Mathematical Biosciences 301 (2018) 185–189
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Mathematical Biosciences 301 (2018) 185–189 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs 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 phyl- lotaxis. 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 para- stichies 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.