The Fibonacci Series

Fibonacci

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:

• Take any three adjacent numbers in the sequence, square the middle number, multiply the first and third numbers. The difference between these two results is always 1.
• Take any four adjacent numbers in the sequence. Multiply the outside ones. Multiply the inside ones. The first product will be either one more or one less than the second.
• The sum of any ten adjacent numbers equals 11 times the seventh one of the ten. Mesoamericans thought the numbers 7 and 11 were special.

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 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.

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:

• The "golden rectangle" is the "most beautiful" rectangle, and was deliberately used by artists in arranging picture elements within their paintings. (You'd think that they'd always use golden rectangle frames, but they didn't.)
• The patterns based on the Fibonacci numbers, the golden ratio and the golden rectangle are those most pleasing to human perception.
• Mozart used f in composing music. (He liked number games, but there's no evidence he ever deliberately used f in a composition.)
• The Fibonacci sequence is seen in nature, in the arrangement of leaves on a stem of plants, in the pattern of sunflower seeds, spirals of snail's shells, in the number of petals of flowers, in the periods of planets of the solar system, and in stock market cycles. So pervasive is the sequence in nature (according to these folks) that one begins to suspect that the series has the remarkable ability to be "fit" to most anything!
• Nature's processes are "governed" by the golden ratio. Some sources even say that nature's processes are "explained" by this ratio.

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.

• 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 are 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.

• 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 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 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

  • Why did he incorporate it into the smaller end of the building only?
  • Why is the rectangle of the floor plan in ratio 7/16 = 0.4375 with reciprocal 2.286? Wouldn't he have made the ratio f? (There are some interior details of the floor plan that do happen to come close to the golden ratio, but none would be visibly apparent to anyone standing inside.)
  • The parthenon sits on a hill, and none of its rectangular features are seen as rectangles from the ground.
  • Phideas used columns that taper toward the top, giving an illusion often employed by architects. It makes the structure seem taller. Doesn't that defeat the supposed purpose of using the golden rectangle as the most attractive rectangle?

  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. We heard a few years ago that stock market analysts were using the Fibonacci series to predict future stock prices. Well, it may do just as well as other foolish methods for predicting the future. They might as well throw dice. Gives you a lot of confidence in the "expertise" of your broker or investment company, doesn't it?

Order in the eye of the beholder.

Peacock poppycock.

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.

You don't need biology to produce spirals similar to 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 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.

Fibonacci? Not! T-shirt design.

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.

Compounding nonsense

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. [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 says 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.

Dame's Rocket. 4 petals. Usually blue or white. This is an unusual variegated bloom.	Lilly, 6 petals.	Starflower (Trientalis borealis). 7 petals.
Defying Fibonacci. All of these are flowers commonly seen in the wild.

Black-eyed Susan. 14 petals.	Gazania. 16 petals.	Count them yourself.

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 7, 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] correlation is far more than just a chance arrangement. It is one more example of God's marvelous mathematical arrangement of His creation. The fact that it is not perfect reveals that although Adam's sin affected the whole creation (Romans 8:22), yet God in His goodness has not allowed sin to overcome all the marks of His great handiwork (Psalm 19:1)."

• "A most interesting divergence in the chart is that of the Earth. As the next planet in the series after Mars, its number should be 8:21, but it isn't... It is my opinion that this anomaly is evidence of God's showing the uniqueness of planet Earth in relationship to the whole cosmos."

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

Fudge p.

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".

• The areas of mathematically similar objects are proportional to the square of their linear dimensions, their volumes are proportional to the cube of their linear dimensions. Gravitational and electric field strengths obey an inverse square relation to distance. Radiation intensity obeys an inverse square relation to distance from a point source. All of these have an underlying reason: the geometry of the universe is very nearly Euclidean, and therefore these results are dictated by that geometric fact. It doesn't suggest there's anything mystical about the powers "2" and "3".

• The ratio of the circumference to diameter of a circle, p, pops up in formulas for many geometric relations about round objects. A favorite obsession of numerologically-inclined folk is to look for p in man-made structures such as the Pyramids of Egypt. Look and ye shall find—if you are willing to select data and fuge a bit.

The five regular solids corresponding to the five elements.

• The five regular Pythagorean solids have faces of similar shape, either triangles, squares or pentagons. These are also known as the "Platonic solids". The tetrahedron, octahedron, cube, icosahedron and dodecahedron have 4, 6, 8, 20 and 12 faces respectively. There are no other such solids. Only one of these, 8, is a Fibonacci number. Johannes Kepler, when still in the mystical mode of thought, tried to fit these numbers to "explain" the orbital sizes of the planets. He had to fudge things too much to fit his model to reality so he wisely abandoned the project. Only when he rid himself of mystical correspondences was he able to formulate a mathematically correct set of three laws of planetary motion. These laws implicitly embodied what we now know as the conservation of angular momentum.

• The reason f shows up in nature has to do with constraints of geometry upon the way organisms grow in size. Irrational numbers (those that cannot be expressed as a ratio of integers) are often revealed in this process. The well-known irrationals are Ö2, f, e, p and any multiples or products of them. To make matters more interesting, these are related. For example, phi is f = (Ö5 - 1)/2. And the Euler relation, e^ip = -1 relates e, i and p where i = Ö(-1). The natural processes that display irrationals are not governed or caused by f in order to achieve some desired purpose or result, but rather they are constrained by the geometry of the universe and the limitations imposed by that geometry on growth processes.

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 Musings

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 21^st 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.

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.

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 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 lottery numbers. He could provide no data supporting that. Then he claimed they 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.

• Here's an excellent site about phylotaxis.
• Misconceptions about the Golden Ratio by George Markowsky.

Endnotes.

[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 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.

• Livio, Mario. The Golden Ratio, the story of phi, the world's most astonishing number. Broadway Books, 2002. The history of the number phi and its curious mathematical relationships. Also includes accounts of numerous phi-fixated individuals.

• Walser, Hans. The Golden Section. The Mathematical Association of America, 2001. History, fractals, geometry, paper folding, sequences, regular and semi-regular solids. Lots of fun.

• Stewart, Ian. Life's Other Sectret, The new mathematics of the living world. Wiley, 1998. Very readable account of mathematics in nature.

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