g m m,n Case 2. Suppose that m < n. Then we have Dom(f a , b ) ∩ Dom(g a , b ) = m n m n [ ] 2π(a+b) 2πb m+n , n . The function g a , b is increasing and f a , b is decreasing at m n

m n

2πb [ 2π(a+b) a b (θ) < g a b (θ) = 1 for θ = ,n m+n , n ], and we have f m m,n the Intermediate Value Theorem applies.

18

2πb n .

By (15),

Lemma 15.

g a , b (θ) < f a , a+b (θ) < g a+b , b (θ) m n

m m+n

(16)

m+n n

whenever they are well defined and f a , a+b (θ) > 0. m m+n

g a , b (θ) < f a+b , b (θ) < g a , a+b (θ), m n

m+n n

(17)

m b+n

whenever they are well defined and f a+b , b (θ) > 0. m+n n

Proof. If r = g a , b (θ), then we have d(z m , 1) < d(z m+n , 1) by (9), which implies m n the first inequality in (16). If r = g a+b , b (θ), we have d(z m , 1) > d(z m+n , 1), m+n n which implies the second inequality in (16). If r = g a , b (θ), then we have d(z n , 1) < d(z m+n , 1). If r = g a , a+b (θ), then m n

m m+n

d(z n , 1) > d(z m+n , 1). These imply (17).

a b The covering of V[ m , n ] by S(m), S(n), S(m + n) is described as follows. Let √ a b −1θ z = re ∈ V[ m , n ]. If

r ≥ f a , a+b (θ) m m+n

and

r ≥ f a+b , b (θ), m+n n

we have d(z m+n , 1) ≤ d(z m , 1) and d(z m+n , 1) ≤ d(z n , 1), and hence z ∈ S(m + n). If f a+b , b (θ) ≤ r ≤ f a , a+b (θ), m+n n

we have d(z , 1) ≤ d(z m

m+n

m m+n

, 1) ≤ d(z , 1), and hence z ∈ S(m). If n

r ≤ f a , a+b (θ) m m+n

and

r ≤ f a+b , b (θ), m+n n

we have z ∈ S(m) or z ∈ S(n). a b a b Lemma 16. Let ( m , n ) ⊂ (− 12 , 12 ) be a Farey interval and suppose that ( m , n ) 6= 1 1 (− 2 , 2 ). There exists a unique θ ∈ Dom(f a+b , b ) ∩ Dom(f a , a+b ) such that m+n n m m+n f a+b , b (θ) = f a , a+b (θ). m+n n

m m+n

Proof. First note that f a+b , b (θ) = f a , a+b (θ) m+n n

m m+n

⇔ f a , b (θ) = f a , a+b (θ) m n

m m+n

⇔ f a+b , b (θ) = f a , b (θ). m n

m+n n

m If it is the case, we have σ(z , 1) = σ(z n , 1) = σ(z m+n , 1) and z ∈ S(m) ∩ √ S(n) ∩ S(m + n) for z = re −1θ , r = f a , b (θ). m n Case 1. Suppose that m > n. We shall show that there exists a unique θ ∈ a+b 2a+b , m+n ] such that f a , b (θ) = f a , a+b (θ). Dom(f a , b ) ∩ Dom(f a , a+b ) = [ 2m+n m n m n m m+n m m+n The function f a , b is increasing, and the function f a , a+b is decreasing in m n

m m+n

19

a+b 2a+b the interval [ 2m+n , m+n ]. Moreover, we have f a , a+b (θ) < f a , b (θ) = 1 at

θ =

a+b m+n ,

m n

m m+n

and f a , b (θ) < f a , a+b (θ) = 1 at θ = m n

m m+n

2a+b 2m+n . The Intermediate 2a+b a+b θ ∈ [ 2m+n , m+n ] such that

Value Theorem applies to see that there is a unique f a , a+b (θ) = f a , b (θ). m n m m+n Case 2. Suppose that m < n. We shall show that there exists a unique θ ∈ a+b a+2b Dom(f a , b ) ∩ Dom(f a+b , b ) = [ m+n , m+2n ] such that f a , b (θ) = f a+b , b (θ). m n m n m+n n m+n n The function f a , b is decreasing, and the function f a+b , b is increasing in m n

m+n n

a+b a+2b the interval [ m+n , m+2n ]. Moreover, we have f a+b , b (θ) < f a , b (θ) = 1 at

θ =

a+b m+n ,

m n

m+n n

Value Theorem applies to see that there is a unique f a+b , b (θ) = f a , b (θ). m+n n

m n

m+n n

and f a , b (θ) < f a+b , b (θ) = 1 at θ =

a+2b m+2n . The Intermediate a+b a+2b θ ∈ [ m+n , m+2n ] such that

m n

Figure 7 is a stylized picture of the graphs of g a , b , etc. m n Let { √ } F a , b = re −1θ : r = f a , b (θ), θ ∈ Dom(f a , b ) ∩ D \ R m n

m n

m n

√

a b , n ] = re −1θ the unique point be the ‘graph’ of f a , b in D \ R. Denote by P [ m m n such that θ ∈ Dom(f a+b , b ) ∩ Dom(f a , a+b ) and r = f a+b , b (θ) = f a , a+b (θ). m+n n

m m+n

m+n n

m m+n

a b It is a node point in F. The spiral lattice P [ m , n ] has three disk parastichy √ a b numbers m, n, m + n, as in Figure 4 (a). Denote by Q[ m , n ] = re −1θ the unique point θ ∈ Dom(f a , b ) ∩ Dom(g a , b ) and r = f a , b (θ) = g a , b (θ). It m n m n m n m n a b is an intersection point of F with the boundary of V[ m , n ]. The spiral lattice a b Q[ m , n ] has two disk parastichy numbers m, n and a quadrilateral Voronoi tiling, as in Figure 4 (b). Let Q[0, 12 ] = Q[− 21 , 0] = 0 be the origin.

Theorem 17. The van Iterson graph F ∪{0} is connected and simply connected. a b a b Proof. Let ( m , n ) ⊂ (− 21 , 12 ) be a Farey interval such that ( m , n ) 6= (− 12 , 12 ). a b a b In (14), cl(V[ m , n ]) ∩ S(m) ∩ S(n) is a subarc of F a , b connecting Q[ m , n] m n a b a b with P [ m , n ], cl(V[ m , n ]) ∩ S(m) ∩ S(m + n) is a subarc of F a , a+b connecting m m+n

a a+b a b a b Q[ m , m+n ] with P [ m , n ], and cl(V[ m , n ]) ∩ S(n) ∩ S(m + n) is a subarc of a+b b a b F a+b , b connecting Q[ m+n , n ] with P [ m , n ]. This shows that F ∪ {0} has a m+n n binary tree structure, which completes the proof.

8. Discussion The logarithmic spiral lattice is the simplest nonlinear geometric model of ideal spiral phyllotaxis. It has some common features with the linear lattice in the cylindrical model, that it has a symmetry of a transformation group (parallel translations in the cylindrical model, and similarity transformations in the centric model), and that the parastichy numbers are globally constant in the 20

r 1

a−b m−n

a+b m+n

a m

●

g a , a+b (θ)

● ●

m m+n

● ●

●

f a+b , b (θ)

● ● ●

g a+b , b (θ) m+n n

f a , a+b (θ)

m+n n

●

g a , b (θ)

b n

m m+n

g a , b (θ)

m n

m n

θ 2π

0

Figure 7: A stylized picture of the graphs of g a , b , etc. Note that the orientation is reversed m n

when it is mapped to D by the mapping (θ/2π, r) 7→ re

21

√ −1θ .

whole space (the cylinder or the punctured plane). Due to this strong symmetry, the origin is an accumulation point of infinite sites, which is{not realistic. } √ √−1θk ke : k = 0, 1, . . . More plausible geometric models are the parabolic spirals {√ } { √ } √ [18, 21] or k + 12 e −1θk : k = 0, 1, . . . [4], and the Archimedean spiral ke −1θk : k = 0, 1, . . . . The parabolic spiral has a good condition that the area provided for each site point seems bounded from below and above. In any case, the parastichy numbers are not global constants, and their global analysis is another subject of mathematical phyllotaxis. We can also consider the ambient spaces such as the cone, the sphere, or a rotation surface with positive curvature [14, 16]. Appendix: Background materials C is the complex plane. C∗ = C \ {0} is the punctured plane. D = {z ∈ C : |z| < 1} is the unit disk. D∗ = D \ {0} is the punctured disk. Denote the fractional part of x ∈ R by − 21 < hxi ≤ 21 , where [[x]] := x−hxi ∈ Z. a b , n ) with rational endpoints is called a Farey interval if An open interval ( m a+b a m, n > 0 and mb − na = 1. The number m+n is called the Farey sum of m and a a+b b n . We may use the terminology by Adler [1] to call the subintervals ( m , m+n ), a+b b a b ( m+n , n ) the extensions (or ‘children’) of ( m , n ). The contraction (or ‘parent’) a b a−b b a b−a of ( m , n ) 6= (0, 1) is defined by ( m−n , n ) if m > n, or ( m , n−m ) if m < n. Let x = a0 +

1 a1 +

1 a2 +···

= [a0 , a1 , a2 , · · · ], a0 ∈ Z, ai ∈ Z+ , i ≥ 1

be the continued fraction expansion of x ∈ R, where Z+ denotes the set of positive integers. Define the sequences {pj }j≥−1 and {qj }j≥−1 by p−1 = 1, p0 = a0 , p1 = a0 a1 + 1, pj+1 = aj+1 pj + pj−1 , j ≥ 1; q−1 = 0, q0 = 1, q1 = a1 , qj+1 = aj+1 qj + qj−1 , j ≥ 1. Let pj,k = kpj + pj−1 , qj,k = kqj + qj−1 for j ≥ 0, 0 ≤ k ≤ aj+1 . Note that pj,0 = pj−1 , qj,0 = qj−1 , pj,aj+1 = pj+1 , qj,aj+1 = qj+1 . The fraction pj /qj = [a0 , a1 , · · · , aj ], j ≥ 0, is called a principal convergent of x, and pj,k /qj,k = [a0 , a1 , · · · , aj , k], j ≥ 0, 0 < k < aj+1 , is called an intermediate convergent of x. See [10] for the details of Farey sequence and continued fractions. The convergent pj,k /qj,k is the Farey sum of pj /qj and pj,k−1 /qj,k−1 , for j ≥ 0, 1 ≤ k ≤ aj+1 . For j even, (pj /qj , pj,k /qj,k ) is a Farey interval containing x, and for j odd, (pj,k /qj,k , pj /qj ) is a Farey interval containing x. Conversely, if (a/m, b/n) is a Farey interval that contains x, then we have a/m = pj /qj , b/n = pj,k /qj,k for some j even and 0 < k ≤ aj+1 , or a/m = pj,k /qj,k , b/n = pj /qj for some j odd and 0 < k ≤ aj+1 . For z ∈ C and r > 0, D(z, r) = {ζ ∈ C : |ζ − z| < r} denotes the disk centered at z with the radius r. A family of disks {Dj }j in the plane is called a packing if they have no overlaps, i.e., int(Dj ) ∩ int(Dk ) = ∅ for Dj 6= Dk , and each disk Dj is tangent to some Dk . 22

A tiling of a two dimensional manifold X is a family T = {Tj }j of topological ∪ disks Tj ⊂ X which covers X without gaps or overlaps, that is, X = j Tj and int(Tj ) ∩ int(T { k )}= ∅, j 6= k. Each Tj is called a tile. Let Λ = z j j∈Z , z ∈ D \ R, be a spiral set in C∗ . The Voronoi region for the site z j , j ∈ Z, is defined by { } Tj = Tj (z) := w ∈ C∗ : |w − z j | ≤ |w − z k |, ∀k 6= j .

The family T (z) := {Tj (z)}j∈Z is a polygonal tiling of C∗ . Two Voronoi regions Tj , Tk are called adjacent, or edge-adjacent for clarity, if the intersection Tj ∩ Tk contains at least two points and Tj 6= Tk . The line segment `(z j , z k ) joining the distinct sites z j and z k is called a Delaunay edge if it satisfies the Empty Circumdisc Property, i.e., there exists a closed disc D ⊂ C∗ that contains no point in Λ except z j}, z k , or equivalently, { there exists a closed disc D ⊂ C∗ such that ∂D∩Λ = z j , z k and int(D)∩Λ = ∅. (It is called a ‘strongly Delaunay’ edge in [5].) For distinct z1 , z2 , z3 ∈ C, let ) ( z1 − z2 , ∠(z1 , z2 , z3 ) := Arg z3 − z2 where −π < Arg ≤ π denotes the principal argument in C∗ .

Lemma 18. Let T = {Tj }j∈Z be a Voronoi tiling of C∗ for the spiral lattice √ { } Λ = z j j∈Z , z = re −1θ ∈ D\R. For distinct j, k ∈ Z, the following conditions are mutually equivalent: (i) The Voronoi regions Tj , Tk are adjacent. (ii) The line segment `(z j , z k ) is a Delaunay edge. (iii) If ∠(z j , z i1 , z k ) > 0 and ∠(z k , z i2 , z j ) > 0, we have

∠(z j , z i1 , z k ) + ∠(z k , z i2 , z j ) < π Proof. [20, Lemma 1]. Acknowledgments. We would like to thank Christophe Gol´e and Pau Atela for helpful comments and fruitful discussions. The anonymous referees helped us to fix some mistake and polish the paper. We also thank Smith College for the hospitality extended to Yamagishi during his sabbatical stay in 20152016. This work is partially supported by JSPS Kakenhi Grant 15K05011, and the Joint Research Center for Science and Technology of Ryukoku University. [1] I. Adler, The consequences of contact pressure in phyllotaxis Journal of Theoretical Biology 65(1) (1977) 29–77. also in: I. Adler, Solving the riddle of phyllotaxis, World Scientific (2012), pp.91–139. [2] I. Adler, D. Barabe and R. V. Jean, A history of the study of phyllotaxis, Annals of Botany 80 (1997) 231–244. 23

[3] P. Atela, C. Gol´e and S. Hotton, A dynamical system for plant pattern formation: a rigorous analysis, J. Nonlinear Sci. 12 (2002) 641–76. [4] K. Azukawa and T. Yuzawa, A remark on the continued fraction expansion of conjugates of the golden section, Mathematics J. of Toyama Univ. 13 (1990) 165–176. [5] S.-W. Cheng, T.K. Dey and J. Shewchuk, Delaunay Mesh Generation (Chapman & Hall) (2012). [6] R.O. Erickson, The geometry of phyllotaxis. in: The Growth and Functioning of Leaves, ed. Dale J E and Milthorpe F L. Cambridge: Cambridge University Press, (1983), pp.53–88. [7] E. Freeman, Spiral Lattice, GeoGebra script (2016) http://tube.geogebra.org/material/simple/id/2975083 . [8] C. Gol´e and L. Grecki, unpublished note (2007). [9] H. Hellwig, T. Neukirchner, Phyllotaxis, Die mathematische Beschreibung und Modellierung von Blattstellungsmustern, Mathematische Semesterberichte 57(1) (2010), pp.17–56. [10] W. J. LeVeque, Fundamentals of Number Theory (Massachusetts: AddisonWesley) (1977). [11] L. S. Levitov, Energetic Approach to Phyllotaxis Europhys. Lett. 14(6) (1991) 533–9. [12] R. Maksymowych and R. O. Erickson, Phyllotactic change induced by biggerellic acid on xanthium shoot apices, American Journal of Botany 64 (1977) 33–44. [13] M. F. Pennybacker, P. D. Shipman, and A. C. Newell, Phyllotaxis: some progress, but a story far from over, Physica D 306 (2015) 48–81. [14] J. N. Ridley, Ideal phyllotaxis on general surfaces of revolution, Math. Biosci. 79 (1986) 1–24. [15] F. Rothen and A.-J. Koch, Phyllotaxis or the properties of spiral lattices. II. Packing of circles along logarithmic spirals, Journal de Physique France 50 (1989) 1603–1621. [16] J.-F. Sadoc, J. Charvolin and N. Rivier, Phyllotaxis on surfaces of constant Gaussian curvature, J. Phys. A: Math. Theor. 46 (2013) 295202. [17] G. van Iterson, Mathematische und mikroskopisch-anatomische Studien u ¨ber Blattstellungen nebst Betrachtungen u ¨ber den Schalenbau der Miliolinen, Verlag von Gustav Fischer, Jena, (1907).

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[18] H. Vogel, A Better Way to Construct the Sunflower Head, Math. Biosci. 44 (1979) 179–189. [19] R. F. Williams and E. G. Brittain, A geometrical model of phyllotaxis, Austr J. Bot. 32 (1984) 43–72. [20] Y. Yamagishi, T. Sushida, and A.Hizume, Voronoi Spiral Tilings, Nonlinearity 28 (2015) 1077–1102. [21] F. R. Yeatts, Another look at parastichies, Math. Biosci. 144 (1997) 71–81.

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Highlights for the article “Spiral disk packings” by Yamagishi and Sushida. Shows that van Iterson's metric satisfies the axiom of distance function. A nontrivial proof that the disk packing parastichy is Voronoi parastichy. Shows that the phase diagrams of spiral disk packings and Voronoi tilings are dual.