The Fibonacci Series
Leonardo of Pisa (~11701250), also known as Fibonacci, wrote books of problems in 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:
The Fibonacci sequence is but one example of many sequences with simple recursion relations. The Fibonacci sequence obeys the recursion relation P(n) = P(n1) + P(n2). In such a sequence the first two values are arbitrarily chosen. They are called the "seeds" of the sequence. When 0 and 1 are chosen as seeds, or 1 and 1, 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 value of 1.6180339887...., called "phi", whose symbol is ø or φ. Sometimes the Greek letter "tau", τ, is used. A striking feature of this sequence is that the reciprocal of φ is 0.6180339887... which is φ  1. Put another way, φ = 1/φ + 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 φ. They are called "generalized Fibonacci sequences". The ratio φ = 1.6180339887... is called the "golden ratio" or "golden mean". A rectangle that has sides in this proportion is called the "golden rectangle", and it was known to the ancient Greeks. The golden rectangle is the basis for generating a curve known as the "golden spiral", a logarithmic spiral that is fairly wellmatched to some spirals found in nature, and this fact is the source of much of the popular and mystical interest in this mathematical subject.
Note: Writers on this subject sometimes concentrate on φ and some on 1/φ as the ratio of interest. This is no "big deal" for when you have a ratio of two values, say A and B, which is a comparison of their sizes, the reciprocal of the ratio A to B is just the ratio of B to A.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 nextbest known is one of the Lucas sequences: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199 ... It has the seed values 1 and 3, and the same recursion relation as the Fibonacci series. The ratio of adjacent values approaches φ for large values. Other examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Jacobsthal numbers, and a superset of Fermat numbers. See: Lucas Sequence. All are mathematically interesting, so why is it that only the Fibonacci numbers get all the attention from Fibonacci fanatics? How about a different recursion relation, say P(n) = P(n2) + P(n3)? With three seed numbers 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 1.3247295.... whose reciprocal is 0.7545776665... [Note that its reciprocal is not one smaller than itself, contrary to what you might have expected. However, this ratio and its reciprocal do add to nearly 2 (Approximately 2.08 if you are fussy.) Isn't that close enough to count as "mystical"? Typically, for all of these sequences, the first few values of the ratios of successive numbers seem to have no consistent pattern, but farther along to larger values they converge to ratios that are nearly constant, and after about n = 30 have settled down to values constant to about 10 decimal places.
Fibonacci Foolishness.A search of the internet, or your local library, will convince you that the Fibonacci series has attracted a lunatic fringe of Fibonacci fanatics who look for mysticism in numbers and in nature. 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.
Golden Spiral Hype.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 fits a particular pattern found in nature, even if that pattern is 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. The examples from nature that you find in books often have considerable variations from the "golden ideal". 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.Many books about Fibonacci numbers feature dust jackets with pictures of spirals from nature. This helps sell the books, because people like pretty pictures.
Order in the eye of the beholder.Sometimes the authors who write "geewhiz" science books for the layman engage in "Fibonacci fakery". We cite a few examples.
Nautilus shells. Consider the commonly seen assertion that shells of the Chambered Nautilus conform to the golden spiral. The photo on the right shows one that has been sawed carefully to show the inner chambers. For comparison, the actual golden spiral is shown on the left. Clearly this creature hadn't read the books! If these two were superimposed they wouldn't match no matter how they were scaled or aligned. 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.
The Peacock's tail. This peacock is teasing the mysticallyinclined mathematicians. The spots on its tailfeathers 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 us seriously and incorporate it into a web site—or a textbook. The Chameleon's Tail. This is the tail of the smug chameleon, whose picture is higher up on this page. He's telling us something with that curled tail. Something like "I can create something close to a golden spiral without a degree in higher mathematics. It's simple. Just start with a tail that's essentially a long slender cone, and wrap it tightly. The result is just as good as that chambered nautilus shell everybody makes such a fuss about."
Takehome experiment. Using modeling clay, make a slender, long cone and wrap it like the chameleon's tail. See, you, too can easily create a wonder of nature.This reveals the simple secret of sprials in nature. They often result from growth with constraints. As the nautilus grows, the open end of its shell increases in diameter, at a nearly constant rate. It is constrained to curve around the existing shell. The result is a spiral curve, something close to a logrithmic spiral, which is a Fibonacci spiral.
Common garden snails' shells show a nice spiral, resulting from the way their shell grows.
Takehome experiment. Another spiral sometimes found in nature is the Archimedian spiral. You can construct such a spiral on paper. Mark a center point. At some distance mark another point. Then make an angle from this point, centered at the center point, of 45°. Mark a point at a radius an amount x smaller than the radius of the first point. Keep doing this. As you rotate another 45° make the radius x smaller each time. Result: a nice spiral, but it's not a golden spiral. And remember that nature's spirals aren't always Fibonacci spirals, and sometimes not even close.Here's an example of how the Fibonacci fanatics fool themselves and fool others. This picture from a website shows a chameleon's coiled tail with superimposed golden spiral, to make the point that they are "alike". But the comparison clearly shows that the two are not the same shape—the match is quite poor.
The spirals of pinecones and sunflower seedheads are pretty, but can we say they are golden spirals? The part we see of each spiral doesn't wrap around the center even once, much less than more than once. So we have only a short segment of spiral, which could be fit to any number of different mathematical curves. The golden spiral has segments of varying curvature, and short segments of it can match many other curve segments rather well, from circles to ellipses to parabolas. The claims of phylotaxis for plants does not predict golden spirals, only, in some cases, the number of spirals around 180°. We'll have more to say about this later.
Spirals Everywhere.Spirals naturally result from geometry. Consider the woven straw basket or hat. Straw of roughly constant width is woven together starting at the center, always keeping adjacent strips closely packed. The result is quite naturally a patern of spirals.
In all cases spirals result from simple rules of layering and packing, filling a two or three dimensional space.
Golden Ratio Obsessions.Navels. We read that you can reveal φ by measuring the height of a person and the height of the person's navel, measured from the floor. The ratio of navel height to total height is supposed to be φ. And with the current interest in navels, the implication is that this is one indicator of attractive bodily proportions. Has anyone checked real people? In the interest of science I checked that assertion for a large sample of the most popular swimsuit models. This should check the claim that bodies judged "beautiful" should have the ideal characteristics of form, including the ideal navel height. [It's a tough job, but someone has to do it.] The results averaged 0.58±0.01, with rather small variation. So much for that myth. Can anyone claim that a gorgeous swimsuit model is any less attractive if she wears a one piece suit that covers her navel commpletely? That observation exposes the silliness of any claim about Fibonnaci navels. 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 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 1/φ = 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 φ. 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 φ 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. Rectangles. The golden rectangle has side lengths with ratio φ, and is supposed to be the most pleasing or attractive rectangle. Studies with real people judging rectangles of various ratios have shown that rectangle preferences vary widely, but most people prefered a ratio of 3/4 = 0.75. The size of our recently published book, Science Askew has dimensions of 222±0.5 × 142±1 mm, with ratio 0.64. That's only about 3.5% higher than 1/φ. Coincidence? Art and architecture. Some authors claim that artists and architects have throughout history deliberately incorporated φ into the proportions of their work. And oftencited example is the Parthenon.
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 "fie". But we must ask, if φ 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.
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 π in measurements of the Egyptian Pyramids. Some call them "pyramidiots".
Certainly, the oftenrepeated 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, mathematician. 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. If the ancient architects deliberately included specific mathematical ratios and shapes in their structures it may have had nothing to do with aesthetics, but with a desire to achieve some mystical harmony with nature. But this hypothesis has as little support as the aesthetic hypothesis. Stock Market Shenanigans. Investors often seek a "holy grail" mathematical method for predicting the stock market. Some stock market analysts and playyers use the Fibonacci series to guide their investments. Well, that method may do just as well as other foolish methods for predicting the future. They might as well throw dice or read tea leaves. This gives you a lot of confidence in the "expertise" of your broker or investment company, doesn't it? Fibonacci in Nature?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 and pineapples. [1]We have noted above that not all spirals in mathematics or in nature are golden spirals. Likewise, spirals can be produced by nonbiological 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.
Takehome experiment. Homemade spirals. 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 "geewhiz" articles about Fibonacci numbers. (Some sunflower seed heads spiral in patterns more closely matched to one of the other Lucas sequences.) 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 phylotaxis is really not a universal law but only a fascinatingly prevalent tendency. Spirals can be observed in the bracts of pinecones, the numbers of clockwise and anticlockwise 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. What might those numbers be? Look at reasonably small numbers in the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21. Ratios 2:3, 3:5, 5:8, 8:13, 13:21 match the "adjacent" rule. What's left? 2:5, 3:8, 5:13, 8:21 4:7 4:9, 7:9 7:10, 7:11 9:11. Some of these are ratios of alternate members of the Fibonacci sequence. Some might be found adjacent in a Lucas sequence. Readers are invited to send me pictures of pinecones with these ratios.
Homemade peackock tail fan. Cut strips of cardboard that will be fastened at one end and 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. Are they Fibonacci spirals or something else?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 something profound about nature: Nature does not require intelligence or a purposeful designer to produce a pattern that we recognize as "orderly". 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.
Compounding nonsense.It's not hard to select examples that seem to support the notion that nature's patterns are built on φ. 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 φ. If we also include ratios of these ratios we can play with 1/3, 3/8, 8/21, 5/13, 5/21. 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. We can even throw in the approximate value 1.6^{2} = 34/13 and its reciprocal 13/34 for good measure. And if none of these suit our purpose, we can always try one of the Lucas sequences.
Creationist FlimFlam.One can hardly escape the observation that the Fibonacci fanatics display an almost religious conviction that all of nature is somhow based on or guided by the numbers of the Fibonacci sequence and the golden mean. They reinforce this belief by seeking examples that "fit" their conviction, and ignoring all that don't.Here's an example of a flimflam 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. [2] His first blunder is to introduce the Fibonacci numbers, 1, 2, 3, 5, 8, ..., then he 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 asserts that many things we use are (approximately) patterned after the golden rectangle. In this list I've added measurements (starred) 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. Older computer screens have ratio 1.333 as did movie screens until Cinemascope and Panavision formats popularized widescreen ratios of 2.666 in the 1950s. Computer screens are now wider, often, 13 × 8.5 inch, with a ratio of 1.53. 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. [3] Mustard, Dames' Rocket 4 Not a Fibonacci number. Lily, Tulip, Hyacinth 6 Not a Fibonacci number. Starflower 7 Not a Fibonacci number. Gardenia 8, 9 or 10 petals. 9 and 10 are not Fibonacci numbers. Greek Anemonie (various) 14 or 15 Not Fibonacci numbers. BlackEyed Susan (some) 14 Not a Fibonacci number. Mountain Laurel 10 Not a Fibonacci number. Gazania 16 Not a Fibonacci number.I have sometimes seen a Blackeyed Susan with 13 petals (a Fibonacci number), but that must be a freak of nature. :) See my picture below. Actually this plant has many varieties, with various numbers of petals. One might wonder why so few flowers have 9, 11 or 12 petals. None of these are Fibonacci numbers. Some of the flowers that have these numbers of petals have doubled petals, thus 3×2 = 6 (lily), and this doubling is revealed in their structure. This is often cited by those who want to support the notion that nature prefers Fibonacci numbers. But how do you account for the Starflower, with 7 genuine and equal sized petals?
Many trees have flower parts (stamen, pistil) with no petals. In the mustard family is the colorful 4petaled Dames Rocket, a garden escape flower that is prolific along roadways and fields in the early summer in the USA. All of these pictures are of common flowers, found in fields and roadways and in your gardens. None are exotic rarities. Anyone who accepts the "geewhiz" assertions that nature's flowering plants prefer Fibonacci numbers is simply not very observant, and rather gullible.
I have only recently begun paying attention to flowers with a number of petals larger than six. So my own picture collection is missing the numbers 9, 11, 12 and many of the larger numbers. I welcome reader submissions of such pictures, prefereably with identification of the name of the plant.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 cherrypicking 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 flimflam! 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 blamed on 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 drivel 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 arbitrarily chosen. I remember that in school we were expected to memorize how many moons each planet had. 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?
Conclusion
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".
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. Some patterns in nature are significant, but many are purely accidental (patterns in tealeaves, for example) and have no deeper meaning or significance.
Miscellaneous MusingsWe have seen that the Fibonacci sequence is not the only sequence that converges to φ. There are also many other mathematical sequences that start out with the conventional Fibonacci numbers, but as the sequence is extended, converge to something else. These are lots of fun, too, and are called "Fibonacci Forgeries". A search of the web will reveal dozens of sites explaining them.So which of these is the mystical mathematical foundation of nature? Silly question! This raises a skeptical thought. When I was young, we 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 formulae 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 now, in the beginning of the 21^{st} century, have dimwitted "experts" in educational testing begun to realize that these are frauds, for this very reason. I remember one satire of educational tests which asked for the completion of the series O T T F F S S _ _ _. Is a person who "sees" the answer the testmaker 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". I could make a case for the answer, "C S E", as in "Oh, That The Fierce Fires Should Swiftly Consume Such Exams."
Historical precedents.
The Greek Pyhthagoreans held the view that certain numbers and geometric figures were especially significant and carried mystical importance. They were "more perfect" than others. Nature was in a continual state of seeking perfection and never quite achieving it, here on earth. But in the heavens all was perfect, and therefore, they thought, the motions of heavenly bodies in the sky were based on a system of perfect circles. Modern mathematical mystics seem to be playing the same game. Nature "wants" to make Nautilus shells perfect Golden Spirals, but none of the actual shells perfectly match the spiral. Nature "wants" all flowers to have petal numbers that are Fibonacci numbers, but some exceptions are found. But what is the criteria to determine which things are the perfection that nature strives for and which are not? Simple, the ones most commonly found in nature are the perfect ones. That's circular reasoning, but isn't the circle the perfect geometric figure?
Another somewhat cynical footnote.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 "edbiz". 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 some 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 height of navels, serves that purpose. In doing this, some textbook writers and teachers often display their own shallowness of thought.
Footnote on dubious investment schemes.I remarked that stock traders and investment counsellors these days often use Fibonacci ratios as a guide to guessing predictions. There is even computer software for making market predictions, that claims to use "Fibonacci methods". This is one of the methods used in "technical trading". Using numbers and charts to make predictions justifies the label "technical", though the results would be just as good had the patterns of tealeaves been used. One only has to eavesdrop on the websites and forums these people frequent to discover that many of them still believe in the "magic of numbers". Whole books tout these methods, with testimonials to their success, and these do make money, for those who write the books. One fellow who uses Fibonacci ratios frankly admits that they may not be "magic" but they do make his presentation charts look more impressive to clients. Of course the efficacy of such methods has never been scientifically tested. And why should anyone waste the effort?One such fellow emailed me, complaining about my negative comments. I soon discovered that he was a sucker for all sorts of pseudoscientic numerology. He even tried to tell me how valuable was the Martingale system, popular in 18th century France and still used by some gamblers. It's simple. Each time you win you make the same size bet the next time. When you lose, you double the size of your bet the next time. Of course, any "system" can seem to work in the short run, once in a while. But in the long run (when played for a long time, or many times) it has no advantage, and while your chance of winning in the short run may seem to be improved, your chance of losing big increases the more you play. Statisticians have analyzed such systems and concluded they are deceptions, but gamblers are often susceptible to such deceptions. And what is the stock market, but a gambling game with confounding variables, with the players themselves affecting the odds? Then this guy tried to tell me that Fibonacci numbers show up more often in winning lottery numbers. He could provide no data supporting that. Then he claimed Fibonacci numbers show up more often in the digits of phone numbers in the phone book. Well, duh? Of the ten digits 0 through 9, six are Fibonacci digits (0, 1, 2, 3, 5, 8) and four are not (4, 6, 7, 9), so Fibonacci digits should show up about 60% of the time if the digits were equally probable. No great mystery there. The only example he could produce, from his own "extensive research", was a set of 200 phone numbers, 65% of the digits being Fibonacci digits. That's well within the limits of error for that small size sample. [5] Some say that you can increase your success in the stock market by rolling dice or throwing darts to make your choices. Such investments will, in the very long run, averaged over many investors, do as well as if you used a broker, and you won't have to pay the broker's fee. I am sure there are brokers who shun mystical and magical formulas, but I remain unconvinced that even they earn their large fees.
More Fibonacci Fakery.This picture was emailed to me by someone who could not remember the source. It is a clever deception. I suspect the picture was delibrately embellished to make fun of Fibonacci foolishness. If the person who created this would like credit, please get in touch. The water path isn't a Fibonacci spiral, but someone has cleverly superimposed the golden rectangle on part of it so that at first glance looks like it is. Look carefully. That large inner rectangle should be a square, but it is wider than high. The rectangle in the upper right corner is almost a square. If you tried to cheat by reducing the horizontal magnification, you could make the large one square, but the smaller one would no longer be nearly square. I think that's clear forensic evidence that someone was deliberately cheating. The "handdrawn" appearance of the rectangles seems contrived to disguise the cheating. (The artist should have used computerdrawn clean straight lines and perfect geometry.) Such cheating is often done with the pictures of Nautilus shells seen in books (see above). A physicist would conclude immediately that this can't be a golden spiral, nor any of the textbook spirals common in physics. Textbooks of important mathematical spirals show pictures and equations for the Spiral of Archimedes, the logarithmic or equiangular spiral, the hyperbolic or reciprocal spiral, the parabolic spiral, and my favorite, the involute of a circle. The reason is simple. Such spirals are radially unbiased. In this picture gravity provides a bias (and distortion) compared to a similar process happening in a horizontal plane. Also, the source of the water, her wet hair, isn't stationary. She produces this dramatic picture by flinging her head and body rapidly upward and backward. This sort of cheating is what I object to. If one wants to be honest one might say this picture "suggests" a golden spiral. The "fact" of it being "something like" a golden spiral tells us nothing useful about it. Deeper inquiry might lead to design of a water hose rotating about a fixed vertical axis to experimentally investigate whether this process could produce something closer to the Fibonacci spiral. I have no reason to predict that outcome with certainty. It might just be an involute of a circle. I've challenged some mathematicians with this problem, but they don't take the bait. See: The Skater's Spiral for another interesting spiral. With so many sprials that do have physical significance, I wonder why these don't capture public attention.
Fibonacci's Rabbits.
A reader complained that I never mentioned Fibonacci's rabbit problem. It appears in chapter 12 of his Liber abbaci (1202), among problems dealing with food and money. You might have thought this example was an important part of his writing, but it was apparently no more important to him than any of the other problems he discussed. Yet modern geewhiz popular books would lead you to think that it was his greatest contribution to mathematics and science.
HOW MANY PAIRS OF RABBITS ARE CREATEDThe monthly populations turn out to be the Fibonacci series, starting with 2. That's it. End of story. Does this tell us anything about rabbits in the real world? Hardly, for rabbits don't necessarily choose to follow the arbitrarily imposed rule that they must bear a single pair every month. (At the end of the year you'd have 377 highly inbred rabbits, if they followed the rules.) Fibonacci's problems, in the style of the time, used concrete examples rather than abstract numbers, and these were chosen for audience appeal, not scientific significance. The problems were posed using only words, as were the solutions, for our modern abstract notation for algebraic equations hadn't yet been invented. An anonymous book of around 1290, Livero de l'abbecho, clearly derivative of Fibonacci's book, has the same problem posed with pigeons. [Devlin, p. 135.] Nearly all the problems in Fibonacci's books derive from earlier sources. The books were intended for instruction, not as original contributions to mathematics. The rabbit problem dates back at least to the Sanskrit grammarian Pingala (~450~200 BCE) in his Chandahshastra (The Art of Prosody) about the rhythms of music and speech. [Devlin, p. 145.]
The golden ratio in art.I just checked an art supply store to see what size mounted blank canvas they offer for artists. I find 8x10, 6x8, 11x14, 9x12 and 12x16 (all in inches), with ratios 0.8, 0.75, 0.786, 0.75 and 0.75 respectively. Apparently they, and their customers, haven't heard that the golden ratio 0.618 is supposed to be the most artistically pleasing. Of course artists can always remount the canvas to another size. How many do? Check your nearest art gallery.
Web links.
Endnotes.[1] Someone told me of a theory of phylotaxis in plants. The leaves branch from the main stem so that upper leaves don't shade the lower leaves from the sun, and this causes the radial angle of the leaf stems to follow Fibonacci numbers. This argument is lost on me. The sun moves across the sky, so the ideal leaf position in the morning would be different at other times of the day. Now it is true that some (but not all) plants "follow the sun". That doesn't argue for the leaf (and branch) stems having a particular numeric pattern of emergence from the main stem. Then there are plants that live entirely in the deep shade of a forest and receive little or no direct sunlight, only diffuse light from all directions. Yet the emergence pattern of leaf stems is still present in many such plants. What use is the "shadowing" theory to explain them? Now consider the often cited example of the sunflower. The large seed heads do follow the sun during the day, but this has nothing to do with either (1) the arrangements of seeds in spiral patterns, or (2) the number of flower petals around the seed head (many), or (3) the arrangement of their leaves. Some web sites claim the number of sunflower petals is 34, a fibonacci number, some say it is on average 34. Do you suspect there's some variation in the number of petals?But on a more basic level, why should plants prefer cycles of 3, 5, 8 (Fibonacci numbers) rather than the adjacent numbers 4, 6 and 7 (not Fibonacci numbers)? I have never seen a proof that leaf cycles of 3, 5 and 8 are more efficient at utilizing sunlight than 4, 6 or 7. One would think seven would be ideal. Students learn, from years of schooling, to accept and regurgitate bogus textbook assertions and specious explanations without question and without insisting on hard evidence and mathematical proofs, and without even questioning other implications of inadequate explanations. Some say that flowers that have a nonFibonacci number of petals must have one of the Lucas numbers instead. (1, 3, 4, 7, 11, 18, 39, 47, 76, 123...) So why might a particular one of the infinite number of Lucas series fit one plant and not another? The fibonacci fanatics have no answer. If you are willing to play this silly game that way, any number can be a part of a mystical pattern in nature. The rules are simple: Find a pattern in nature. That's always possible, for nature has many regularities in time and space. Then choose (or invent) a mathematical pattern to fit. Then bask in the aura of mystery and wonderment at the "fact" that nature "chose this pattern". The hypothesis of mystical numbers, is circular, selffulfilling, and trumps common sense. [2] Impact is a free publication of the Institute for Creation Research, an organization that does no scientific research, but promotes "Creation Science", or "Creationism", a collection of religious ideas promoting itself as "scientific". Shortly after I wrote this web page, stimulated to write it by Wison's outrageous article in that publication, I noticed that they stopped sending me the magazine. I had been receiving it regularly for more than 10 years previously. Coincidence? Or could it be because I had never sent them a "donation" to support their work? [3] Why is zero a Fibonacci number? You can choose 0 and 1 as seeds to generate the Fibonacci sequence, or 1 and 1, or 1 and 2, or any other two numbers of the sequence, and the subsequent sequence is the same. It's a matter of definition. If we define the seeds to be the smallest integers that generate the sequence, and if zero is an integer (which mathematicians assure us it is), then certainly zero meets the definition of a Fibonacci number. [4] Flowers with large numbers of petals larger than 13 have inconsistent numbers of petals from one bloom to another. One cannot make a case for nature's petal number preferences in such flowers. [5] Feb 2010. Reader Andrea Zimmerman suggests a very good reason why digits of phone numbers might not be random.
I loved the example at the end about the digits of phone numbers being mostly Fibonacci. There is even more explanation as to why this is true. It was designed that way. 0 and 1 as the middle digit had a specific meaning, and the more populous the area, the lower digits were used as the first and second since with rotary phones it took more time to dial a 9 than a 3. Most area codes have stayed with the area they started in even as we got away from rotary phones. See: Area Code History .This historic "rotarydialerbias" would favor smaller digits. Most of the onedigit Fibonacci numbers 0, 1, 2, 3, 5, 8, are the smaller value digits. This effect would indeed make Fibonacci numbers more common digits in phone numbers than would be expected if they were randomly distributed. As available phone numbers get "used up" in high population areas the residual bias may be small, but it could account for the frequency of Fibonacci digits in phone numbers being somewhat higher than the naive expectation of 60%. When something appears to be magic or miracle, look deeper and you may find the real reason.
For further reading.These books and web documents are good reading, they explain the math, and they don't promote fantastic and mystical interpretations.
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