Shared by Christian Bolstad
Mer Fibonacciporr.
by Donald E. Simanek
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:
-
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.
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:
-
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 good evidence that 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 even 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.
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
- 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 or its reciprocal?
(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. 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 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 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] 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
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 fudge a bit.
-
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,
eip = -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
We 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, 
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.
- Technical failure, by Buttonwood From The Economist print edition, Sep 21st 2006. Exposes the delusion that Fibonacci ratios are a magic key to prediction of future financial markets.
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.
- 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 some 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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