The mathematical daisy - Stanford Universityrr631pd8806/rr631pd8806.pdf · NewScientist 17December 1981 793 Ishall come back to the significance of the precise \ angle of 137-50776.

792 New Scientist 17 December »98l

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The mathematical daisy"

The inner florets of a daisy-and many other plant parts-grow in spiral patterns that are fundl^^linked to the theory of numbers and the golden ratio known to the mathematicians of ancient Gr^Hi

Robert Dixon Why do plants often appear tofollow particular patterns as they

grow? How do they know which pattern to follow? Canwe explain the patterns using the theory of numbers,and if so, why? These questions had been intriguing mefor some time when I realised that modern computergraphics provides the ideal way to draw out patterns ofplant growth according to specified mathematical rules.Thus with the aid of the computer I have investigatedthe problem of spiral growth in a variety of plants.

The patterns into which plants grow represent opti-mum solutions to the geometric problems they face: ofhow to pack all the constituent parts into a structurethat can grow continuously, or how to occupy a volumeof air and receive sunlight with maximum economy.Botanists and mathematicians use the term "phyllotaxis",whose original meaning is "leaf-arrangement", whenclassifying and analysing the arrangements of any repeti-tive parts of a plant, including florets, seeds, petals,branches and so on. One elementary and widespreadpattern of such parts in plant life is the formation of aparticular type of spiral, named' after the medievalItalian mathematician, Leonardo Fibonacci.

I first looked at this pattern as it occurs in compo-site flower heads, such as daisies. The way the floretsare packed together on a daisy head looks somewhatsimilar to the structure of a honeycomb from a beehive,but the differences are significant. Instead of a symmetri-cal array of identical units, as in the honeycomb, thedaisy is a collection of florets all at different stages ofgrowth^ What you see if you look carefully at a daisyRobert Dixon has a degree in mathematics and now works as a freelanceartist. This article is based on work reported in an articleunder reviewforpublication in Leonardo. The author would like to acknowledge help andinspiration from Alan Senior, Ensor Holiday. Keith Laws, Cliff EdwardsandProfessor Coxeter.

Figure 1 A daisy-head (far left) anda sunflower showspiral patternscurling clockwiseand anticlockwisefrom the centre.The florets, andultimately the seeds,are generatedat thecentre, one afteranother in a singlesequence.Eachfloret is pushedfrom the centre asnew ones

form,

andas older florets growin she. The singlespiral of growthgenerates apparent"secondary" spirals.Count thenumberof spirals you see ineither direction. Thenumbersyou willfind are Fibonaccinumbers—membersof the series1,12J,5,8,1321, . . .

are spirals, running both clockwise and anticlockwise(Figure 1). If you count the number of spirals in eitherdirection you will find that the total is not any number,but a number that is a member of the so-called Fibonacciseries: 1,1,2,3,5,8,13, . . . , where the three dots indicatethat the series continues indefinitely.

Fibonacci published his influential Liber Abaci' in 1202.It is a book on the abacus in which he set down thecase for adopting the Arabic system of notation fornumbers instead of the Roman—29 instead of XXIX, andso on. He also posed a problem about the breedingpattern of rabbits whose periods of maturation andgestation are both one month. If each pregnancy yieldsone new pair, and if you start with a single newbornpair on 1 January, how many pairs will there be onthe first day of subsequent months? Answer: a series ofnumbers which has since been named after Fibonacci.

But why do the spirals in daisies occur only in thequantities given by the Fibonacci series? Of course thisis really two questions rolled into one: why spirals, andwhy Fibonacci numbers? The simple numerical patternin the Fibonacci series does not suggest an obviousexplanation for "Fibonacci spirals" in plants. In

fact,

despite the almost perennial appearance of fresh studiesdating back to the 1830s, when a rigorously mathematicalapproach to the puzzle was first introduced, it has proveddifficult to provide a completely satisfactory explanation.

The most successful line of analysis begins with theobservation that the different spirals that you see onthe daisy head are formed rather by a single spiralsequence of florets. And in this sequence the floretsfollow each other at an angle of 137-50776 .°. Thesingle spiral is the "primary" spiral; the apparent spiralsthat correspond to the Fibonacci numbers are "second-ary" spirals.

New Scientist 17 December 1981 793

I shall come back to the significance of the precise\ angle of 137-50776 . . . °; suffice it to say that this

special angle not only produces the apparent secondaryspirals, but also provides a uniquely flexible design forthe harmonious and efficient spacing of plant partsrepeated around a central stem, even as they grow. Themathematical difficulties lie in turning these intuitivelyspatial ideas into a sound theory.

While the analysis and proofs are difficult, the basicgeometric ideas of a spiral sequence and a regularangular interval are relatively simple. It is possible toconsider the pattern visually, and in particular computer-ised drawing allows one to construct the mathematically-idealised pattern with great speed and accuracy. Butmore of this shortly.

Besides the daisy head, other oommon examples ofFibonacci phyllotaxis—plant growth on Fibonacci spirals—include the arrangement of the sunflower's seeds, thepine cone, the petal sequence in a rose or a lotus, thesequence of leaves on a thistle, the fruit partitions ofa pineapple, and the succession of twigs branching fromthe stem of a pear tree. The form of a daisy head is aflat disc with a central point, while other examples takethe forms of a cone or a cylinder. This variety of formsis unified, however, in the continuous spatial transforma-tion from one to another: a cylinder becomes a cone byreducing the radius of one end to a point; a cone becomesa flat disc by reducing its height to zero. And whena flat disc is extended to form a cylinder, the patternof spirals on the disc becomes a pattern of helices.

The pattern of leaves growing from a stem reflects an figelementary predicament that plants face—that of how thito occupy space, collect sunlight and breathe, in the most reieconomic way. To begin with, a plant grows along an foiaxis, extending its occupation of space along one line .to gather more and more sunlight. Then periodically *it sprouts leaves which branch out from the stem tooccupy the surrounding space. But in which directiondo they sprout? At every point on the stem the planthas 360° around the stem to choose from. In response cto this choice, plants have evolved several systematicbranching patterns, each species following one or other.These patterns represent the relatively few optimumsolutions to the geometric problem. One such patternis the spiral/helical succession of branches at every137-50776 . . .°, or Fibonacci phyllotaxis (Figure 2).

To see why this is an ideal angle, or to see why anyinterval of spiral succession is better than any other,consider the Sun's-eye view of the plant after severalleaves have sprouted. Each leaf needs its own equalplace in the Sun. So each new leaf seeks a directionfrom the stem not already occupied by a lower leaf.And one by one the available gaps are further divided,as the circle becomes crowded with more and moreleaves. The significance of 137-50776. . .° is that it is theangle which most evenly and gradually divides the circle.

You might wonder what would happen if other angleswere used as the repeated interval, and so query theuniqueness of 137-50776 . . .". First, any exact fractionof a whole circle has the drawback of repeating itselfafter a finite sequence of directions. This leaves the fullcircle of directions with gaps which can never be closed.

But what happens if each new member of the sequencefalls between the direotion of two previous members?If it falls exactly mid-way between its two neighboursthen the sequential interval is again an exact fractionwith its consequent drawback. On the other hand, it isno good if the new member is too near either of itsneighbours.

With an interval of 137-50776 . . .", each new memberof the sequence divides the gap between its two pre-ceding nearest neighbours in the so-called "goldenmean". Describing this angle of 137-50776 . . ." in de-grees disguises its identity somewhat. It is an expressionof Euclid's "golden section", but in a less traditionalform—the golden section of the circle. Three geometricfigures that traditionally express the "golden ratio" arethe golden section of a straight line segment, the goldenrectangle, and the regular pentagon. Figure 3 shows allfour examples together.

A rectangle whose proportions are in the goldenratio,that is, whose long side is r times the length of theshorter, is a golden rectangle. The diagram shows itsinherent pattern of recursive squares. Remove a square

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from a golden rectangle—the square which fits theshorter side—and you are left with another, smallergolden rectangle. And so on indefinitely. The sequenceof squares spiral forever inwards towards a limitingpoint—following a logarithmic spiral. Conversely if youadd the square to a golden rectangle you get a largerrectangle, which is itself a golden rectangle. The squareis an addition which leaves the shape of the goldenrectangle unchanged, while changing its size and orienta-tion. The addition increases the size by the same pro-portion every time, giving a pattern of uniform growth.

In a regular pentagon a diagonal, the line joining anytwo non-adjacent corners, is r times as long as a side.Not for nothing was the pentagram, the continuous starformed by all five diagonals of a regular pentagon, thesymbol of membership to Pythagorean brotherhood.

The golden section of a circle divides the whole cir-cumference into two arcs, the longer being t timesthe shorter. Again, the ratio of the longer to the shorterpart is the same as that of the whole to the longer.But unlike the straight-line segment which has two endsand needs only one cut, the circle has no end and needstwo cuts. The golden section of a circle subtends a"golden angle" at the centre—which is none other than137-50776 . . .°.

So there is an ideal angle that guarantees that theparts of a plant—be they seeds, leaves, florets, twigs orpetals—which branch out in spiral succession will fallinto place in an ideal way, and this angle is the goldensection of a circle. But why do the Fibonacci numbersturn up in daisies and other plants? The answer to thisquestion lies in the number theory of the golden ratio.

Certain geometric quantities cannot be exactly repre-sented in whole-number ratios. This discovery troubledthe Greeks, and continues to this day to pose a trickyhurdle in our mathematical education. Classic examplesare the relative sizes of the circumference and diameterin a circle, given by ir = 3-1426 . . . , and the relativesizes of diagonal and side in a square, given by </2 —1-414 . . .—post-Greek mathematicians describe suchquantities as irrational numbers (no insanity implied).

So, the circumference of a circle is approximatelythree times the diameter, but not quite—22/7 is a betterapproximation;31 416/10 000 is much closer, but still notexact. The decimal notation, 3- 1416 . . . , expresses thislast approximatingratio of whole numbers, and also indi-cates in the three dots that better and better approxi-mations can be obtained indefinitely by finding the valueof more and more decimal places. Another such geometricquantity which cannot be exactly expressed in wholenumbers is the golden ratio, t. Like </2, but unlike

ir,

r is the solution to an algebraic equation, namelyx' = x + 1.

In decimal notation, and for all practical purposes,t = 1-618034 . . . But if we express r as a decimalwe miss the fundamental truth—that t can be derived,or at least approximated, by coupling Fibonacci numbers(2, 3, 5, 8, 13, . . . ). Thus a rough approximation tothe value of t is given by 3/2, the first two numbersin the Fibonacci series greater than 1. But 5/3 approxi-mates more closely; 8/5 even more closely, and so on.The fractions in this series, 3/2, 5/3, 8/5, 13/8, 21/13,. . . , approach successively closer to the limiting valueof t, while alternating above and below its ultimatevalue.

In this sequence of fractions whioh converge on thevalue of r, the numerators (above the lines) and thedenominators (below them) are from the Fibonacciseries. The irrational value of t therefore finds a com-pletely ordered sequence of approximations in the wholenumbers. Moreover, these whole numbers are the num-

Figure 4 Repeatedsteps of 2/5 of a turn on a circle give fivepositions (top); on a spiral, five rays emerge (left). But a stepof slightly less than 2/5, creates five secondary spirals (right)

bers which turn up in daisies and so on. But how donumerators and denominators become spirals?

In Figure 4 I try to explain the answer to this lastquestion. The first diagram shows the sequence of posi-tions reached by a moving point circling clockwisethrough regular discrete steps of 2/5 of a whole turn.Mter five steps—the number given by the denominator—you come back to where you started, after which yourepeat the same five positions over and over. The nextdiagram shows what happens if at the same time asrotating clockwise you steadily increase the radial dis-tance from the centre, forming a spiral sequence. Themoving point now reaches a new position every time,but they are restricted to five directions from the centre.The positions together form a pattern of five straightrays. If the fraction of rotation had been, say 3/8 of aturn instead, the pattern would have been eight straightrays. The denominator gives the number of rays.

Figure 5 An "electronic" daisy drawn by computer graphics.The single spiral sequence starts at thecentre, each floretfollowing every 137-50776 . . .". The spiral windsmore than100 times before it reaches the periphery. The outer 14floretswhich wouldbe the first 14 formed in aplant—have beenreplcuced by numbers 0-13. These florets lie on the primaryspiral which connects themin order. The florets grow beforereaching a maximumsize; as they grow theFibonacci numbersof secondary spiralsemerge

New Scientist 17 December 1981 795

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The third diagram of figure 4 modifies the patterna stage further. Instead of an exact 2/5 of a turn Ihave used an angular interval which is nearly, butslightly less than 2/5. The resultant pattern approximatesto the previous diagram, but the five straight rays becomefive spirals, displaying a slightly out-of-phase effect. Thisis how secondary spirals emerge from the single primarygenerating spiral sequence. The figure also shows thatthe number of secondary spirals is the denominator ofthe fraction to which the angular interval approximates—2/5. Similarly you generate Fibonacci spirals when theangular interval is 1/t of a turn, the golden angle.

Spiral growth—on a cone (left) and a succulent

Computer graphics provides a powerful means of ex-ploring patterns like Fibonacci phyllotaxis. The com-puter turns theory directly into a drawing—Figure 5shows such an "electronic daisy". The tiny pentagramsrepresent the florets, in a sequence that begins at thecentre and spirals steadily outwards at each turn of thegolden angle. I allowed the florets to grow non-uniformly(increasing exponentially at first, but slowing down toan upper limiting size) and then formulated the primaryspiral to enclose an area which is always in step withthe accumulating area of florets. The results revealsecondary Fibonacci spirals and a quasi-regular space-filling array.

Also, the spirals of contacting florets change from onepair of Fibonacci numbers to another. This correspondsto the increasing number of florets needed to packaround the growing circumference as the growth rate ofthe florets slows down. The 8 by 13 system at the centregives way to a 13 by 21 system, followed by a 21 by 34system at the periphery. While the growth rate isexponential (increasing by the same proportion fromone term to the next) the Fibonacci numbers of con-tacting florets stays the same. But the smooth way inwhich the contacting systems switch from one set toanother without the pattern departing too far from aregular packing is perhaps the crucial and unique pro-perty of Fibonacci phyllotaxis. The comparable phenome-non using non-goldenanglesis always less regular.

Having seen the computer demonstrating how a plantcan achieve such an efficient arrangement by using thegolden angle, I was eager to see what happens whenyou use other angles. Figure 6 shows what happenedwhen I repeated the drawing of the electronic daisy butwith different irrational angles, namely, 1/ir and 1/e(e = 2-718 . . . ). The units I am using here are wholeturns.

Again, one spiral sequence makes a pattern of second-ary spirals. They occur in numbers which are not Fibo-nacci numbers, but the denominators of the relevant

Figure 6 "Electronic" daisies based on non-golden angles—llrr,

left,

and1 le.. In each of these the approximating wholenumberfractions are indicatedby the numbers of secondary spirals

approximating whole number fractions. For example, thefirst example in Figure 6 uses the familiar irrationalnumber tr, in the form of the fraction of a turn l/?r.The first whole number fraction to approximate to thisis 1/3, followed by 7/22; thus three spirals dominate thecentre of the pattern, slowly switching to 22 spirals.

The mathematics of the electronic daisy describe thepath followed by the computerised pen-point across thedrawing paper plane. This is not to be confused withthe path along which a flower grows. Each drawingrepresents a sort of snap-shot in time. To represent thewhole growth of the flower requires a sequence of draw-ings with good continuity from one to another. Theycould be shown in rapid succession on a single screen,making an animated film. A continuous sequence ofFibonacci patterns of this kind represents an ideal path-way of quasi-regular packing, along which a flower canflow—a pathway of least resistance. □

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