The Fibonacci Series
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 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 constant value of 1.6180339887...., called "phi", whose symbol is f or j. Sometimes the Greek letter "tau", t, is used. A striking feature of this sequence is that the reciprocal of f is 0.6180339887... which is f - 1. 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". The ratio f = 1.6180339887... is called the "golden ratio". A rectangle that has sides in this proportion is called the "golden rectangle", and was known to the ancient Greeks. The rectangle is the basis for generating a curve known as the "golden spiral", a logarithmic spiral that is fairly well-matched 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 f and some on 1/f 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 Lucas sequence is the next-best known: 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. [Some books start this series with the seeds 2 and 1, and the rest of the series follows 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 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.... 0.7545776665... whose reciprocal is 0.7545776665... [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 ratios of successive numbers seem to have no consistent pattern, but farther along they converge to values 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 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.
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 "gee-whiz" 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 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. 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." 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. You could make such a thing yourself. Take children's modeling clay and roll it into a uniformly tapered long cone. Then, starting at the pointed end, wrap it around itself.
Common garden snails' shells show a nice spiral, resulting from the way their shell grows.
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 afficianados 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. Golden Ratio Obsessions.
Navels. We read that you can reveal f 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 f. 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 this 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. 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/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.
Rectangles. The golden rectangle has side lengths with ratio f, 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 x 142±1 mm, with ratio 0.64. That's only about 3.5% higher than 1/f. Coincidence? Art and architecture. Some authors claim that artists and architects have throughout history deliberately incorporated f into the proportions of their work. And often-cited 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 "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.
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. Stock Market Shenanigans. Investors often seek a "holy grail" mathematical method for predicting the stock market. Some stock market analysts use the Fibonacci series to guide their investments. Well, it may do just as well as other foolish methods for predicting the future. They might as well throw dice or read tea leaves. Gives you a lot of confidence in the "expertise" of your broker or investment company, doesn't it? Fibonacci Sequence 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.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.
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 "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 numbers 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.
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 something 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 < a> 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 nonsenseIt'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. [1] 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 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. 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. [2] Mustard, Dames' rocket 4 Not a Fibonacci number. Lily, Hyacinth 6 Not a Fibonacci number. Starflower 7 Not a Fibonacci number. Black-eyed Susan 14 Not a Fibonacci number.I have sometimes seen a Black-eyed 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.
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. 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 "gee-whiz" 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, 10, 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 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? 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. Miscellaneous MusingsWe have seen that the Fibonacci sequence is not the only sequence that converges to f. 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! But 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 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." Later, when I went to the university, I realized that folks who major in "Education" are (with some exceptions) those who wouldn't be able to survive the rigors of a real academic discipline. And many 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 which asked for 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". I could make a case for the answer, "C S E", as in "Oh, That The Fierce Fires Should Swiftly Consume Such Exams." A 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 "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. Footnote on dubious investment schemes.I remarked that stock traders and investment counsellors these days often use Fibonacci ratios as a guide to guessing their 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 tea-leaves 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 this fellow 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 longer 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, and 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 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. 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. 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. Web links.
Endnotes.[1] 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?[2] Why is zero a Fibonacci number? You can choose 0 and 1 as seeds to generate the 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. For further reading.These books are good reading, they explain the math, and they don't promote fantastic and mystical interpretations.
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