.

Fibonacci  Numbers

and how they are related to flowers, pine cones, pineapples, palm trees, suspension bridges, 

spider webs, dripping taps, CDs, your savings account, and quite a few other things.

If you wish to find out about the properties of the Fibonacci numbers, I strongly recommend

Dr Ron Knott's excellent website about the Fibonacci numbers and the Golden Section.

 

Back to Home Page    Back to Flowers    Back to  Nature's Maths

 

Here are some photographs showing the day by day flowering of a sunflower. As the flowers open, in an inward wave, the neat spirals are obscured by the wayward growth of the florets. But after the florets have withered and set seed, the neat pattern reappears, albeit somewhat enlarged. From this observation we see that whatever the reason for the tidy spirals, it has nothing to do with the flowers or with attracting the bees that are seen in two of the pictures. The wave of maturing runs inward because the youngest florets are in the middle, and the oldest are at the edge.

 

And here is a picture of a part of a sunflower head after setting seed. In the centre of this particular head, the arrangement is not as tidy as it is near the edge. Note the difference between this array and the array of ommatidea in the eye of an insect, or the cells in a honeycomb. The insect eye does not grow out from a centre like the flower of a plant, and the honeycomb is built up from the edges. So these two constructions can make use of hexagonal close-packing, which is denied to the plant because of the way the growth begins.

Here are some pictures of other plants.

 

Two curved lines have been drawn on the first photograph of a thistle head - one spiralling out clockwise, and one spiralling out anti-clockwise. There are thirteen of the first kind, and twenty-one of the second. In the second photograph the situation is reversed. The third photograph shows a teazle flower head, which is longer than the almost spherical thistles. In the fourth picture there are twenty-one spirals and thirty-four spirals. The shapes of the individual flowers varies considerably, more than we see in the cells of honey bees. Why do you think this happens? More cylindrical versions of these patterns are seen in the arrangement of leaves on plant stems, and in the scars on some tree trunks.

These are not random numbers - they are members of the following sequence -

1  2  3  5  8  13  21  34  55  89  144  233  377  610  987  1597  2584  4181  etc.

 

This sequence is known as the Fibonacci series, and is well known in mathematics.  Each number is the sum of the previous two. The ratio of successive pairs tends to the so-called golden section (GS) - 1.618033989 . . . . . whose reciprocal is 0.618033989 . . . . . so that we have the formula 1/GS = 1 + GS.

Plants do not know about this - they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144.  Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.

Why do these arrangements occur? In the case of leaf arrangement, or phyllotaxis, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space.  

This is well described in "Patterns in Nature" by Peter S Stevens (Peregrine Books) ISBN 0 14 055 114X, and, more recently, in "Life's Other Secret" by Ian Stewart (Penguin Books) ISBN 0 14 025876 0. See also the older classic in this field, "On Growth and Form" by D'Arcy Thompson.  

So nature isn't trying to use the Fibonacci numbers: they are appearing as a by-product of a deeper physical process. That is why the spirals are imperfect. The plant is responding to physical constraints, not to a mathematical rule.

The basic idea is that the position of each new growth is about 222.5 degrees away from the previous one, because it provides, on average, the maximum space for all the shoots. This angle is called the golden angle, and it divides the complete 360 degree circle in the golden section, 0.618033989 . . . .  

If we call the golden section GS, then we have

1 / GS = GS / (1 - GS) = 1.618033989 . . . .

If we call the golden angle GA, then  we have

360 / GA = GA / (360 - GA) = 1 / GS.

Here is a picture of a houseleek, taken with a wide aperture to help to identify consecutive leaves by defocussing the others. The yellow lines show the axes of these. The angle between the yellow lines was measured from a print as about 220 degrees, as compared with GA = 222.5 degrees. The measurement was probably measurable with an inaccuracy of around 2 degrees. Measuring two leaves of one plant is of course not a scientific test. The next pictures show flowers on a cactus.

The green numbered diagram represents an idealised cross-section through a plant, showing leaves that have started to grow out from the centre. We see that the successive leaves are displaced around the circle. What is the angle of displacement that gives the most room for growth?

The next picture shows a set of slices through a celery stem. We see the transition from a thin section that fits neatly at the bottom, to a section further up that provides stiffness in both dimensions. Note the asymmetry at the bottom of the stem. Does this help in the close packing?

 

 

The next picture simplifies the situation by representing each leaf by a circle.

 

Each centre is connected to the next by a red line. The colours are changed as the numbers grow, to suggest the progression. What happens if we add more centres?  Can you guess?  See the next picture.

 

 

You are bound to ask if this is a fluke. So in the next two diagrams, the angle between consecutive leaves has been changed from the golden angle by plus 0.1 % and minus 0.1 % respectively, or by factors of 1.001 and 0.999 if you prefer.

 

Which of the three pictures looks most like the middle of a daisy or a sunflower? The first one, obviously. The plant uses the golden angle, not because it is a philosopher, a mathematician or an aesthete, but because it packs the most into the smallest area.

Perhaps you don't believe that the original scheme can generate these patterns. The next diagram includes some of the numbers that let us count the leaves. Start at 200, for example, and follow them round.

 

If we start at the circle marked 200, and go along the apparent spirals in eight directions, we see the following series -

. . . 184  192  200  208  216 . . .

. . . 174  187  200  213  226 . . .

. . . 158  179  200  221  242 . . .

. . . 132  166  200  234  268 . . .

These are arithmetic progressions with steps of 8, 13, 21 and 34 respectively, all Fibonacci numbers.

Larry Albertelli has kindly supplied two good pictures that can be downloaded here, a PDF and a Postscript file.

These spirals don't actually look like those in the original diagram of the green leaves. And this is where we find a relationship between a sunflower and a compact disc. Let's look at another diagram.

We can clearly see spirals like those in the thistles at the top of this page. If we count the spirals we get 13 in one direction and 21 in the other - two Fibonacci numbers.

But we have been cruelly deceived. There are no Fibonacci numbers in this diagram, no Fibonacci numbers in a thistle, no Fibonacci numbers in a celery, and no Fibonacci numbers in a palm tree. Well, there are, but they are not in the plan. 

If you don't believe this, try writing a computer program that will join up the generated points to make the spirals that we see. It's easy to write a program to generate the diagrams as the plant does it, but not so easy to do it as we think we see them.

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. We must also remember that however pretty these patterns look, they can never be perfect: their symmetry is broken twice, once at the outside where the first flower or leaf appeared, and once at the centre, where the last one grows. Only if the pattern were infinite in both directions could it be perfectly symmetrical.

 

What is the reason for all these series of numbers?

 

Here is a picture of a part of a 78 RPM gramophone record. The undulations of the groove represent the collective velocities of the molecules in the air. They are analogous to a graph of the actual motion, so the system is called an analogue recording system.

The next picture shows some samples on a simple wave, such as a digital recorder might make in order create data for a CD. For each sample a digital word is produced and stored. A digital oscilloscope uses the same idea.

In a CD player there is an analogue-to-digital converter that converts the digital data into analogue signals, with the help of some filtering. You can see that from the black data points, it should not be difficult to re-create something very similar to the red sine.

So far, so good. Now look at the next picture.

Although the signal frequency has been increased by a factor of ten, the samples do not reflect this. The period of the sine represented by the black data is actually a little longer than before. The signal frequency is too high for the sample-rate, and it has fooled the system.  

The signal on a CD has been digitised at about 44 kHz, so that the highest frequency component that could be recorded correctly is at 22 kHz. A signal at 23 kHz would be heard as 21 kHz, 24 kHz as 20 kHz, and so on. These false frequencies are called aliases.

You see this in "western" films, when the wagon wheels may appear to rotate in the wrong direction. You can see it in films in which the propellers of an aircraft start at zero revolutions per minute and then speed up. The propeller will start in a normal manner, and then appear to slow down and go backwards, corresponding to negative frequencies.  

This happens because the moving picture is made of many still pictures which are projected in succession. Despite of all the skills of the movie industry, nobody can defeat the sampling theory, which requires there to be at least two samples per cycle at the maximum signal frequency. You might try to defeat the weird rotations by having randomly spaced spokes, but the result would almost certainly be bizarre and unnerving in some other way.

Another kind of aliasing occurs in films in which people are trying to escape. A searchlight sweeps round at regular intervals, and the escapers time their short runs between hiding places to coincide with the dark periods. The people operating the searchlight should sweep so fast that the escapers could not get from one piece of cover to the next between the flashes of light. Or they could move the beam in a pseudo-random sequence. But that would spoil the story.  Sometimes maths and science have to be ignored in the interests of art.

What happens in sampling systems is that the frequency spectrum is folded up and packed into a bandwidth of half the sampling frequency. And that is what we see in plants. These Fibonacci numbers are aliases of the real processes and frequencies.

The next three pictures show the result of using every 5th point, every 8th point, every 13th point, and every 21st point, in a series of cycles whose frequency is related to the sampling-rate by the golden ratio. Note how the lines straighten out as the spacing increases, corresponding to large periods.

In the diagram above, the horizontal black line marks off half a cycle of the blue wave, and 17 cycles of the red wave. So the number of red cycles covered by one blue cycle is 34, a Fibonacci number.

The next picture shows output frequency (y axis) versus input frequency (x axis) for a digitiser. Only the input frequencies up to FN, the Nyquist frequency, have any chance of being recorded correctly.  FN is a half of the sampling frequency.

Note that if a signal consists of a burst of three cycles of a 1 kHz sine, a sampling rate of 2 kSa/s will not suffice. The bandwidth of such a signal is more than 1 kHz, because the signal effectively comprises a 1 kHz sine, amplitude modulated by a 3 ms sharp edged pulse. The side-bands needed to characterise this are extensive.

 

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The picture and download below show examples of real and visual aliasing, as the signal frequency increases. It shows a green wave at the top, with a white sampled wave below. The white sampled wave is never under-sampled, but you will see visual aliasing, because the eye-brain system "joins up the dots" in the most obvious way, which is not always correct.

The red line represents sampling at a tenth of the rate of the white graph. It becomes under-sampled quite early in the demonstration, and it shows the behaviour corresponding to the first triangular cycle above, from 0 to 2 FN.

Click here - and select "Run the program in the current location" you can see what happens. You should see something like the picture above. The effects of increasing frequency on the visual appearance and on a sampled signal will be seen.  The frequency will increase and decrease alternately. You can quit the program by pressing "q". Note the red lines have no physical meaning: they are interpolations to make the result easily visible.  But in a real sampled data stream, only the sample exist, any interpolation, however sophisticated, is a guess.

Click here - and select "Run the program in the current location" you can see what happens when a series of pulses is sampled. This program illustrates both visual aliasing and actual aliasing.

The diagram below shows the result of sampling a sine at 1.618034 (1/GS) of the sine frequency, and then joining up every Nth point. N runs from 1 at the top to 25 at the bottom. This corresponds to a point revolving around the centre of a plant, with a new leaf starting to grow at every new golden angle.

The curves corresponding to the Fibonacci numbers 2 3 5 8 13 and 21 are drawn in black. They clearly have lower frequencies, and longer periods, than the ones around them, which is why these numbers appear in plants so readily. We don't see the other periods very well, because they are under-sampled, and look like noise.

 

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The next diagram represents the results for N =2 to N = 55, again with the Fibonaccis in black, this time including 34 and 55. The vertical and horizontal scales are smaller than before.

The next picture shows the spectrum of alias periods for N = 1 to 300. We see clearly the peak in red at the Fibonacci numbers 2 3 5 8 13 21 34 55 89 144 and 233, corresponding to the long wavelengths above. Between these, at about the golden section points, there are subsidiary peaks, and at the golden section of those, even smaller peaks. This is an important diagram.  Using only the golden section, GS, and no other parameters, it generates the Fibonacci numbers.

Just to show how special this is, the next picture was made by the same program, but with the sampling frequency increased by one fifth of one percent. The original pattern is replaced by a spectrum which looks a bit more like a classical pattern of evenly spaced harmonics, albeit with strange amplitudes.

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Finally, below, the diagram represents the range of values N = 0 to 1000, including the F numbers 377, 610 and 987, rescaled by dividing the verticals values by N, to show the multiple golden sections more clearly. These are shown by blue horizontal lines. The short blue lines mark the two golden section (GS) points in each segment. If the length of a long blue line is taken as 1, then the three segments have lengths GS2, GS3 and GS2 respectively.  GS2 and a GS3 add to GS.

The values are -

GS     0.618033989 . . . . 

GS2   0.3819659 . . . . . . 

GS3   0.2360678 . . . . . .

The diagram is rather like a one-dimensional fractal. Any small part of it looks a mess, but on the large scale the structure appears. You can find a structure like this in the central movement of Bartok's fourth string quartet, and in some other works by the same composer. Erno Lendvai has written about this in several books.

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So all the spirals that we see in plants are aliases of the true growth pattern, which is very hard to detect, especially in flowers like sunflowers and daisies. The eye and brain are completely fooled by the aliases.

This picture shows very crudely how the leaves of a palm grow out of the top and slowly move out as the trunk grows. At a certain point they die, leaving scars that encircle the trunk, the oldest being nearest the ground.

The position of each leaf is marked by a wider part in the scar. The positions of these wider parts look random at first, but on inspection you can see that they follow trends like those in the diagrams of every Nth leaf shown above.

As scars on a tree trunk these positions are meaningless. but as relics of leaves growing for maximum space their message is clear.  In the left picture below, the spacings 5 and 8 are marked. In the right picture 5, 8, 13 and 21 are marked.  The trend agrees with those in the calculated diagrams. The larger the step, the more parallel to the axis, and the longer wavelength of the yellow curve.

Palm6.jpg (62884 bytes)    Palm7A.jpg (65000 bytes)

These long wavelengths correspond to the long periods of the previous diagrams. The existence many short periods in those diagrams are the reason why the bands on a palm look like a random jumble, or noise, as an engineer would say, like the fractal diagram shown earlier, and unlike the order that we seem to see in a sunflower.  In the palm the eye-brain cannot easily see the pattern. In the sunflower the eye-brain is practically forced to see it. Yet palm and sunflower are telling us the same story. One looks like prose, and the other like poetry.

Fibonacci helices can be seen on pine-cones, pineapples, and teazles. They can be seen in the leaf arrangement, or phyllotaxis of many plants.

That is the power of maths and science - they can relate things that look very different. See the Tyger page. And once a few brilliant people have worked out general rules, we can all benefit from applications of those rules to useful devices, or from making existing devices more efficient.

The earliest steam engines were not invented by scientists, but they could never have become efficient without the science of thermodynamics.

The invention of early radio valves did not require much knowledge of physics, though it was dependent on high vacuum techniques, which were developed for scientific research. But the development of today's solid state devices could never have happened without the deep understanding of matter provided by quantum theory. This theory explains so many things that it may be at present our greatest bringer-together of phenomena. 

The next diagram is a simulation of a part of the trunk of a palm tree. Apart from the numbers needed to specify the width of the trunk and the spacing of the rings, the only parameters used were the golden angle and the direction of winding. The dark areas represent the trunk, while the pale areas represent the leaf scars. 

The alternate F numbers give sines of alternate starting polarities. The yellow lines show the start of four sines of different wavelengths.

The next diagram is similar to an earlier one, except that the leaves completely encircle the centre. GA is the golden angle. Imagine that after a certain point these shapes do not expand any more, but just become scars on the trunk, and you have the explanation of the scars on a palm. 

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