Here is a program that will generate sets of 500 points with a given angular separation.

First the program chooses random numbers, from which it generates sets of 500 points, each set having a particular angular spacing, chosen at random between 0 and 360 degrees.  In every case, the space is not filled efficiently; you will see some form of bunching, as in the example below.

Then the program uses the golden angle so that you can see how no obvious pattern emerges, and how the space is filled without apparent bias.  That is very odd – a random filling is in a sense more symmetrical than the rational ones, in that it looks the same everywhere.  On the other hand, a random set has no symmetry, in that it is unique everywhere.  No operation can superimpose it on itself except the identity operator.  Of course, this pattern is not random, because it is based on the recurrence of a particular angle.  In this it resembles chaotic patterns, which are also not random, however much they may appear to be. 

The definition of "random" is in fact not easy to frame, because in principle any finite set of numbers can be fitted exactly by a formula that contains enough parameters.  That is the clue – if we can describe a set using fewer elements than are contained in that set, it cannot be classed as random.  An extreme example is the Mandelbrot set, which is exceedingly complex, but is generated by a simple rule.  This is all very well, but suppose we define the "real numbers", of which there are infinitely many.  The definition of a real number is both finite and simple, yet the real numbers follow no pattern.  This is the sort of place where authors like to "leave it as an exercise for the reader."

Click here – and select "Run the program from the current location" you can see what happens.  You should see something like the picture below.

So the whole Fibonacci thing is an accident.  The rule is that the leaves or florets grow for maximum space.  The rest – Fibonacci numbers, spirals, pretty patterns – follows automatically.

In this context, the Fibonacci numbers are like the magic numbers in nuclear physics.  The difference is that if plants had never existed, the Fibonacci numbers would still have interesting properties, whereas the values of nuclear magic numbers are dependent on the properties of nuclear forces, and are not in themseleves interesting.  Very far from the floor of the valley of stability the values may even be slightly different from the normal ones.

Furthermore, if the universe could have been created with slightly different fundamental constants, the nuclear magic numbers could have been different, but any life-form that could grow like plants would still show the GS and Fibonacci numbers.

Perhaps the whole of physics is like the plant numbers.  Perhaps the many elegant laws that so many people have struggled for so many years to create are just an accidental reflection of a few deep rules that we do not know.  For example, from the use of the Lagrangian function, many laws can be recovered.  The laws of refraction and reflection can be deduced from the simple rule – light takes the path of least time.

This beautiful heron needs no knowledge of maths, botany, genetics, physics or engineering of the palm on which it sits and watches for fish and little crabs.  Where does the idea of beauty or elegance come from?  Whether you admire a motor bike or an equation, you probably have some idea of what you find admirable.  What would life be like without that sense?

Back to "maths".  Bees do not try to make hexagonal cells as such.  Their method uses the minimal material compatible with the strength of the honeycomb and with minimal time spent in perfecting, but not over-perfecting, the work.  The picture shows an old nest of a Polistes, a solitary wasp.  If you find an old wasp nest in your attic, you will find out how light it is, for its size.

Again the maths is not inherent as maths, it is only a by product, just as a football is made spherical because that is the shape that rolls and flies best.  As a bonus, inflating a slightly non-spherical ball may tend to make it more spherical.

You can find the Fibonacci numbers by counting the helical series of scars on many small coniferous trees, on their cones, and in the helical arrangement of leaves on plants.

If you count the parts of different daisies, thistles, sunflowers, cones, etc, you will find different pairs of Fibonacci numbers.  How can this happen?  It happens because the change in radius between successive parts is different in each plant.

This section through a cabbage shows leaves cut through at various angles relative to their axes.  Starting at the bottom left, and calling that leaf zero, you might be able to convince yourself that leaves 3, 5 and 8 are thicker, meaning that they have been cut fairly near their axes.

The diagrams below show what happens as the ratio of successive radii varies.  You can count the spirals to see whether Fibonacci numbers appear, and if so, which ones.  A previous diagram generated the Fibonacci numbers using only the golden section.  

Using only one more parameter, we can select which two are used in an actual growing plan.  If the distance of a floret from the axis of a sunflower is d, the growth factor is the factor by which d grows during the time between its inception and that of the next leaf.  The resemblance of the system to a set of logarithmic spirals suggests that d remains fairly constant during the growth of the entire flower.

But in the thistles shown at the top of the page, in teazles, in pine-cones, in pineapples, and in many other examples, d decreases with the age of the plant part, and the growth enters the third dimension.

We must remember that these diagrams are only snapshots in time.  The flower does not grow in spirals: the florets just expand outwards from the centre.  Click here to run or download a simple demo.

The leaves of many plants are arranged on the stems in helices that seem to be based on small Fibonacci numbers.  This seems reasonable in terms of their original growth points.  But once they are spread out along the stem, the spacing requirement disappears.

Of course, if the stem grows with no twists, the pattern will remain.  Perhaps there is also a physical reason.  What is the optimal distribution of leaves on a stem to minimise the shading of any one leaf by all the others?  To calculate this we could assume isotropic illumination for simplicity.

But it is not an easy calculation.  If we use a Monte Carlo technique, we might need a vast number of data, if the variations are small, in order to see them in the noise.  Yet a small minimum in a variable would be enough to gain an evolutionary advantage.  Every square millimetre of leaf that is less well illuminated that another part represents energy wasted in producing it.

Here are some diagrams showing phyllotaxis reduced to two dimensions, by unrolling the 360 degrees of a cylinder on to a plane.  The labels show the angle between successive leaves, and the number of leaves per turn of the helix.  Do you think that shading is affected by phyllotaxis?

Some people think of science as "reductionist".  By showing that many phenomena can have the same simple cause, science can certainly reduce the number of credible origins of these effects, and it may reduce the number of ways in which we think the world could have been created and constructed.  

But the world remains exactly as before – nothing physical has actually been reduced – all we have done is viewed another way of looking at the worl.  You can listen to Mozart’s symphonies without knowing that the first movements are very likely to be in sonata form, and you don’t need to know about fugues or rondos either.  But it doesn’t do any harm to know what a composer was up to.  

And you certainly wouldn’t enjoy a football match or a cricket match or a game of chess very much if you didn’t know the rules.  If you watch a game for the first time, you will very rapidly try to work out some of the rules, or you will ask someone near to you to explain them.  That’s exactly what scientists do when they observe parts of nature.

We cannot all go to distant places to see exciting plants and animals, but the ones we find around us at home are in many ways just as interesting.  If we insist on dividing the world into "interesting" and "uninteresting" parts, then we are indeed reducing it to small regions of interest and huge areas of "background".  And can we really understand the rare without exposure to the common?

When we see a peregrine falcon stooping, a hare running, or a  rhinoceros gracefully trotting, we cannot of course understand the thousands of details of the structure and motion, let alone the vast number of aspects of their digestion, blood flow, respiration, chemistry, and so on.

But the bits we can see stand for all the others, and we sense the perfection for purpose.  We see the total mastery of the air that the peregrine or buzzard possesses.  We see the cunning and speed of the hare.  We  see the economy of the the rhinoceros, and although we cannot measure its centre of gravity, we feel sure that it is gliding along without wasteful up and down motion.  Of course, when a rhinoceros is charging rather than trotting, efficiency may be sacrificed for speed.

Perhaps the admiration excited by Torvill and Dean was partly attributable to their ability to achieve continuity in many differential coefficients of the motion.  They certainly had something that no others had at that time.

Many people are also impressed by a well designed machine, whether it be a car, a motor-cycle, or an aircraft.  The same fitness for purpose comes through.  

We can also be thrilled by a great stroke in cricket, a brilliant try in rugby, or a well-executed goal in soccer, irrespective of the state of the game.  These are poetry, but not in words; mathematics but not in symbols.

Common leaf arrangements on stalks are 2 + 2, 2 + 3 and 3 + 5, where the pairs of numbers refer to the right-handed and left-handed helices of leaves.

The composer Bartok Bela was very interested in many aspects of nature and science.  Some of his scores have the Fibonacci numbers actually marked at the relevant bars, and one composition, the fourth quartet, has 2584 beats in it.  Many other composers apparently used numbers, but if, like Bartok, they did not leave any writings about it, the possibility of coincidence cannot be disproved.

Click here to see an elegant diagram created by Jean-Pierre Hébert, inspired by Max Bill’s "Fifteen Variations on a Single Theme".  The apparent spiral, where the lines are closer together, expands in a ratio that tends to somewhere around 1.64.

Here is a curve which crosses the X-axis at the Fibonacci numbers –

In the blue corner above we see a graph of the Fibonacci numbers, and in the yellow corner below, alternate ones are labelled in green and yellow.  The spiral shown earlier was generated using two continuous mathematical functions which included these points.

If we now look across to the green corner to the right, we see two curves which resemble the trends of the two sets of Fibonacci numbers.  These two curves are the hyperbolic cosine (cosh) in red, and the hyperbolic sine (sinh) in green.  Both curves are generated from exponential curves, which are the same as the curve of compound interest.  These curves also describe the overall behaviour of decaying quantum mechanical systems such as radioactive nuclei or fluorescing atoms.

The next picture shows the first ten Fibonacci numbers, superimposed on the curves of cosh and sinh, which have been scaled by the numbers shown in order to fit the Fibonacci numbers.  What are these two scale factors?

It turns out that XScale = 1/ln(Phi) and YScale = 1/cosh(1/XScale), where Phi is the golden number 1.618033989….  If this is correct, it means that we have a means of calculating values in between the Fibonacci numbers.  A similar thing happens with the factorial numbers.  The continuous function that includes them is called the gamma function.  Suppose we generalise the idea of a Fibonacci number to include these intermediate values.  The properties of the Fibonacci numbers should apply to these new numbers as well.  For example, if we label the scaled cosh and sinh curves C and S respectively, we should expect to find that because consecutive Fibonacci numbers are related by F_N = F_N-1 + F_N-2, the curves are related by the following generalizations –

          C(x) = S(x – 1) + C(x – 2) = S_Ycosh(S_Xx)   where S_X = ln(Phi) and S_Y = 1/cosh(S_X) 

          S(x) = C(x – 1) + S(x – 2) = S_Ysinh(S_Xx)    where S_X = ln(Phi) and S_Y = 1/cosh(S_X)

and x need not be an integer.  This turns out to be the case, as we can see from the following tables.

If the curves C and S, which are rescaled hyperbolic functions, behave like this, we might expect similar relationships between the hyperbolic functions.  What we find are the following equations;

          cosh(x) = sinh(x – d) + cosh(x – 2d)

          sinh(x) = cosh(x – d) + sinh(x – 2d)

where d = ln(Phi) where Phi is the golden number 1.618033989 . . . .

Scaling the cosh and sinh functions in different ways would lead to different values of d.  The scaling that makes the curves include the Fibonacci numbers of necessity forces d to take the value unity.

In the next two diagrams, the real (Re) and Imaginary (Im) parts of phi^N have been separated (where phi = 1 / Phi = Phi – 1) showing how the Fibonacci numbers are related to Phi.  These curves are related to Binet’s formula for the Fibonacci numbers, which is as follows –

F(N) = (Phi^N  – (-Phi^N))/sqrt(5).

There is something very peculiar indeed about these extensions of the Fibonacci numbers to non-integral values of N, and that is that there are several of them.  That means that we may have to wonder which is the best one.  Let’s remind ourselves of the three that we have; they are

          the Binet formula

          the two-curve cosh and sinh formulae

          the one-curve exponential + damped sine formula.

The Binet formula gives a single value of F(N) for each N, but for non-integral N the result is a complex number, as it is for complex N.

The cosh and sinh formulae provide two real numbers for each real N.

The last curve, exp+sine, also provides one real number for each real N.

One common feature of all three methods is that for all N the results obey the Fibonaci relationship F(N) = F(N-1) + F(N-2), and probably other Fibonacci relationships as well, so we are in the peculiar position of being able to interpolate a set of numbers in three different ways, with no obvious means of preferring any one of them.  Each has a disadvantage –

Binet creates complex numbers, with real values only at the integers.

Cosh-sinh creates double values with integral values at alternate integers on each line.

Exp+sine oscillates, creating zeroes about halfway between the Fibonacci numbers at negative N.

It appears that an apparently simple and straightforward operation like interpolation is not simple and not straightforward.

By replacing cos(N) by sec(N) we can create yet another curve, shown below.  There is probably no limit to the number of curves that can be made to connect the Fibonacci numbers.  Do any of them have enough mathematical interest to be considered as an extension of the Fibonacci numbers?

Finally, we could abandon any attempt to use a closed formula: we could use a method such as sin x / x or cubic spline instead.  These would of course produce an oscillatory curve.

If cosh and sinh can be related to Fibonacci numbers, what about cosine and sine.  There are many parallels between the two sets of functions.  The graph below shows what happens when we use different values of d and x in the equations 

          cosh(x) = sinh(x – d) + cosh(x – 2d)

          sinh(x) = cosh(x – d) + sinh(x – 2d)

The value of d runs from -1 to +1 from left to right.  The vertical stripes are made by varying x and plotting the difference between the left and right sides of the equations, red for cosh and blue for sinh.

Let’s now try these equations – 

          cos(x) = sin(x – d) + cos(x – 2d)

          sin(x) = cos(x – d) + sin(x – 2d)

This is what we find –

Yes – there are some places where the equations balance, an infinite number of places in fact, because the functions are cyclic.  Can you calculate the values of d for which the equations work?

Cosh and sinh are made by adding and subtracting two exponentials, one increasing and one decreasing.  The cosh curve is seen in the shape of a uniform flexible cable with nothing attached to it.  It is then  called a catenary curve, which only means the curve of a chain.  The cables of a suspension bridge follow this curve before the hangers and the deck have been attached.  In practice, engineers in some areas may use the term catenary for any hanging structure, such as the overhead electrical supply system for a railway.

The catenary can also be seen between two parallel circular loops of wire that have been dipped in a soap solution.  The catenoid of revolution is a surface of minimum area, which is the result of the system settling into a state of minimum energy.  It is a special case of a family of such curves.  Some are called nodoids and some are called unduloids.  

The unduloids are seen in spider webs, on the threads that do the trapping.  Along these threads are little sticky blobs which are shaped rather like lemons.  They are formed of a viscous fluid which has gathered itself into blobs with minimum energy.

A sphere has the least area for a given volume, so why aren’t the blobs spherical?  The reason is that there are forces between the molecules in the blob, and forces between blob molecules and thread molecules.  It is the total energy that has to be minimised.  So although the spider makes sticky blobs that have a mathematical shape, these are a by-product of the physical forces.

We have seen here a connection between the Fibonacci numbers and a suspended cable and a spider’s web.  But these connections are spurious: they have no physical or mathematical meaning, being pure coincidence, and they lead to no new insights.  True connections are deep, and lead to new knowledge.

Bees make honeycombs based a hexagonal array of cells.  This structure minimises the mass of wax used for a given volume of cells.  But it is mathematical for a different reason from the maths of a spider web.  No physical force affects the wax, apart from cohesion and gravity: the bees must make the shapes themselves.  

In fact if you look carefully at a web and a honeycomb, you will probably find that the shapes in the web are more perfect than those in the honeycomb.  Evolution has probably got the bees to a point where further improvement might be lost in the noise of individual variation.  On the other hand, the evolution of the bees’ technique may not have stopped.

The pink corner , top right of the previous diagram, repeated for convenience, shows a sinusoid.  It doesn’t apparently look like an exponential, yet the two curves are intimately related.  For example, there is a simple formula –

exp(ix) = cos(x) + i sin (x)

If you write down the infinite series for sin, cos and exp you will see the resemblances and the relationships.  See also Sine and Exponential

And so the catenary of a hanging cable is related quite closely to the sinusoids of its oscillation in the wind, though their physical origins are quite different.

Can you discover a smooth curve that passes through all the points in the blue diagram at top left?

This peacock’s tail carries a large number of coloured spots.  The white lines suggest the possibility that they might be arranged on spirals.  But there is no reason to suppose that they are Fibonacci spirals.  The tail is composed of independent feathers, and is not rigid.  On the other hand, in order to maximise the visibility of the spots, they would be regularly, and not randomly, arranged.  Almost any pattern with polygonal symmetry would allow spirals to be drawn.  Symmetry might well be attractive to female peacocks, just as symmetrical tails are attractive to female swallows.

Why?  If the possession of "good" genes is to be signalled visibly, the signal should be easily recognisable.  Symmetry certainly qualifies on that score.  A more subtle possibility is that symmetry is less probable than asymmetry, in that there are more ways of being asymmetrical.  Symmetry implies control, just as tidiness does in peoples’ clothes and hairstyle.  

A good example is the discipline of soldiers on a parade ground.  The movements they make have a high degree of symmetry, from which any deviation is apparent.  No doubt the same people could be trained to perform far more elaborate movements, but it would be harder to detect imperfection.  In the same way, a female swallow might more readily detect deviation from perfection if the "ideal" is a simple.

Here are some patterned feathers.

Fibonacci won’t work for these: what might work is the type of theory put forward by Alan Turing, which allows the possibility that a great variety of patterns may be related by the same simple rules.

Some people have tried to find the golden ratio and the Fibonacci numbers in the human form, and have based the proportions of paintings on them.  But in view of the huge range of body shapes and proportions, it is hard to see how the golden ratio could be consistently expressed in a body, and hard to think of a reason why it should be.

If you measure a lot of pictures, you may find a few in which parts of the figures have been placed at mathematically calculated positions.  But when you find that different parts of the body have been used for this in different pictures, it suggests that the idea is artificial.  

If you measure a lot of abstract paintings, such as those by Mondrian, in which the proportions of the picture are among the important features, you don’t usually find any golden ratios.  The pictures have been adjusted to please the artist, and this clearly hasn’t led to the golden ratio being used.

In other words, golden ratio probably appears in pictures, not because it is pleasing, but because people believed a theory about it.

According to Ernö Lendvai, Béla Bartók used Fibonacci numbers in some of his music, probably because of their occurrence in nature.  It is hard to believe that he thought that people could discern times accurately enough to notice these proportions, especially as he himself did not always play the piano at the metronome speeds given in his own scores.  These numbers were perhaps just a way of giving structure to the pieces.

However, Bartok was very interested in nature and science.  These numerical references seem to start with the Fourth String Quartet, which has 2584 crochet beats in it, as well as numerous GS structures, and they seem to end with the Divertimento for Strings.  The ratio of the totals of the crotchet beats in the 5th and 4th string quartets is very close indeed to 8 : 5.  The ratio of the beats in the 5th quartet to the specified playing time is very exactly 5 : 2.  But how can we prove that any of these examples is more than coincidence?  You can find "meaningful" numbers almost anywhere, if you look, but for real meaning, they must result from some theory outside themselves: numbers alone prove nothing.

The story of the Fibonacci numbers in plants is a fascinating one.  It can stand as symbol for the great number of phenomena that must occur in any life-form.  It has the advantage that most of us can understand its basic reason, whereas even something as familiar as the flight of a common insect is still not well understood, even by specialists.  

Like many other topics nature has so many levels that we can all find something of interest,  In even the simplest form of life, many features have evolved to create something that is near to the current best compromise between  many conflicting requirements, so what we can easily observe are only the more superficial features.  But we can  bet that nature is neat and elegant all the way down.  Notice that we write "near to the best" and not "the best", because the goalposts are constantly moving.  Climates change, and all the other species change, so that any one species never quite reaches equilibrium.

In a complex life-form, thousands of different features have been fine tuned over millions of years to create what we currently see.  Given that there are probably tens of millions of species, any knowledge we have can only be representative of the whole. 

It is as if we are in vast dark art gallery, looking at huge pictures with only a box of matches for light. The difference is that we believe that all our glimpses of nature can in principal be explained by a few very simple ideas, even though in many cases we can never know how it happened  –

Evolution by natural selection

Transmission of characters via DNA as the genotype

Modification of the expression of the genotype by many factors, producing the phenotype

As with many other ideas, such as general relativity, QED, QCD, and even classical physics, the ideas can be very simple, but the working out of the consequences can be formidably difficult.

Simulated adder patterns

Fluorescence     Spider Webs     Surface Tension     Suspension bridges

A  Bit  of  Fun

Here is a plan for a city based on 21 clockwise spirals and 21 anti-clockwise spirals.

Imagine that to go into the city you spiral anti-clockwise, and to go out you spiral the other way, so that all the roads are one-way.

What are the potential advantages and disadvantages of a completely planned city, when designed by one person, several individuals, one team, or several teams?  What if all the industrial areas were designed by one team, all the museums by another, etc?  Or you could ask each team to design a variety of areas.

The next diagram shows how external roads might be connected to the city.  What are the advantages and disadvantages of this idea?

Has any planned city been successful?  Bournville?  Brasilia?  Chandigarh?  Crawley?  Espoo?  Kyoto?  Milton Keynes?  Stevenage?  Welwyn Garden City?

Has any culture built cities or towns based mainly on curves?  The use of a rectangular grid is not uncommon.  Chinese and Japanese cities have been built, not only on rectangular grids, but in locations and orientations considered to be favourable.  Kyoto, for example, is sheltered to north, east and west by hills, and its grid is aligned N-S and E-W.  The temples are based on rectangular alignments as well.  

Remains of Roman settlements in Britain show clearly that a rectangular grid was used, and many modern towns are built in the same way.  

Partly because of the ubiquity of steel frames and concrete, many modern towns are dominated by rectangles in all three dimensions.

Click here for Ron Knott’s comprehensive web-site about Fibonacci numbers and Phi.

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html some very neat maths . . . .

          http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html more about nature

          http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat2.html more about nature

          from here you can download an excellent interactive demo about growth of

          seed-heads, and you can download Geometers Sketchpad, which you need to run it.

          http://www.keypress.com/product_info/sketchpad3.html Geometers Sketchpad

http://www.math.smith.edu/~phyllo/  phyllotaxis with applets

http://www.cpsc.ucalgary.ca/projects/bmv/papers/phyllo.sig92.html phyllotaxis model

http://mathworld.wolfram.com/P/Phyllotaxis.html more on phyllotaxis

http://documents.wolfram.com/v4/MainBook/2DGraphics/G.2.8.html neat diagram

http://www.templejc.edu/precalc/media/units/unit1/topic3/explorations/explor3a-fibonacci/exfibonacci3a.html exploring phyllotaxis

http://math.holycross.edu/~davids/fibonacci/fibonacci.html

http://www.zometool.com/deepzome/golden.html

http://forum.swarthmore.edu/advanced/robertd/fibboard.html

http://math.bu.edu/DYSYS/FRACGEOM2/node7.html

http://thunder.indstate.edu/~telles/ppoint2/tsld001.htm

http://www.sdstate.edu/~wcsc/http/fibhome.html

http://www.geocities.com/CapeCanaveral/Lab/5833/cycas.html

http://www.baacks.com/Fibonacci/fibonacci.html

http://www-history.mcs.st-and.ac.uk/~gap/Manual/C046S020.htm

http://www.isepp.org/Math Alive/ian.htm

http://www.isepp.org/Math Alive/ma1.htm

http://plaza.ufl.edu/myklad/bartok.html

http://www.bayarea.net/~kins/AboutMe/Bartok/BartokStuff.html

This and many other topics are discussed in Professor Ian Stewart’s exciting book "Life’s Other Secret" – Penguin – ISBN 0 14 025876 0.

An older but beautiful book is "Patterns in Nature" by Peter S Stevens – 

Peregrine – ISBN 0 14 055 114X

An even older, but deservedly famous, book is "On Growth and Form" by D’Arcy Wentworth Thompson.