© ATM 2012 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions. SpiRalS, SpotS and continUed FRactionS Robert dixon explains ... there is a wealth of number theory to be found in a famous spatial pattern approximated by plants.
22 n his book On Spirals, archimedes Here the area swept out is ka 4 demonstrates from first principles what Because the area swept by a moving radius we now call integral calculus in polar i overlaps itself once it exceeds one revolution, coordinates. He finds the area under a curve archimedes performed subtractions to find swept out by a radius rotating about a fixed that the areas of successive coils are in the pole. His one example is the curve r = kθ, the proportions 1:6:12:18:24:... the radius of the archimedes spiral. it is the spiral whose radius archimedes spiral increases by the same amount increases uniformly with turning. divide the sector each turn, and so the area enclosed each turn of the spiral θ = 0 to θ = α into n small sectors of a increases with radius. Following in archimedes’ . angle n each small sector has an area between footsteps, Fermat applied the same calculations an inner and an outer circular sector. in radian to his spiral , and found that the areas measure, the ith outer circular sector area is of its successive coils are in the proportions ki22a 3 k23a 1:2:2:2:2:... an equal area is enclosed each turn, 3 .the sum of all these is Σi² times 3 . 2n 2n and the same may be said for any part of a turn. Knowing a number series, the sum of squares, it is the spiral whose area increases uniformly here solves a geometric problem, the area under with turning. to put this equal area property in an nnn32 a curve. Σi² = ++. So the approximating equivalent rule, the Fermat spiral traverses equal area is k2a3 1 1 1 . this has the limit annuli in equal angles. a + + 2 k ka23 6 4n 12n Whereas the archimedes spiral is a simple coil 6 as n → ∞. and the equiangular spiral is the curve of mollusc
a) b) c)
Figures 1(a)-(c): polar integration, the archimedes spiral and the Fermat spiral.
archimedes shows that the sum of inner circular shells, we can look for the Fermat spiral in botany: sectors has the same limit, thus ‘exhausting’ the for example, in the whorls of florets on a mature difference to prove it is the spiral area. of course, daisy head. the reason for this becomes clear he did not have algebraic notation. nearly two if we learn to draw ‘Vogel daisies’, after Helmut thousand years later, a decade before the birth of newton, pierre de Fermat was reading this Vogel, who first used computers to draw them. work by archimedes and practising the first steps (See Figure 2.)the idea is to, a) draw a single in algebraic notation. He must have become Fermat spiral of small equal circles at constant intrigued by the use of number series to find angular interval, α; and to b) explore what values areas, and wondered what spiral would yield to of α gives the best distribution of the circles. the simplest non-trivial series of all - not Σi², but Without any circles overlapping, how much of the Σi. the answer is , the Fermat spiral. plane can be covered? What is the pattern of gaps?
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can we achieve a close-packing? With unit every irrational can be written uniquely as an circular measure, α can be any real value between infinite simple continued fraction: 1 0 and 1. in botany, α is called divergence, the α = [a ; a , a ,....= a + 0 1 2 0 1 angle between successive like parts - leaves, fruits, a1 + branches - about the axis of a stem. By choosing 1 a2 + different values for α in a computer graphic daisy, a3 + ...
a) b)
c) d)
( 15- 3) ( 5- 1) Figures 2a to d: computer daisies, with a 22, π, , and = 7 3 2 .
different patterns of scattering of the circles will all ai are positive integers, except that ao may p be zero. the th principal convergent is the finite result. For rational α = /q, the pattern will be q radial i pi spokes. only q directions will be occupied. this is continued fraction, [a0; a1, a2,…, ai] = /qi, given by remedied by choosing irrational values for α. With recursion formulae:
irrational α, no two circles in the spiral can occupy p0 = a0, p1 = a1a0 + 1, pi = ai pi-1 + pi-2
the same direction. the pattern will be a mesh q0 = 1, q1 = a1, qi = aiqi-1 + qi-2 which at a succession of radial distances resolves into sets of symmetrically spaced spirals, some sets For example, π = [3; 7, 15, 1, 292, ...] So, the spiralling one way, some the other. the numbers principal convergents are: of spirals in each set are the denominators of the closest rational approximations, the convergents of α. these can be found as follows.
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principal convergents approach their limit - π in this case - alternately from below and from above. Between two consecutive principal convergents pi-2 pi on the same side of the limit, qi-2 and qi there are ai − 1 intermediate convergents on that side, (npi-1 + pi-2) given by /(nqi-1 + qi-2), n = 1, 2, 3,..., ai – 1.
For example, with π, because a2 = 15, there are fourteen improving underestimates between 3 333 25 47 69 /1 to /106, namely /8, /15, /22, and so on to 311 /99, but because a3 = 1, there are no improving 22 355 overestimates from /7 to /113. this theory can be visually represented in a c) Figure 3: the Klein lattice diagram, for α = √2 diagram named after Felix Klein. on a unit square lattice, let any line drawn from lattice point (0, 0) 1 x the answer is α = [0; 1, 1, 1, ...]. Why? Well, it to infinity represent α = /gradient, or / . thus, the y can be shown that the size of gaps in each of the line for any rational α = p/ goes from (0, 0) to q successive sets of spirals showing up in the mesh (p, q), and from there to (2p, 2q), and so on forever is a function of the a’s in the continued fraction through the infinity of lattice points at all multiples i of α, and that size is minimised by minimising of the distance √(p2 + q2). the line for an irrational each corresponding a. Uniquely among all α, on the other hand, goes from (0, 0) to infinity i irrationals, with all a = 1, there are no intermediate without ever passing through another lattice i convergents. the principal convergents are point. We next imagine such a line as a taut string 0, 1, ½, , , , , ... these fractions are anchored at infinity, and all the lattice points as composed of consecutive Fibonacci numbers, upright pins. taking the end at (0, 0), we imagine Fi F . a computer graphic daisy drawn with this the taut string swept first one way then the other, / i+1 value of α will show concentric sets of spirals in in the plane. in each case, the string coming into Fibonacci numbers, as found in botanical daisies. contact with the pins - lattice points - forms a polygon. the polygon vertices are at each of the in fact, [0; 1, 1, 1, ...] = ½(√5 – 1), none other than the golden section, τ. continued fraction theory (pi, qi) corresponding to the principal convergents p i/ . elsewhere the string may touch other lattice formulates the degree of approximation to a given qi points along the polygon sides, between polygon irrational for each of its principal convergents,
vertices: these correspond to the intermediate and this degree is poorest with smallest ai’s. the convergents. golden ratio is the ‘slowest converger’. it is the most remote from a fraction of numbers up to Figure 3 takes the case of α = √2. any given size of denominator. You might say it is √2 = 1 + (√2 – 1) = 1 + 1/(√2 + 1) the most irrational. as a point on the open string = 1 + 1/(2 + (√2 – 1) = 1 + 1/(2 + 1/(√2 + 1)), etc., in music, it must be the least harmonious. But in which gives √2 = [1; 2, 2, 2, ...]. the mod 1 arithmetic of serial packing or spacing all quadratic irrationals are periodic continued round an axis, it is the best divergence. fractions, and vice versa. the principal the discovery of Fibonacci numbers in botanical convergents for α = √2 are , , , ,... phyllotaxis - ‘leaf arrangement’ - dates back to the the intermediate convergents are , , ,... 18th century. this is seen in the numbers of spirals or helices in compound flowers or fruits forming So, when drawing a Fermat spiral of equal circles a disc or cone or cylinder - for example, daisy, at constant divergence α, what irrational value of sunflower, pine cone, pineapple - or is seen in the α gives the best spread? numbers of leaves and revolutions in a spiral
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succession of leaves about a stem before close on coxeter’s heels, however, irving adler returning to the same leafing direction.ar ound looked at the puzzle quite differently. Knowing 1830, the botanist and crystallographer brothers that divergence in actual plants is neither precise l. Bravais and a. Bravais interpreted all these nor fixed, he devised a morphogenetic model. occurrences of Fibonacci numbers as equivalent adler considered what happens to a helical to the geometric fact of golden section divergence. packing of equal circular discs on the surface of thereafter, for a century and a half, the subject circular cylinder if the girth of the cylinder steadily was studied time and again in a string of lovely expands. Starting with discs that are quite large in monographs, but it remained a mathematical relation to the cylinder diameter, and with a more mystery until 1972, when H.S.M. coxeter brought or less arbitrary initial divergence, he shows that the above number theory to bear upon it, taking his a process of continuous settling to maintain close inspiration from the Klein lattice diagram. packing as the cylinder diameter expands leads on that square lattice diagram coxeter draws the to a changing divergence which converges in line for τ = [0; 1, 1, 1, ...], which is the line y = X/τ. oscillating fashion towards the golden section of a then he applies a shear to the whole diagram, circle. Growth in plants is not a crystal formation (x, y) → (x – τy, y) to bring this line into vertical. of static parts added to a static form over time, the square lattice is transformed into a but growing parts added to a growing form over parallelogram lattice. this lattice can now be time. typically the parts are added in sequence divided into identical vertical bands one unit wide. at the tip of a stem, which is conical, and which continually extends into a cylindrical stem.
Figure 4. coxeter’s demonstration for cylindrical phyllotaxis
there is one lattice point from each row in each the parts are born near the apex of a cone and band at the same position. taking the band migrate to a place on a cylinder. the disc form of x = 0 to x = 1, the lattice points have coordinates a daisy head is the opposite extreme to a cylinder, (-yτ (mod 1), y), y = 0, 1, 2, 3, .... this is an a flattened cone. in adler’s model, as the girth unrolled cylindrical phyllotaxis with golden section of the cylinder steadily expands in ratio with the divergence. in this way coxeter demonstrates size of the circular discs, the discs continually that α = τ is the only fixed irrational divergence rearrange themselves. divergence continuously which ensures ‘tait’s condition’, that after any changes accordingly. there is a wiggle in given leaf direction about a stem, successively divergence, and adler derived its formula. in his nearer returns to that direction in subsequent leaf idealised model, it is a chain of quadratic arcs. directions will occur in strictly alternating fashion, first one side, then the other, because it is the only irrational without intermediate convergents.
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,
for
divergence α is formulated as a function of rise, in a mature daisy all the florets are fully grown h, the increase of height on the cylinder between and equal. prior to that, all or some of the florets, successive discs, with unit cylinder circumference. which were generated sequentially in time, are still growing. they form an approximately exponential adler’s cylinder is an exquisite demonstration sequence, which packs in an approximately of automatic generation of golden section equiangular spiral, θ θcotβ. in the life of a divergence, on a growing cylinder of equal r = k = e daisy flower, the spirals begin equiangular and parts. it does not directly model a collection of grow into Fermat; all, of course, approximately unequal parts growing on a growing cone or disc. speaking. and come to think of it, as well as an Yet, coming after adler, Stéphane douady and angular wiggle, there must also be a radial shake, Yves couder investigated disc phyllotaxis, first as packing passes from triangular to square and physically with electrically charged oil drops on back again. a metal plate, then by computer simulation. on a horizontal plane disc, a large number of small equal discs, which are mutually repulsive and free to move, are added at the centre one by one. douady and couder found that this sufficed for the automatic formation of Fibonacci phyllotaxis, and their numerical graphs of divergence show a wiggle. Moreover, it is apparent that although the optimal Vogel daisy - fixed divergence - may have the least gaps, it does not have no gaps - see figure 2d.a further improvement of packing would be gained by some sort of wiggle twist. Somewhat surprisingly, adler’s exact formula for cylindrical phyllotaxis supplies a good approximation for this, as shown in Figure 5.
robert Dixon, author of mathographics, Dover Publications, New york, 1991; formerly lectured pure mathematics and statistics at City and east London College, London; he has written a book on modern Art in Higher education.
Reading M.S. Mahoney, The Mathematical Career of Pierre de Fermat, princeton, 1973 archimedes, The Works of Archimedes, ed. t.l. Heath, dover publications, 2002. Figure 5. immature = still growing. R.V. Jean and d. Barabé (eds.) Symmetry in Plants, World Scientific, 1998 i. Stewart, Life’s Other Secret, allen lane, 1998
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