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| Fibonacci |
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Leonardo of Pisa (1170-1250), nickname Fibonacci, was born in Pisa, Italy. He made many contributions to mathematics, but is best known by laypersons for the sequence of numbers that carries his name:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ...
This sequence is constructed by choosing the first two numbers (the "seeds" of the sequence) then assigning the rest by the rule that each number be the sum of the two preceding numbers. This simple rule generates a sequence of numbers having many surprising properties, of which we list but a few:
This is but one example of many sequences with simple recursion relations.
The Fibonacci sequence obeys the recursion relation P(n) = P(n-1) + P(n-2). In such a sequence the first two values must be arbitrarily chosen. They are called the "seeds" of the sequence. When 0 and 1 are chosen as seeds, or 1 and 2, the sequence is called the Fibonacci sequence. The sequence formed from the ratio of adjacent numbers of the Fibonacci sequence converges to a constant value of 0.6180339887.... called "phi", whose symbol is f or j. Sometimes the Greek letter "tau", t, is used.
Probably the most striking feature of this sequence is that the reciprocal of f is f + 1, or 1.6180339887.... Put another way, f = 1/f - 1. This is true whatever two seed integers you use to start the sequence, this result depends only on the recursion relation you use, not the choice of seeds. Therefore there are many different sequences that converge to f. They are called "generalized Fibonacci sequences".
It's easy to invent other interesting recursion relations. Some have been interesting enough to mathematicians that they carry the names of their originators.
The Lucas sequence is the next-best known: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199 ... It has the seed values 1, 3, and the same recursion relation as the Fibonacci series. [Some books start this series with the seeds 2, 1, and the rest follow just the same.] The ratio of adjacent values approaches f for large values.
How about a different recursion relation, say P(n) = P(n-2) + P(n-3)? With seed numberss 0, 1, 1 we get the series 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9 .... The seeds, together with the recursion relation, uniquely define the sequence. The ratio of successive terms P(n-1)/P(n) converges to 0.7545776665... whose reciprocal is 1.3247295.... [Note that its reciprocal is not one smaller than itself, contrary to what you might have expected.]
Typically, for all of these series, the first few values of the ratio series seem to have no consistent pattern, but soon converge to values that are nearly constant, and after about n = 30 have settled down to values constant to about 10 decimal places.
A search of the internet, or your local library, will convince you that the Fibonacci series has attracted the lunatic fringe who look for mysticism in numbers. You will find fantastic claims:
Of course much of this is patently nonsense. Mathematics doesn't "explain" anything in nature, but mathematical models are very powerful for describing patterns and laws found in nature. I think it's safe to say that the Fibonacci sequence, golden mean, and golden rectangle have never, not even once, directly led to the discovery of a fundamental law of nature. When we see a neat numeric or geometric pattern in nature, we realize we must dig deeper to find the underlying reason why these patterns arise.
The "golden spiral" is a fascinating curve. But it is just one member of a larger family of curves/spirals collectively known as "logarithmic spirals", and there are still other spirals found in nature, such as the "Archimedian spiral." It's not difficult to find one of these curves that fit patterns found in nature, even if those patterns are only in the eye of the beholder. But the dirty little secret of all of this is that when such a fit is found, it is seldom exact, and considerable variations from the "golden ideal" are seen in nature. Sometimes curves claimed to fit the golden spiral actually are better fit by some other spiral. The fact that a curve "fits" physical data gives no clue to the underlying physical processes that produce such a curve in nature. We must dig deeper to find those processes.
Sometimes the authors who write "gee-whiz" science books for the layman engage in "Fibonacci fakery". We cite a few examples.
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In fact, the drawing on the left isn't quite correct. It was found on a web site, and is constructed with circular arc segments within each square. This curve has curvature discontinuity wherever it crosses into another square. The actual Fibonacci spiral has smoothly changing curvature. The difference would hardly be noticable to the eye at this scale.
This diagram reveals how to subdivide the golden rectangle. Draw a square within it. The rectangular area left over is a smaller golden rectangle. Draw a square within it, and continue doing this. Then fit the points with a smooth curve as shown to get something that at least looks superficially like the golden spiral.
However, to save the hypothesis, some might assume that the reason some women wear high heels and some men think that's attractive, is to bring the navel height/body height ratio closer to the ideal.
This navel claim is often illustrated with Leonardo DaVinci's notebook drawing Universal Man, also called "Vitruvian Man", for it was drawn to illustrate a book by Vitruvius. The navel/height ratio in this picture happens to be 0.604, somewhat smaller than f = 0.618. The text accompanying the picture says nothing about this ratio, nor about the distance from navel to feet. The text contains no mention of f. There's no suggestion in the picture that Leonardo was doing anything more profound than relating the man to a circle and a square. In fact, it seems that Leonardo forced the proportions of the man to fit those geometric figures. [Click on the picture for a larger version to print out, and measure it yourself.] Had Leonardo wanted to incorporate f into the picture, he could easily have moved the navel's position up a bit. The fact that he didn't do so tells us that he didn't see any reason to.
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One internet source says that the Greek letter "phi" is used for the golden mean because the Parthenon's architect was Phideas. Funny, we thought phi honored Fibonacci, since the first syllable of his name is pronounced "fi". But we must ask, if f was so important to Phideas, then
It's very hard to find a photo, drawing or painting of the Parthenon seen straight on, to show that alleged beauty. I think most will agree that the most attractive aspect is the usual one that shows two adjacent sides, with perspective. The original Parthenon is crumbling, but there's an exact scale replica in Nashville, Tennesee. Let's look at a photograph of it.
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One of these has been stretched vertically 20%, or reduced vertically 20%. Which one do you find most pleasing? Most dramatic? Most like the real thing?
We wonder why the golden rectangle includes the whole facade, and not just the visually dominant rectangular portion of it. How do you decide whether the lintel should be included? To a scientist this looks like imposing one's own assumptions on the data by making choices without sound reasons. We also wonder how accurate are the measurements used? To a student of pseudosciences it looks like the same sort of exercise engaged in by those people who think they can find p in measurements of the Egyptian Pyramids. Some call them "pyramidiots".
Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and the golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 B.C., showed how to calculate its value. —Keith Devlin, architect.
Some modern artists and architects have deliberately incorporated the golden mean into their works in ways that are more obvious than we find in earlier art. The claim that the result is more pleasing than if they'd used a different ratio is still suspect.
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This peacock is teasing the mystically-inclined mathematicians. The spots on its tail-feathers seem to form spiral patterns. Are these "golden" spirals or some other kind of spiral? The exact mathematical equation of the spiral depends on just how far the bird chooses to fan his tail. He can make it be whatever you like. Does this pattern tell us any scientifically important fact about avian biology? Very unlikely.
Or could it be that there's some mystical or genetic connection bewtween peacocks and sunflowers? We shouldn't mention such possibilities, or someone might take it seriously and incorporate it into a web site—or a textbook.
Phylotaxis. The dictionary defines Phylotaxis as the history or course of the development of something. In biology it generally refers to how a living thing develops and changes over time. This is one part of nature where the fibonacci sequence and related sequences seem to show up uncommonly often, and it's legitimate to inquire why. The interesting cases are seedheads in plants such as sunflowers, and the bract patterns of pinecones.
We have noted above that not all spirals in mathematics or in nature are golden spirals. Likewise, spirals can be produced by non-biological processes if the discrete elements which make up the spiral are laid down according to some simple rules. The problem for biologists is to find those rules. Merely asserting that "nature seems to prefer fibonacci numbers (most of the time, in certain particular cases) isn't an explanation.
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You don't need biology to produce spirals such as those found in sunflower seeds. Here hardware store washers have been laid out on a sheet of refrigerator magnet material. Starting at the center, washers are laid out in a string wrapping around the center, leaving minimal space between each wrap and the previous one. After about six or seven wraps additional spiral patterns develop, just as in the sunflower, spiraling steeply outward, curving away from the center. The reason is simple. The growth pattern of the seed head (and our constructed spiral) is such that it is biased to povide reasonably close packing of the seeds (or washers) consistent with the growth processes. [Photo ©2003 by DES] |
Homemade sunflower pattern. The photo above shows washers laid out in a string, starting at a center. Each washer touches the previous one, and each wrap around the center just touches the previous wrap. No pattern is obvious at first, but after a number of wraps, a pattern of additional spirals emerges. The pattern depends on the radius of the wrap relative to the radius of the washers.
The sunflower seed head is an example of botanist William Hofmeister's 1868 observation that primordia form preferentially where the most space is available for them. They also must form where they attach efficiently to the rest of the plant, and this is a geometric consideration. The pattern can also be modified by moisture and nutrient conditions that affect the size of forming seeds. The pattern of seeds seldom comes out perfectly matched to the golden ratio in the sunflower, but when it is very close, those are the seed heads that get photographed for "gee-whiz" articles about Fibonacci numbers. (Some sunflower seed heads spiral in patterns more closely matched to the Lucas sequence.)
H S M Coxeter, in his Introduction to Geometry (1961, Wiley, page 172), says:
it should be frankly admitted that in some plants the numbers do not belong to the sequence of f's [Fibonacci numbers] but to the sequence of g's [Lucas numbers] or even to the still more anomalous sequences 3,1,4,5,9,... or 5,2,7,9,16,... Thus we must face the fact that phyllotaxis is really not a universal law but only a fascinatingly prevalent tendency.
Spirals can be observed in the bracts of pinecones, the number of clockwise and anti-clockwise spirals usually being two intgers that are adjacent Fibonacci numbers (5 and 8, for example). It's a fun game to look for cones that have spirals that do not match this pattern, just as we children used to look at clover leaves untill we found one with four leaves instead of the usual three.
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| Fibonacci? Not! T-shirt design. |
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Homemade peackock tail fan. Cut strips of cardboard that will be used for a fan. Paint colored spots on each strip, but in such a way that the spots alternate. When these are spread out parallel to each other, a pattern of straight lines, is seen, like the patterns in a field of corn. But when these are held at one end and fanned out, a pattern of spirals results.
Spirals of many kinds can be constructed by application of a simple repetitive rule to govern the placement of objects, as in the washer example above. This tells us somthing profound about nature. It does not take intelligence or a purposeful designer to produce a pattern that is recognizable as "orderly" by intelligent beings. A small set of very simple rules can produce such order. This is demonstrated in the mathematical studies of "cellular automata" and in John Horton Conway's "Game of Life". But part of that rule set is the underlying geometry of the playing field, which puts contraints on what physical processes can do.
It's not hard to select examples that seem to support the notion that nature's patterns are built on f. But if that doesn't work for a particular case, some of these folk start using the ratios of sizes of the first few values of the Fibonacci series, before the ratios begin to clearly converge to f. These ratios are 1, 0.5, 0.6, 0.618, 0.75 with reciprocals 1, 2, 1.5, 1.6, 1.618. In fractional form we have approximately 1/2, 2/3, 3/4, 4/3, 1, 3/2, 5/3 and 2 to make mischief with. Let's see how a flim-flam artist manipulates these. If we also include ratios of these ratios we can play with 1/3, 3/8, 8/21, 5/13, 5/21. We can even throw in the approximate value 1.62 = 34/13 and its reciprocal 13/34 for good measure.
Here's an example of a flim-flam artist at work. Fred Wilson, Extension Specialist in Science Education at the Institute for Creation Research (ICR), wrote a paper titled "Shapes, Numbers, Patterns, and the Divine Proportion in God's Creation." (Impact #354, December 2002). It's full of specious religious drivel, which we will spare you.
His first blunder is to introduce the Fibonacci numbers, 1, 2, 3, 5, 8, ..., then he then tells us that for any two members of the sequence the ratio of the smaller to the larger is "very close to 0.618." Actually that's only true for pairs with values larger than 55. Then comes a statement, italicized yet, "This ratio is only true for this set of numbers". That's flat out false. This ratio is also found in the convergence of the Lucas series and all series with the same recursion relation as the Fibonacci series. They are all called "generalized Fibonacci series". But the early numbers and their ratios, are very different. For example, the Lucas series, 1, 3, 4, 7, 11, 18..., gives us the ratio 3/4, which we didn't have in the Fibonacci series. Choose other seeds and you get lots more ratios to play with!
Wilson says that many things we use are (approximately) patterned after the golden rectangle. In this list I've added measurements and ratios that Wilson didn't mention. He referenced his assertions to a popular book!
credit cards 5.4 x 8.5 cm 0.635
playing cards 5.8 x 9 cm 0.644 (bridge size)
6.4 x 9 cm 0.711 (poker size)
postcards 9 x 14 cm 0.643 (US Postal)
light switch plates 2.75 x 4.75 inch 0.579
writing pads 3x5, 5x7 0.6, 0.714
3 x 5 inch cards 0.6
5 by 8 cards 0.625
Clearly he's willing to consider these ratios "close enough" for his purposes. He conveniently doesn't mention 3.5 x 5, 5 x 7 and 8 x 10 inch standard size photographic prints, nor 8.5 x 11 inch and 8.5 x 14 inch office paper. Computer screens have ratio 1.333 as did movie screens until Cinemascope and Panavision formats popularized wide-screen ratios of 2.666 in the 1950s.
And what's the point anyway? These proportions are often determined by the measurement system in use, and, in the case of photographic and writing paper, the practical need to cut large sheets into smaller ones without waste.
Wilson asserts that great artists of the past have "employed the golden proportion in their works". He says (without proof) that they did this deliberately when dividing their easel "into areas based on the golden proportions" to determine the placement of horizons, trees, and so on. Obviously he hasn't a wide acquaintance with great art works.
Wilson cites numbers of petals on flowers.
lily 3 violet 5 delphinium 8 mayweed 13 aster 21 pyrethrum 34 helenium 55 michelmas daisy 89
These examples associate with Fibonacci numbers. But Wilson neglects to mention these others:
many trees 0 This is a Fibonacci number, though he didn't tell us that. mustard, dames' rocket 4 Not a Fibonacci number. hyacinth, corn lily, solomon's seal 6 Not a Fibonacci number starflower 7 Not a Fibonacci number
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| Dame's Rocket. 4 petals. |
Little False Solomon's Seal. Lily family, 6 petals. |
Starflower (Trientalis borealis). 7 petals. |
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| Defying Fibonacci. All these are flowers commonly seen in the wild. | ||
Many trees have flower parts (stamen, pistil) with no petals. In the mustard family is the colorful 4-petaled Dames Rocket, a garden escape flower that is prolific along roadways and fields in the early summer in the USA.
It gets better. Wilson says that studies of phylotaxis show that the arrangement of leaves around a plant stem conform to Fibonacci numbers.
elm 1/2 beech and hazel 1/3 apricot and oak 2/5 pear and poplar 3/8 almond and pussy willow 5/13 pine 5/21 pine 13/34
Wilson is selecting cases again, using ratios of the first few members of the series and neglecting plants that have other ratios. But there's a reason. He's leading up to something, as we shall see. He wants to demonstrate a relationship between these numbers and the periods of planets of the solar system! He compares each planet's period (in round numbers!) to the period of the planet adjacent to it, starting with the planets most distant from the sun.
elm 1/2 Uranus beech and hazel 1/3 Saturn apricot and oak 2/5 Jupiter pear and poplar 3/8 Asteroids almond and pussy willow 5/13 Mars pine 5/21 pine 13/34 Mercury
Unfortunately Pluto, Neptune, Venus and Earth don't fit this scheme. It's rationalization time! And his rationalizations are lulus:
This is classic pseudoscientific mystical flim-flam! After all this flummery, Wilson has the audacity to say "To think that the times of revolution of the planets around the sun correlates with the arrangement of leaves around stems on plants is also an amazing phenomena."
"Incredible" would be a better term, i.e., "not credible". Wilson wants to have it both ways! Anything that fits is evidence of God's creation, Anything that doesn't fit is evidence of that thing being "special" in the eyes of god. Other things that don't fit are due to Adam's sin. It's scary to realize that someone with this warp of mind is charged with the science education of students at ICR, who may end up certified to teach sciences in high schools. This sort of specious argument is entirely typical of the pseudoscientific garbage regularly spewed out by creationists. And then they wonder why real scientists do not take them seriously.
[2007 update] Now that astronomers have deleted Pluto from the family of true planets, Wilson may be able to say "I predicted that!". It does illustrate that some names and labels in science are partly arbitrary. I remember that in school we were expected to memorize how many moons each planet has. Such an exercise seems rather pointless now that our space probes have found so many more moons, and more and more each year. I liken it to the pointlessness of elementary school excercises that have students look at flowers and pinecones to discover the Fibonacci ratios in them. Haven't schools anything better to do with class time?
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| Fudge p. |
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It's not difficult to find examples of most any pattern or mathematical relation you want. Then some people make the mistake of supposing this reveals some mystical governing principle in nature. This is reinforced by ignoring equally important cases that don't fit the pattern. If the fit isn't very good, approximate or fudge the numbers. If some things remain that ought to fit but don't, just rationalize a reason why they are "special cases".
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| The five regular solids corresponding to the five elements. |
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Folks addicted to mystical mathematics are really motivated by a belief that there's something "magical" about certain combinations of numbers. They are obsessive pattern seekers. Pattern recognition can be a useful trait, if not carried to the point of believing that every perceived pattern represents something profound or mystical.
So which of these is the mystical mathematical foundation of nature? Silly question!
But this raises a skeptical thought. When I was young, one commonly saw questions on "intelligence" tests that gave six or seven numbers or letters of a sequence, and you were asked to supply three or four entries to continue the sequence. When I first encountered these, in high school, I thought to myself, "This is an unfair question!" Why? Because any finite string of numbers or letters can be the starting point of an infinite number of different sequences, all with a different recursion formula. Who is to say which of these formula is the "right" answer, or the "most intelligent" answer or even the "best" answer? I thought to myself, "Are these test writers idiots, that they don't realize that obvious fact? Maybe they should have their intelligence measured—properly." Only later, when I went to the university, did I realize that folks who major in "Education" are (with few exceptions) those who wouldn't be able to survive the rigors of a real academic discipline. And they often lack creativity and imagination.
Only now, in the beginning of the 21st century, have dim-witted "experts" in educational testing begun to realize that these are frauds, for this very reason. I remember one satire of educational tests suggesting the completion of the series O T T F F S S _ _ _. Is a person who "sees" the answer the test-maker had in mind really more intelligent than one who doesn't? Is the response D T B acceptable? (It should be: "Otters Tend To Frolic Freely, Sliding Smoothly Down The Bank.") Sounds as good to me as the desired answer, "E N T" for "One Two Three Four Five Six Seven Eight Nine Ten". Oh, that the fierce fires should surely consume such exams.
The internet is cluttered with "educational" sites that have documents on "Fibonacci numbers in nature" and similar topics. A corresponedent suggested a reason for this, which resonated with my own experience in the "ed-biz". Perhaps this is the result of the current climate in education in which teachers are under great pressure to pander to student feelings and interests. Students continually ask for reasons for studying academic subjects—reasons that will convince students of the relevance of that subject to their own narrow interests and egocentric perspective. The idea of being interested in something for its own sake is a foreign concept to them. So teachers go out of their way to "invent" relevance, even if it is a fragile and tenuous relevance. To show that some part of mathematics is relevant to nature, art, or the location of navels, serves that purpose. In doing this, textbook writers and teachers often display their own shallowness of thought.
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