Mathematical Biosciences 301 (2018) 185–189

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Mathematical Biosciences

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Visibility in a pure model of golden spiral phyllotaxis T Burghard Herrmann

Freie Universität Berlin, ERG, Kaiserswerther Str. 16-18, Berlin 14195, Germany

ARTICLE INFO ABSTRACT

Keywords: This paper considers the geometry of plants with golden spiral phyllotaxis, i.e. growing leaf by leaf on a spiral with Divergence angle golden divergence angle, via the simplest mathematical model, a cylinder with regular arrangement of points on its Fibonacci numbers surface. As is well-known, Fibonacci numbers appear by means of the order of parastichies. This fact is shown to be Golden ratio a straightforward application of logical consequences to a particular model with respect to pure visibility.This Morphogenesis notion is very similar to that of contact parastichies. The 3-D cylindrical model of golden spiral phyllotaxis ab- Pattern recognition stracts from the form of leaves and identiﬁes them with points. Pure visibility is speciﬁed in the 2-D representation Parastichies so that common sense parastichies can be scrutinized. The main Theorem states that the orders of the purely most visible parastichies are Fibonacci numbers.

1. Introduction reciprocal, i.e. ψ =≈1/Φ 0.618. Of course, the angles ψ · 360° ≈ 222.5° and ψ2 · 360° ≈ 137.5° are of golden ratio. Normally the golden angle is We consider the patterns of spirals you can actually see on fruits, defined to be the smaller one. Exceptionally, as it is more convenient in inflorescences, or on the stem of a plant that grows leaf by leaf from a this case, the divergence angle is fixed to be ψ · 360° ≈ 222.5°. shoot apical meristem with constant divergence angle like conifer cones, The letters i, m, n always denote positive integers. pineapples, or grasstrees. The subject is likely to be of general interest, The sequence of Fibonacci numbers 1,1,2,3,5,8,13,21,⋯ is re- even to people outside plant science, as the spirals you see in phyllo- cursively defined from FF12==1, 1 by adding the last two elements, tactic patterns are some of the most striking patterns in nature. e.g. F3 =+11, F9 =+13 21 and F10 =+21 34. Background from previous work is annexed and (A.6) provides a new The golden ratio and the Fibonacci sequence are closely related. For argument. large values of n, on the one hand, Φ is approximated by FFn+1/ n and, on n One of the most popular examples of golden spiral phyllotaxis is that the other hand, Fn is approximated by Φ/ 5(cf. Binet’s formula in of a sunflower head. Unfortunately, the centric disc, as analyzed in [8]). particular by Vogel [17], is not the simplest model. A very general and even more complex model is that of a solid of revolution, which is 2.2. Normalization and the pattern parameter ξ outstandingly analysed by Ridley [14] by means of 3-D infinitesimal calculus. The present paper considers the simplest model: a cylinder The stem is assumed to be a cylinder with radius r and successive with regular arrangement of points on its surface, as introduced in the leaves are assumed to have a constant height difference h, called plas- 1830’s by Auguste Bravais and his brother. It fits already for a wide tochrone distance by Turing in Saunders [16] or rise by Adler in [1]. range of plants. Cylindrical plants are easily measured by radius or Radius r and the plastochrone distance h are lengths measured in some diameter, height, and divergence angle. standard unit of length, e.g. mm. The ratio h/(2πr) is dimensionless and, referencing [1,2,7,14,17], it is called the normalized rise. Nor- 2. The cylindrical model malization means taking the circumference as unit. Of course, the nor- malizied rise h/(2πr) in unit 2πr is identical with the originally mea- This section describes the simplest mathematical model of golden sured rise h. Furthermore, Coxeter [3] and Ridley [14] apply the spiral phyllotaxis. reciprocal of the normalized rise, i.e. ξπrh= 2/, which proves to be very expressive. It is proposed to be called the pattern parameter. The pattern 2.1. Golden ratio and golden angles parameter ξ is a dimensionless counting quantity, that indicates the number of leaves to make a pattern that is one circumference in height. Let Φ denote the golden ratio ( 51)/21.61+≈8 and ψ its Coxeter [3] uses capital letter N instead of ξ. We prefer the Greek letter

E-mail addresses: [email protected], [email protected]. https://doi.org/10.1016/j.mbs.2018.05.003 Received 12 December 2017; Received in revised form 25 March 2018; Accepted 4 May 2018 Available online 05 May 2018 0025-5564/ © 2018 Elsevier Inc. All rights reserved. B. Herrmann Mathematical Biosciences 301 (2018) 185–189 of Ridley [14] as ξ itself looks like a spiral pattern. For a specimen with radius r = 23 mm and rise h = 0.725 mm it holds 2πr ≈ 145 mm and ξ ≈ 200, see the conifer cone in Fig. 3(A). The biologically significant range of the pattern parameter ξ starts from low numerals, e.g. ξ ≈ 8 for Dracaena Marginata on the bald stem where it sheds its bottom leaves. The biologically significant range goes up to the five-digit area, e.g. in Fig. 4 the stem of a grasstree shows granularity similar to that of a sunflower head.

2.3. 2-D representation

Nature has no preferred direction of rotation, as has been observed indeed for conifer cones of a single tree, cf. caption of Fig. 7 in Rutishauser and Peisl [15]. Even in the literature the spirals of Vogel [17] are opposed to those of Mitchison [10]. In the present paper, the chirality or handedness of spirals depends on fixing the large golden divergence angle 222.5° counterclockwise (or, equivalently, 137.5° Fig. 2. 2-D representation of golden spiral phyllotaxis for four different values ψ clockwise). This is convenient as it corresponds to turns in the of the pattern parameter ξ. The point of the mth leaf has an additional hash mathematically positive sense. mark if m is a Fibonacci number. The points P(0, 0) and P(1, 0) both are marked Cut at some starting point, the front view of the cylinder and its by a cross. unrolled surface are shown in Fig. 1. The axes are normalized as the unit corresponds to the circumference. Let α be the fractional part of iψ i 2.4. Pure parastichies which represents the angular position of the ith leaf. The first values, correct to three decimal places, are α ===0.618,αα 0.236, 0.854, 123 Given a positive interger m, there are right- as well as left-handed and α = 0.472. The length of an arc of α turn on a circle with radius r is 4 i helices for connecting every mth subsequent leaf on the surface of the α 2πr. Thus, normalization has the advantage that it adopts α as it i i cylindrical model. They are represented on the unrolled surface by the stands. The 2-D repesentation of the cylindrical model of golden spiral following line segments: phyllotaxis with pattern parameter ξ consists of the unrolled surface of the cylinder where the leaves are described by points Pi with normal- • (right-handed) from P(0, 0) up to the right toward Pm, ized Cartesian coordinates • (left-handed) from P(1, 0) up to the left toward Pm. P = Pα(,/) iξ. ii Parastichies of order m are directed helices that take the shorter way, i.e. parastichies of order m are right-handed if α < 0.5 and left- As ξ is the reciprocal of the normalized rise, i/ξ stacks the normalized m handed if α > 0.5. The chirality of m is the handedness of the para- rise i times. The starting point has two representations, P(0, 0) on the m stichies of order m. Fig. 1 shows the representing line segments of the bottom left and P(1, 0) on the bottom right. Reconstructing the cy- parastichies and thus the chirality for some Fibonacci numbers. linder, the reader should think of the surface being rolled up such that Note that there are m distinct parastichies of order m. Hence, both points and the dashed lines coincide. The dashed line is centered in counting them allows to determine the order of parastichies in the the front view. absence of labels as in Fig. 2. Further 2-D representations are shown in Fig. 2 for some special values of the pattern parameter ξ. 3. Pure visibility

This section considers the cylindrical model of golden spiral phyllotaxis.

3.1. Vicinity

Looking at Figs. 1 and 2 there are imaginary lines between serried points. In particular, in Fig. 2 for ξ = 55 and ξ = 377 the reader can recognize a densely dotted line sloping about 45° increasingly from the origin P(0, 0). In Fig. 2 for ξ = 21 and ξ = 144 the reader can recognize a densely dotted line sloping nearly 45° decreasingly to the point P(1, 0). Other less densely dotted lines can also be recognized. As a HEURISTIC: The lesser the distance between dots the more visible is a dotted line. Let δ(m) be the distance between subsequent points on the parastichies of order m calculated by the length ℓ of the representing line- Fig. 1. Cylindrical model with golden divergence angle, radius r, and pattern segment as follows parameter ξ = 12.05. The ith leaf is identiﬁed with the point labeled by i for i =⋯1, 2, , 13. The red helix or line, respectively, connects even-numbered ℓ(PPα (0, 0),mm ) if< 0.5 δm()= ⎧ leaves up to the right. (Left) Front view of the 3-D model. The starting point, ⎩⎨ℓ(PPmm , (1, 0)) if α> 0.5. where the surface is cut, is centered and marked by a cross. (Right) 2-D representation of its unrolled surface where the back side is shown greyed out. The same distance as δ(m), the Euclidean distance between neighboring The unit on the axes corresponds to the circumference of the cylinder. The line points, is discussed in the references listed above. Turing uses the term segments with dash-dotted extensions represent parastichies of Fibonacci order lattice for the set {Pi | i ≥ 1} and the representing line-segment of he- 1, 3, and 8 going up to the left. lices is called a vector.In[16] on page 56 he writes:

186 B. Herrmann Mathematical Biosciences 301 (2018) 185–189

Fig. 3. (A) Front view of a conifer cone with numbered scales. (B) Front view of the cylindrical model for ξ = 200 as unrolled in D. (C) Front view of a pineapple with numbered carpels. (D) Unrolled surface of B where the back side is shown greyed out.

most two of them since if two numbers are purely most visible then the purely third most visible number has a longer distance by about 1.4, similar to the diagonal of a square. In a more general model there might be three purely most visible numbers when the cylinder is transformed to a cone by widening the radius, cf. Fig. 3 of [10] that illustrates equiangular triangles for certain transition points. In the cylindrical model for other divergence angles three purely most visible numbers are possible, e.g. for 135.9° and 142.1°, as Fig. 4. (Left) Detail of a grasstree with numbered leaf bases (Image without Prusinkiewicz and Lindenmayer [13] Fig. 4.13(a) and (c), respectively, numbers courtesy of Dale [4] Figure 2.5: Measurements of the [yellow] angle show honeycomb patterns with equiangular triangles. between parastichies). (Right) Detail of the 2-D representation with ξ = 21, 500. Note: Visibility to the eye requires that Pm is on the front view of the As calculated below, parastichies of order 55 slope -17° and 89 slopes 40°. cylinder, i.e. αm < 0.25 or αm > 0.75. This property deﬁnes the new Thus, the angle ∠P(0, 0)P P measures 57° and corresponds to the average 55 144 integer sequence [11] which includes Fibonacci numbers Fn ≥ 2 as well measurement of yellow angles in [4]. as Lucas numbers Ln ≥ 11.

3.2. Numerical solution One may define the first principal vector of a lattice as being that which is of shortest non-zero length. This defines it at best with a doubtful sign Let ξ be the pattern parameter of the cylindrical model of golden ...the first three principal vectors will play an important part. They cor- spiral phyllotaxis. The simple algorithm of ranking the lowest values of respond more or less to the “contact parastichies” of other investigators, δ(m) calculates the top-seeded numbers. If two numbers are equal- the correspondence being closest for the parastichies generated by the first ranking they are sorted in ascending order. two principal vectors. For ξψ=≈5/7 12.05 the top-seeded numbers are 2, 3, 1, 5 and The following definition removes the doubtful sign via the cases of they can actually be seen in Fig. 1. Although δδ(2)= (3), the para- δ(m). Pure visibility concerns the pure unmasked model disregarding the stichies of order 2 slope less than those of order 3. Obviously, shape of organs, their overlap and spaces between them. δ(3) < δ(1). The inequality δ(1) < δ(5) can be recognized in rhom- buses like P P P P . Definition 3.1. (cf. Turing’s “principal vectors”) 4 6 9 7 In Fig. 2 for ξ = 21, the top-seeded numbers are 3, 2, 5 where 2 and A number m is purely more visible than i if δ(m)<δ(i). 5 are equal-ranking. Similarly, the second and third place are equal- A number m is purely most visible if δm()= min δi (). i≥1 ranking for ξ = 55 or ξ = 144 or ξ = 377, respectively, where the top- These notions apply to parastichies by means of their order. seeded numbers are 5, 3, 8 or 8, 5, 13 or 13, 8, 21, respectively. The above heuristic justifies the term pure visibility, at least re- Fig. 3 shows specimens of different plants of different size and dif- stricted to the top-seeded numbers, i.e. the first, second, and, less con- ferent form of cells that have the same pattern parameter. (A) shows the spicuous, the third and, occasionally, the forth purely most visible torso of a conifer cone with diameter 46 mm and 200 scales on a total numbers m1, m2, m3, m4. Turing’s first two principal vectors, i.e. the set length of 145 mm. For r = 23 mm and h = 0.725 mm the pattern {m1, m2}, are called a conspicuous pair as in [7]. The term does not mean parameter is ξ = 200. (C) shows the torso of a pineapple with diameter the same as common sense contact parastichies. 106 mm and 72 carpels on its total height of 120 mm. For r = 53 mm, Of course, δ(m) and, hence, pure visibility depends only on the h = 1.66 mm the pattern parameter is ξ = 200. The three top-seeded pattern parameter ξ. For the parameter of Fig. 1 where numbers for ξ = 200 are 8, 13, 5. Departing from the citation of Turing ξψ=≈5/7 12.05, parastichies of order 2 and 3 both are purely most above, the diamond form of scales of the conifer cone omits the second- visible, since ξ has the critical value such that δδ(2)= (3) is minimal. placed 13 as contact parastichy. The hexagonal form of carpels of the Thus, a purely most visible number need not be unique. There are at pineapple confirms 5, 8 and 13 as contact parastichy.

187 B. Herrmann Mathematical Biosciences 301 (2018) 185–189

3.3. Principal result 4. Discussion

Notably, the comparatives and superlatives of Deﬁnition 3.1 are This section raises several issues. purely mathematical. This allows us to prove the following well-known observation. 4.1. Biological signiﬁcance Theorem 3.2. In the cylindrical model of golden spiral phyllotaxis the order The proof of Theorem 3.2 provides deep insight into the relation of of any purely most visible parastichy is a Fibonacci number. the golden divergence angle and Fibonacci patterns. Actually, the re- Proof. Some prerequisites are summarized in Appendix A. Let m be lation between golden ratio and the Fibonacci sequence as mentioned in purely most visible, i.e. δm()= mini≥1 δi (). That m is a Fibonacci Section 2.1 is carried over to phyllotaxis. The proof underpins that number will be shown by Reductio ad absurdum. Assume m is not a Fi- “...the appearance of Fibonacci numbers is nothing but a secondary con- bonacci number. Hence, m ≥ 4 and there is a maximal n such that sequence of the prevailing rule of the constant divergence angle 137.5°” as

Fn < m. Thus, mF< n+1. Of course, F4 = 3 implies n ≥ 4. By (A.2), m emphasized in Okabe [12]. has the same chirality as either Fn−1 or Fn. Thus and since The constancy of the divergence angle is biologically signiﬁcant. It FFmFFnn−+112<<< nn <+, (A.6) applies to either Fn−1 or Fn and yields implies δF()n−1 < δm () or δ(Fn)<δ(m). The latter contradicts δm()= mini≥1 δi () and completes the proof of Theorem 3.2. • events of maximal contact pressure and directional change (cf. [17]) To state what is new explicitly, the proof requires a new argument, and namely, (A.6) which follows from (A.3), (A.4), see Appendix A. • periods when “space and substance are available to induce ﬂoral organs” (cf. Hirmer in [12]).

3.4. Slope of parastichies These events and periods are described in the mathematical model and observed with the variances of the growth process in many plants. Fig. 4 shows a detail of the trunk of a grasstree with diameter Even displacements of the Fibonacci pattern, as shown in the peeled oﬀ 330 mm that grows 20 mm in height at approximately 415 new leaves and unrolled surface of a pineapple in [15], proceed regularly. per year as described in [4]. For r = 165 mm, h = 0.0482 mm the pattern As a CONJECTURE, the purely second and third most visible num- parameter is ξ = 21, 500. The three top-seeded numbers are 89, 144, 55. bers are also Fibonacci numbers. This conjecture does not hold for other Here the diamond form of leaf bases of the grasstree again omits the divergence angles if they have “intermediate neighbors of the y-axis” as second-placed 144 as contact parastichy. described in [3]. Let σm denote the slope of the parastichies of order m. The trigo- nometry of the representing line segment P(0, 0)Pm or, corresponding to 4.2. Common sense vs. pure parastichies the chirality, PmP(1, 0) yields the following formula Let us discuss the relation between common sense contact para- ⎧(/)/mξ αmm=< m /() α ξ if α m 0.5 stichies of a specimen and the top-seeded numbers/conspicuous pair in tan(σm ) = ⎩⎨−−=−−(/)/(1mξ αmmm ) m /((1 α ))if ξ α > 0.5 the corresponding model. Generally, the purely most visible parastichies are contact parastichies. However, the second most visible

(A.1) simpliﬁes the formula for Fibonacci numbers Fn for n ≥ 2 as fol- parastichies need not be in contact as has been shown above for a lows conifer cone and a grasstree. These counterexamples provide conspicuous pairs that are not the same as contact parastichies of ﬂattened n ⎧Fψξn/( ) if n is odd diamond shaped leaves. Similarly, the reader can imagine a pineapple tan(σFn ) = n n ⎩⎨−−−=−FψξFψn/((1 (1 )) )n /(ξn ) if is even with top-seeded numbers 13, 8, 21, 5 where hexagonal carpels show contact for 5, 8 and 13, whereas the third-placed 21 is omitted.

With respect to the grasstree of Fig. 4 let ξ = 21, 500. For F10 = 55 and n ∘ F11 = 89 calculating arctan(Fψξn /( )) yields σ55 =−17. 5 and 4.3. Generalization ∘ σ89 = 39. 5 . Given a plastochrone distance, if the radius increases and, thus, the Finally, we consider “visible pairs” in the sense of Jean [8],i.e. pattern parameter ξ increases then the absolute slope of parastichies pairs of co-prime numbers. Any pair of co-prime numbers naturally decreases. For comparison only, the outer handrail of spiral staircases extends to a Fibonacci-like sequence, e.g. (7, 11) extends to the Lucas slopes less than the inner one. sequence 2, 1, 3, 4, 7, 11, 18, 29, .... As is easily seen, each pair of subsequent numbers of the sequence is co-prime as well. In each case the “fundamental theorem of phyllotaxis” of Jean [8] yields an in- 3.5. Critical values of the pattern parameter ξ terval for the divergence angle such that, roughly speaking, it is a potentially conspicuous pair. Proceeding the Fibonacci-like sequence This section compiles some critical values that originally have been by pairs the intervals are nested and converge to a so-called noble exposed with respect to the normalized rise. For n ≥ 3, the transition angle. In particular, the Fibonacci sequence yields the golden angle ∘ ∘ from one triple of purely most visible Fibonacci numbers FFFnnn−+1,, 1 and the Lucas sequence yields (ψ/ 5 )360≈ 99. 5 .Forthegolden to the next triple FFnn,,++12 F n is speciﬁed in Coxeter [2]. The critical angle, Section 3.5 describes how a potentially conspicuous pair be- value is determined such that the parastichies of order Fn and Fn+1 are comes a conspicuous pair. perpendicular. Thus, they constitute rectangles with diagonals of length Theorem 3.2 generalizes to noble angles where the order of purely

δF()nn−+12= δF (). These rectangles are squarish. The slightly diﬀerent most visible parastichies is a number of an associated Fibonacci-like critical value for the conspicuous pair {,FFnn+1 } satisfying sequence. The proof is reserved for a sequel. δF()nn= δF (+1) is determined in Vogel [17]. Both these critical values are next to the pattern parameter ξF= 2n+1. 4.4. Conclusions and outlook The transition from one conspicuous pair {,}FFnn−1 to the next conspicuous pair {,FFnn+1 }is speciﬁed in Adler [1] by δF()nn−+11= δF (). The mathematical model mimics biology and the notion of pure This critical value is approximated by the pattern parameter ξF= 2n. visibility supports simulation and analysis of packing and shaping. How

188 B. Herrmann Mathematical Biosciences 301 (2018) 185–189 does pure visibility depend on the parameters? This question comes Acknowledgments down to critical values of the pattern parameter ξπr= 2/h. Such thresholds provide a classiﬁcation of golden spiral phyllotaxis con- Finally, the author would like to thank his family, friends, and Peter cerning top-seeded numbers of pure visibility. Fascinating results are Saunders (Department of Mathematics, King’s College, London) for already summarized in Section 3.5 and detailed formulas deserve fur- their support, contributions, and guidance. Thanks to Richard S. Smith ther consideration. (Max Planck Institute for Plant Breeding Research, Cologne) for fruitful Further outlook is foreseen in generalization from golden ratio to discussions. I would also like to thank the participants of the 17th noble numbers as considered by Hellwig, Engelmann and Deussen in International Conference on Fibonacci Numbers and Their Applications [5], generalization from the cylindrical model to centric discs and solids for the opportunity of my talk and stimulating questions. I would like to of revolution, and computer simulation of phyllotactic transition as oﬀer my special thanks to Paul Cull (Computer Science, Oregon State mentioned by Zagórska-Marek and Prusinkiewicz in [18]. University) and the anonymous reviewers for most valuable comments on the manuscript.

Funding

This research did not receive any speciﬁc grant from funding agencies in the public, commercial, or not-for-proﬁt sectors.

Appendix A. The golden sequence

Here are some facts about the golden sequence α123,,,αα… of fractional parts of multiples of the golden ratio. Exercise 31 of Knuth [9] p. 85 (or [6] Lemma 2.1) states that

n ⎧ψnif is odd αFn = n ⎩⎨1ifiseven− ψn (A.1) n Let n ≥ 2. Since ψ < 0.5, (A.1) implies that a parastichy of order Fn is right-handed if n is odd and left-handed if n is even. Thus, for n ≥ 2

Fibonacci numbersFFnn and+1 have different chirality (A.2) Lemma 2.2 of Herrmann [6] states that ifnααiF is odd thenFin ≤< for all n+2 (A.3) ifnααi is even thenFin ≥< for all Fn+2 (A.4)

In the Cartesian plane the length ℓ of the line-segment between two points P(x1, y1) and P(x2, y2) is calculated via the Pythagorean theorem as follows:

2 2 ℓ((,Px1 y1 ),(, Px2 y2 ))=−+− ( x21 x ) ( y21 y ). It is readily seen that the deﬁnition of δ(m) is equivalent to

2 2 ⎧ αmξ+<(/) if αm 0.5 δm()= m ⎨ (1−+αmξα )22 ( / ) if > 0.5 ⎩ mm (A.5) Let n ≥ 3.

If FmFnn<<+2 have the same chirality thenδFδm (n )< ( ) (A.6)

For instance, F6 = 8 and m = 11 have the same chirality, see Fig. 1. We show that (A.6) follows from (A.3), (A.4) and (A.5). α 2 2 Let n be odd. Thus, Fn is right-handed. Let m have the same chirality, i.e. m < 0.5. Hence, by (A.5), δF()n =+ αFn (/) Fn ξ and 2 2 ξ 2 ξ 2 δ δ ﬃ δm()=+ αm (/) mξ .AsFn < m implies (Fn/ ) <(m/ ) , for (Fn)< (m)itsu ces to show αFn ≤ αm. The latter holds by (A.3) and mF< n+2. α 22 Let n be even. Thus, Fn is left-handed. Let m have the same chirality, i.e. m > 0.5. Hence, by (A.5), δF()nFn=− (1 αn ) + (/) F ξ and 22 ξ 2 ξ 2 δ δ ﬃ δm()=− (1 αm ) + (/) mξ .AsFn < m implies (Fn/ ) <(m/ ) , for (Fn)< (m)itsu ces to show 1 −≤−ααFn 1 m. The latter is equivalent to

αFn ≥ αm and holds by (A.4) and mF< n+2. This completes the proof of (A.6).

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