Ghost In The 'Lac
06-16-2009, 12:57 PM
How is mathematics the language of the universe when mathematics didn't always exist? The universe always existed, but not mathematics.
tut tut tut
How is mathematics an invention? Mathematics is based on patterns in nature.
A Natural Ratio
Although appearances of the golden mean in art, architecture, and human proportions are largely unsubstantiated, the golden ratio genuinely appears in many elements of nature.
Nautilus shells, for example, have a logarithmic spiral pattern. The golden spiral is also a logarithmic spiral which is governed by the golden ratio. For every 90 degree turn, the radius of the spiral grows by a factor of the phi. Similar logarithmic spirals (http://ddrive.cs.dal.ca:9999/page/lvl3/42) map the Pergrine falcon’s prey snatching flight path, swirl patterns in hurricanes, and the shape of spiral galaxies.
Hurricane - NASA (http://www.nasa.gov/externalflash/sts115_front/index.html); Galaxy - NASA (http://www.nasa.gov/externalflash/sts115_front/index.html); Pinecone - "Paddy Patterson (http://flickr.com/photos/51035609472@N01)" (license (http://creativecommons.org/licenses/by/2.0/)); Nautilus - "Chris 73 (http://en.wikipedia.org/wiki/User:Chris_73)" (license (http://creativecommons.org/licenses/by/2.0/))
Spiraling Outta Control!
The number of spirals that can be outlined in pineapples, cauliflower, pinecones, cactus spines, and the seeds of sunflowers are all Fibonacci numbers. For a nice demonstration of the 8 spirals one way and the 13 spirals the other way in a pinecone, visit phyllotaxis (http://maven.smith.edu/%7Ephyllo/). Ok, but how do we know that all these occurrences in nature are not purely coincidental? These distributions of the respective units and spirals they form can actually be recreated.
Spin a source point (or growth point) at a constant speed and release a seed every 1/φ ≈ 0.618 turn of the circle. Using this special angle related to φ, you end up not only with spiral patterns (Fibonacci in number), but also you get a distribution of seeds that is the most evenly spaced distribution possible. It looks exactly like the seeds in the sunflower. There is no overcrowding and no strange spaces or holes. The golden ratio results in a perfectly spaced pattern containing a Fibonacci number of spirals. Since 360° times 1/φ approximately equals 222.5° this is the ideal angle, along with 137.5° (the same angle measured from the other direction). To see this in action, see this animation (http://cinderella.de/%7Ejuergen/Christmas/Bild02.html), where you can simulate the growth pattern by setting the angle to 137.5°, and then increasing the number of seeds.
There is a brilliant and logical explanation for all this. Nature is governed by efficiency. It wants to pack the most amount of parts in the least amount of space, but not put everything so close that it can’t function. The golden ratio accomplishes this goal by being the key angle of rotation to make the most efficient, equally spaced packing.
The leaf arrangements of plants also have a spiraling pattern based on the golden ratio. If the plant only produces a leaf about every 0.618 of a turn about the stem (1.618 leafs per turn), then we get the best and most even distribution of leaves. The maximum amount of space is allowed between every leaf that is directly above another one on the stem, so shadowing is minimized. The leaves make the best use of the space they have to capture sunlight in.
Counting on Plants
Because a plant’s growth acts this way, with high dependence on the golden ratio, it makes sense that the Fibonacci numbers would pop up. If you look at the number of petals in flowers, more often than not, they are Fibonacci numbers. Lilies, irises, and trillium all have 3 petals. Buttercups, geraniums, pansies, primroses, rhododendrons, and tomato blossoms all have 5 petals, the most common number of petals for a flower to have. Delphiniums have 8, marigold and ragwort have 13, and black-eyed susans ,chicory, and asters all have 21. Finally daisies often have 34, 55, or even 89 petals.
Ever hear of the Golden Mean? Neither had I, until I was supposed to teach a graphic design course, and started (you know me) to do research on what, exactly, I should be teaching. It's one thing to be able to design things, and quite another to have to teach it to others.
A lot of what I found, gestalt theory and the principles of visual weight, and so on, were really interesting; but the Golden Mean was what really caught my fancy.
(NB: Math following. Don't be scared, all will be clear [and un-mathlike] in the end, I promise. I hope)
The Golden Mean, also known as the Golden Ratio, was developed as a proportional measurement by the ancient Greeks, as a way of making the most pleasing artworks. It was felt to be semi-divine, in that it seemed to show up in Nature as well. The ratio, an irrational number, began as 1.6180339887... and continued onward, pretty much forever. It is found by working out the following algebraic equation:
Essentially, if you take a line that is 1 long, and go from there, you will find that the above equation will work out to that same irrational number, which, when used as a proportional device, allows you to produce varying lengths of lines that are smaller and larger. But I'm not going to explain how, because though I really, really love math, I don't do well with equations, which are difficult for me: at least, to express the near-mystical magic that shows up in numbers.
So: now you have a bunch of varying line lengths. So what?
Well, let's see: take one of these lines and make a square out of it (putting four of them at right angles to each other, remember? I sometimes blank on these little leaps of logic). Then, starting at the center of one side, measure to one corner and draw an arc downward:
Aha! Now we start to have something. If the length of the square is 1, then the length of the rectangle (shown as the Greek letter phi here) is, of course, 1.6180339887... well, you get the point: the Greeks were smart. We call this shape the Golden Rectangle, and you can find it everywhere in Greek and Renaissance art (and elsewhere! Stage height proportions, window shapes, chair backs, believe me, they're everywhere. They are quite pleasing to the eye).
(If you really want to know the equation looks like this:)
http://bp1.blogger.com/_c4twH_sjx5o/RudgSY3TMGI/AAAAAAAAAjs/3TVqRigyh2Y/s400/GM_equation2.png (http://bp1.blogger.com/_c4twH_sjx5o/RudgSY3TMGI/AAAAAAAAAjs/3TVqRigyh2Y/s1600-h/GM_equation2.png)(but don't ask me to explain that part).
http://bp2.blogger.com/_c4twH_sjx5o/RudgSo3TMHI/AAAAAAAAAj0/OLRLoT27blo/s400/ParthenonGoldenRatio.png (http://bp2.blogger.com/_c4twH_sjx5o/RudgSo3TMHI/AAAAAAAAAj0/OLRLoT27blo/s1600-h/ParthenonGoldenRatio.png)Golden rectangles in the proportions of the Parthenon
Okay, onwards. I know this is looking like a lot of math, but bear with me here. Now we come to this guy named Fibonacci (http://en.wikipedia.org/wiki/Fibonacci), born c.1170, who is considered "one of the most talented mathematicians of the Middle Ages" [wiki]. This is the man responsible for the introduction of the Hindu-Arabic numeral system when all of Europe was doing math in Roman numerals (I strongly urge you to read his link, above: it's fascinating). He used what is now known as the Fibonacci sequence - actually a pre-6th-century Indian concept - as an example in his famous and apparently brilliant book about math, Liber Abaci, or Book of Calculation, which is why we have it now. In any case, the Fibonacci Sequence (http://en.wikipedia.org/wiki/Fibonacci_sequence), as it is now known, leads to all kinds of interesting events.
Try this. Draw a square, measuring one unit across (make it a small unit, like a centimeter, or perhaps the distance between binder-paper lines - otherwise you will need big paper). Now draw another square exactly the same right up alongside it (so they are sharing a side). Now, say you take the line along the top of both squares and use that to draw a bigger square, which of course is two units on a side - right? Okay, now moving clockwise (or counter-clockwise, like the picture below) around this construction, draw another square along the side where the edge of the big square and one small square align. Keep going clockwise and keep drawing bigger and bigger squares. The length of each consecutive square should make a sequence, like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... (by now you've probably run out of paper). Viola! Your own personal Fibonacci sequence, right there in your own home! And...lookie there. It sure looks kind of like...is it? Pretty close to a Golden Rectangle, isn't it?
Okay, okay, you say. That's kind of neat. But aren't we just wanking with numbers?
Well, perhaps. But check this out. You can draw an arc, starting with a point at the middle of the two first squares where they touch the third square. The arc goes from corner to corner of each square, so that the two squares together make a semicircle. Then, by expanding the arc and drawing one in each square, working around the structure, you can build a lovely spiral:
http://bp1.blogger.com/_c4twH_sjx5o/RudrTY3TMJI/AAAAAAAAAkE/Uncdd3uEF9A/s400/Fibonacci_spiral_map.png (http://bp1.blogger.com/_c4twH_sjx5o/RudrTY3TMJI/AAAAAAAAAkE/Uncdd3uEF9A/s1600-h/Fibonacci_spiral_map.png)This Fibonacci spiral does not have the two original squares visible
This spiral is one that closely mimics the Golden Spiral (http://en.wikipedia.org/wiki/Golden_spiral), based on the Golden Mean, above. The main difference is that the Golden Mean goes in both directions, both up and down, whereas the Fibonacci spiral only goes upwards from 1 (though you could take it down if you were into math, I'm sure). Both are considered logarithmic spirals (http://en.wikipedia.org/wiki/Logarithmic_spiral), which are found everywhere in nature. Jakob Bernouli, a mathematician from a great family of brilliant people, called the logarithmic spiral spira mirabilis, or "the Miraculous Spiral," so called because the size increases but its shape is unaltered with each successive curve. This kind of spiral shows up in shells, in hurricanes, in the shape of a cat's claw or a wave; galaxies and flowers all work with logarithmic spirals. The Fibonacci sequence can be found many places as well, such as in the ancestry patterns of bees, the branching of trees, the whorls of a sunflower and the fruitlets of a pineapple.
Technically, though, the Fibonacci spiral has a slight wobble; it is not perfect, so not really a proper logarithmic spiral.
Which brings me to something which I find absolutely wonderful: if you chart that wobble on a graph, it begins to look as if it is ocillating around something, some specific number. Guess which number?
You guessed it. Now tell me there's no mystery in 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
26th October 2000 Back to Home Page (http://www.branta.connectfree.co.uk/index.htm) Back to Flowers (http://www.branta.connectfree.co.uk/flowers.htm) Back to Nature's Maths (http://www.branta.connectfree.co.uk/naturemaths.htm)
http://www.branta.connectfree.co.uk/Thistle1_small.jpg (http://www.branta.connectfree.co.uk/Thistle1.jpg) http://www.branta.connectfree.co.uk/Thistle2_small.jpg (http://www.branta.connectfree.co.uk/Thistle2.jpg) http://www.branta.connectfree.co.uk/Teazel_small.jpg (http://www.branta.connectfree.co.uk/Teazel.jpg)
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. Flatter versions of the patterns are seen in many plants like sunflowers and daisies. More cylindrical versions 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
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 "Nature's Other Secrets" 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.
Let's look at the picture below.
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 second 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 picture 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 the two directions, we see the following series -
. . . 132 166 200 234 268 . . .
. . . 158 179 200 221 242 . . .
These are arithmetic progressions with steps of 34 and 21 respectively, both Fibonacci numbers.
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 to do it as we think we see them.
http://www.branta.connectfree.co.uk/HLeek1_small.jpg (http://www.branta.connectfree.co.uk/HLeek1.jpg) http://www.branta.connectfree.co.uk/HLeek2_small.jpg (http://www.branta.connectfree.co.uk/HLeek2.jpg)
What is going on?
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 cowboy films, when the wagon wheels may appear to rotate in the wrong direction. You see it in films in which the propellers of an aircraft start at zero revs 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.
What happens in sampling systems is that the frequency spectrum is folded up and packed into a band which goes only to 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.
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 (http://www.branta.connectfree.co.uk/ALIASING.EXE) - 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 (http://www.branta.connectfree.co.uk/ALIASES.EXE) - 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.
The next diagram represents the results for N =2 to N = 55, again with the Fibonacci's 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.
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.
http://www.branta.connectfree.co.uk/Palm1.gifThis 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.
http://www.branta.connectfree.co.uk/Palm6_small.jpg (http://www.branta.connectfree.co.uk/Palm6.jpg) http://www.branta.connectfree.co.uk/Palm7A_small.jpg (http://www.branta.connectfree.co.uk/Palm7A.jpg)
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 (http://www.branta.connectfree.co.uk/tyger.htm) 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.
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.
Then it 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, in that 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.
Click here (http://www.branta.connectfree.co.uk/GSF.EXE) - 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. 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.
http://www.branta.connectfree.co.uk/SKHeronA_small.jpg (http://www.branta.connectfree.co.uk/SKHeronA.jpg)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 speedway 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.
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.
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.
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.
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 has been reduced. 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 so 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.
When we see a falcon stooping, or a hare running, are 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, 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 with out wasteful motion up and down.
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.
Here is a curve which crosses the X-axis at the Fibonacci numbers -
The spiral part crosses at 1 2 5 13 etc on the positive axis, and 0 1 3 8 etc on the negative axis. The oscillatory part crosses at 0 1 1 2 3 5 8 13 etc on the positive axis. The curve is strangely reminiscent of the shells of Nautilus and snails. This is not surprising, as the curve tends to a logarithmic spiral as it expands.
And this is how the patterns found in flowers are related to the things mentioned at the top of the page, and many others.
The Fibonacci numbers tend to an exponential series. This is related to compound interest in your savings account, and to radioactive decay, and to many other basic processes, such as the fading of fluorescent materials. If you reverse an exponential and add it to the original, you can get a catenary, which is the curve of a suspension cable before the deck is added. It is also a curve which can be seen in liquid films when suspended between two circular frames, and is related to some curves found in the sticky blobs on spider webs. All these topics are found in other pages in this web-site.
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.
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.
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.
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 (http://www.branta.connectfree.co.uk/sinexp.htm)
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?
http://www.branta.connectfree.co.uk/Peacock2_small.jpg (http://www.branta.connectfree.co.uk/Peacock2.jpg)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 Fibonnaci 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.
http://www.branta.connectfree.co.uk/Feather1_small1.jpg (http://www.branta.connectfree.co.uk/Feather1.jpg) http://www.branta.connectfree.co.uk/Feather2_small1.jpg (http://www.branta.connectfree.co.uk/Feather2.jpg)
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, it is hard to see how the golden ratio could be 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 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 probably 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, and numerous GS structures, and they seem to end with the divertimento for strings.
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 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.
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.
Fluorescence (http://www.branta.connectfree.co.uk/fluorescence.htm) Spider Webs (http://www.branta.connectfree.co.uk/spider_webs.htm) Surface Tension (http://www.branta.connectfree.co.uk/surface_tension.htm) Suspension bridges (http://www.branta.connectfree.co.uk/suspension.htm)
Fibonacci Links and Downloads
You can run these programs in place or download them. You can quit each one by pressing "q".
Aliasing of sines (http://www.branta.connectfree.co.uk/ALIASING.EXE)
Physical and visual aliasing (http://www.branta.connectfree.co.uk/ALIASES.EXE)
Packing with random angles and golden ratio (http://www.branta.connectfree.co.uk/GSF.EXE)
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/ (http://www.math.smith.edu/%7Ephyllo/) 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://www.isepp.org/Math Alive/ian.htm (http://www.isepp.org/Math%20Alive/ian.htm)
http://www.isepp.org/Math Alive/ma1.htm (http://www.isepp.org/Math%20Alive/ma1.htm)
This and many other topics are discussed in Professor Ian Stewart's exciting book "Nature's Other Secrets" - 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.
Back to Home Page (http://www.branta.connectfree.co.uk/index.htm) Back to Flowers (http://www.branta.connectfree.co.uk/flowers.htm) Back to Wild Flowers (http://www.branta.connectfree.co.uk/wild_flowers.htm) Back to Nature's Maths (http://www.branta.connectfree.co.uk/naturemaths.htm)
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