©06 The Media Desk
All Nature answers to the Fibonacci Sequence... doesn't it?
[NOTE: This article is part of the Desk's Mystery Series. The Desk means no undue disrespect to mathematicians. Far from being an exhaustive thesis on the subject, this article continues the series of overviews and briefs on the various subjects. In that light- Enjoy ]
[See note about sources and links below- Thank you]
Part of the Desk's Mystery Series.
Ahhh yes. Those wonderful numbers. Featured on 'Did you know that?' kids shows and preached by people who understand logarithms to the rest of us that don't.
Those who believe in the mystical relationship of mathematics to nature point to the nautilus's shell and the way sunflower seeds are arranged in the flower or even to the spiral pattern of hurricanes (and Galaxies) as proof that Creation takes Fibonacci's numbers into account.
In this edition of the Desk's Mystery Series we'll look at the numbers themselves and how they are arrived at. Then we'll examine a few of the claims and perhaps a look at something quite unusual.... the truth of the matter. Along with the usual assorted tangents and side trips and even a patron saint or two. But first:
Some background to the mathematical phenomenon:
The system of numbers is fairly simple. Take 0 and add one to it, then take the resultant answer, one, and add the number immediately before for the sum of two. Then add the first answer, one, to your second answer, two, and you get three. Add the three and the two for five, and continue until your calculator has a seizure. Usually somewhere around the high thirties in the addition sequence when you add 24157817 and 39088169 to get 43245986.
In the academic world of Greek letters and funny symbols the equation that explains the sequence is fairly straightforward: Fn+1 = Fn + Fn-1 (if n is one or greater). From there you can get as complex as you want with various recursions and conclusions and working it out to the Nth degree and get into the "anti-Fibonacci" numbers.
The ratio between one number in the sequence and the next is explained as the "Golden Ratio" although that name doesn't fit neatly into equations. The Greek Letter: φ (phi) does however, and its similarity in pronunciation to the more famous and more useful π (pi, 3.14159...) is probably not an accident of marketing. More on the Ratio later.
The numbers from the sequence most of us have seen in circulation go 0, 1, 1, 2, 3, 5, 8, 13, 21...
If you like REALLY big numbers, ATT's research people have the first five hundred Fibonacci numbers all figured out for you here: www.research.att.com/~njas/sequences/b000045.txt
Just for giggles, the one hundred ninety ninth number is the lineup is:
The Golden Numbers:
From the main sequence you can move on to 'Golden Ratio' equations and the 'Golden Rectangle' in architecture as well as the Spiral mentioned earlier.
The Ratio (φ) is the relationship between any given number (beyond 2) to the number immediately to either side of it in the sequence. Looking through the table on the ATT site you could pick one, like the 199th and do that math and see that the 200th number is 1.618039 (and change) larger than its predecessor.
However, the ratio isn't an exact number. Like π, φ is an Irrational Non-Repeating (and maybe a Transcendental) decimal number that may well go on forever. If you like mind-numbing numbers, go visit the page where they have it worked out to two thousand decimal places at: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html#phi2000 with a link to a page with the ratio to 1.5 billion places. (see hotlinked entry below article)
The Rectangle. Start with a one unit square. Butt another one unit square up against it. Add one and one which gives you a two unit square. Now increase the two unit square by one on each side and join the resulting three unit square up against the ones you already have. Repeat making the one with three units per side five per side according to the sequence, then make one that is eight per side. Put them all into a rectangle together and you have the Golden Rectangle.
Now, exactly why the 'Golden Rectangle' stops with eight units is anybody's guess since the sequence goes on into infinity, but you get the idea. According to those that wax poetic about squares the Rectangle is the 'Most Beautiful' or 'most architecturally pleasing' or some other collection of adjectives. In reality, if you do a photo lineup of unmeasured plain rectangles ranging from square through absurdly elongated most people will tend toward the square or one that is a 1x2 ratio than the 8x13 'Golden Rectangle' if they prefer any at all above the statistically random.
The Propagating Spiral is supposed to fit neatly inside the Rectangle, crossing its primary lines at certain intersections between the units. And, well, it does, mostly. Therefore anything that can be made to approximate the spiral is said to answer to the Ratio and therefore it has to be party to the mathematical phenomenon involving the Sequence and its Golden this and that and Nature favors the Ratio and ......
We'll come back to all this.... moving on.
Right here and right now, we will state clearly and for the record: The Fibonacci Sequence is only one of an entire range of numerical and mathematical sequences ranging from Natural and Prime Numbers, to Integers and Exponents and the Joyce Sequence which requires about two bottles of aspirins and a prayer to Saint Hubert of Liege (one of the two patrons of Mathematicians) just to read the proof. Just for fun go to a search engine and look up the Perrin or Padovan Sequences. Isn't it odd that nobody is all enthralled with them. (This is an injustice that must be corrected!).
All of these are realistic and demonstrable mathematical phenomena and all can be taken to whatever logical extreme you wish (like calculating pi which at last check was out to a Trillion digits, for a couple of billion places see- http://www.angio.net/pi/piquery#find (hotlink below), note the warning on the page for the download time and space required for a 50 million digit π download.)
There are also other mathematical phenomena such as the Polyhedra also called the Platonic solids (the Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron), and the infamous Mobius strip that fall into Geometry to keep the Rectangle and the Spiral company.
OK, that's enough numbers.
How Fibonacci and friends relate to the real world is something we'll chew on when we get to it.
Now about WHO discovered the sequence named for Fibonacci:
Well. Never mind the wishful thinking by academics who like to attach names of real people with dates of birth and death to really neat theories.
The sequence was known in antiquity in India and is documented in some works as far back as 400 BC or so. There is also some evidence that Chinese scholars at least had discovered the relational sequence, whether or not they pursued it further would be the subject of further research not within the scope of this paper.
From India the knowledge of the sequence moved into the Middle East and most likely from there to Italy where the name Fibonacci became attached to it after the death of the man that popularized 'modern' math in Europe.
The Italian Mathematician Leonardo of Pisa (died in 1250), later called Leonardo Fibonacci or just Fibonacci, was the primary force for modernizing the way Europe did arithmetic using Arabic numbers (1, 6, 14) instead of Roman Numerals (I, VI, XIV). With the basic arithmetic in hand Europe was free to move into 'higher arithmetic' such as Geometry, Calculus, Nuclear Physics and Quantum Mechanics with all that entailed.
In short, when you add your golf score, calculate the expansion of the universe or cash your paycheck written out in hundreds, tens, ones then the decimal places....
"782.12 minus the 20.00 overdraft fee, equals 762.12"
....you have ol' Leo to thank for it!
Today's modern computers, even though they are binary (ones and zeros where the number 5 is written 101) still use the basic principals codified by Fibonacci from Arab (and Hindu before them, as he admits in his book) traditions for addition and subtraction.
Fibonacci's flagship work Liber Abaci, "the Book of Calculation" published in 1202 is still one of the mainstays of modern mathematics. It is available for purchase online and is web published on several university math department websites if you go searching for it.
The Claims in Nature
Some of the claims made by various ones are serious, and upon first brush the observer will go 'hhhmmm' and think about it. It looks plausible that the natural object, say the sunflower we mentioned earlier, is reflecting nature's natural selection of the Fibonacci sequence as the solution to the problem.
The examples from nature continue with the famous Nautilus shell and the number of petals on various flowers (buttercup and black-eyed Susan) or the way leaves are arranged on certain tree branches.
Yes, it's true.
But only certain trees and flowers even come close, others have four (like mustard) or six (like the lilly- which true believes say have two sets of three) petals which are NOT Fibonacci numbers, and they seem to be doing quite well. As are the ones with nearly random numbers of petals or leaves depending on the individual plant, like the dandelion and some roses. And only the Nautilus is held up for adoration as the Mollusk With The Number (even if the example most often used has been doctored).
Other crustaceans seem to only answer to the sequence by accident. There are entire families of trees and flowers that absolutely ignore the equation and grow at random or reflect some other natural sequence.
When the adherents pull up pictures of one or another Hurricane and show you the spiral of clouds with the Fibonacci Propagating Spiral superimposed on the clouds you can easily go to the weather web site and pull up fifteen tropical storms that don't obey the rule some being spiraled tighter, such as 2004's Hurricane Charley which came ashore on the Florida Gulf Coast with an eye only five miles across and winds of 120 mph. And some not wrapped as tightly such as Ivan who later that same year had an eye about fifty miles wide.
Perchance maybe you haven't heard all the claims from the plausible to downright kooky from those with almost a religious faith in the sequence and its cousin, the Golden Ratio.
Maybe we should run through a few of them before we continue.
[NOTE: All debunking will wait until after all examples have been run out and saluted. ]
Examples posted without editing. Some formatting changes made for readability. Links will open in new window.
On the page:
The Prehistoric Alignment of World Wonders
"The relationship between the distances from Angkor Vihear to the Great Pyramid and from the Great Pyramid to the Nazcan Hummingbird is also a precise expression of φ (phi)"
From the page under the heading:
Henry "Wild Goose" Niese, the teacher from Eagle Voice Center
"The ratio between each succeeding pair of numbers (1:1.618+) describes the ratio of the "Golden Section", a famous classical proportion, as well as the growth rate of many organisms, as seen in the seedhead of the sunflower, the end view of a pine cone, growth rings of clam shells, etc. The Fibonacci series of numbers is found over and over in plant growth. Looking at the seedhead of a sunflower, one will see that the seeds are arranged in 55 clockwise spirals, and 89 or 34 counterclockwise spirals, numbers adjacent to each other in the series. Likewise, in a daisy, pine cone and other natural forms, their structure is based on a combination of numbers from the Fibonacci series."
The Golden Ratio and Architecture
"Ancient Egyptians, Mesopotamians and Greeks were aware of the beauty of the Golden Rectangle and used it to create many different buildings. One of the most famous and beautiful buildings, built in ancient Greece on the Acropolis, is called the Parthenon. It was created about 2500 years ago....
Golden Ratio appears in several places in the design of the Parthenon. The dimensions of the front of the temple form a perfect golden rectangle, as shown on the diagram above. Also, the spaces between the columns are in proportion to the golden ratio. The golden rectangles can be seen in almost all exterior dimensions of this great masterpiece."
We're all quite breathless now, but two more will suffice:
A quote from the page selling the textbook "A Mathematical Mystery Tour":
"This is no set of scattered "bytes" of math. Rather it is a unified, unfolding journey with ideas linked together by the well-known Fibonacci Numbers and the celebrated Golden Ratio. (An enclosed mind-map of the whole book is a spiral!) After some math effort a student can begin to see how bees are linked to the Great Pyramid in Egypt, how pine cones are linked to our DNA, and how a Greek vase is linked to the earth-moon relationship. (Now in the 2001 supplement there are connections to fractals, the stock market and the musical scale). These are not just poetic math metaphors, they are real, undeniable, numeric trails that leave the learner with goosebumps."
And the Sacred Geometry Home Page
"Leonardo da Vinci used the Golden Ratio in his painting of The Last Supper in both the overall composition (three vertical Golden Rectangles, and a decagon (which contains the golden ratio) for alignment of the central figure of Jesus."
"Sacred Geometry" ? (debunking time)
We can all be fairly certain that the Ultimate Renaissance Man, da Vinci, not only knew about the Rectangle (and the ratio, quotient, solution and the rest of it) that he quite probably used it in his initial outlining of the picture under discussion so it would all fit on the canvas if for no other reason. (We will ignore the pervasive Hoax of the Day- 'the daVinci Code' and its adherents as that has been debunked to death elsewhere.)
The rest in no particular order.
Architecturally speaking, the Parthenon in Athens (or Nashville), which is the Golden Example of the Golden Rectangle only answers to the rule in one specific way on the small end. You can also work it so the space between the columns answers to it, but that seems to be a stretch. If you take the building from the side, or fail to include the pedestal and the roof (which is a triangle sitting on another sort of rectangle), it does not fit the golden equation.
If you consider Continental Drift, the distance between the Great Pyramid and the Nazca Lines in Peru has changed enough since the Pyramid was built that even if both had been constructed with the φ in mind, it wouldn't be accurate now. And if you are measuring it with GPS now and doing the math, you are delusional.
Now about those sunflowers and the bees. To start with, which type of sunflower are you talking about? There are several types of the flowers, and not all of them have seeds that line up perfectly to the prescribed pattern. And even individual plants along a row of flowers, if the seed head doesn't match the spiral the plant doesn't wither away and die.
With bees, the adherents to the natural wonder of the Golden Ratio and related computations like to quote about the ratio of drones to workers to the queen and so on inside the beehive. In reality. If you went through a hundred beehives you might find one or two that were close (within a bee or two of the magic number), as you would expect with random chance. The numbers of bees at any one time who call any particular hive home has more to do with the weather and the species of bee and availability of food than any mathematical number crunching on the part of the bees or the great and ancient powerful pagan god of beehives and number sequences. (With apology to St Barbara, the other patron of mathematicians. Note: Beekeepers have THREE patrons; Ambrose of Milan, Bernard of Clairvaux and Valentine. Mathematicians two.)
At least this guy admits it on his page titled: Pascal's triangle, the shallow diagonals and the sequence
"...nature isn't trying to use the Fibonacci numbers: they are appearing as a by-product of a deeper physical process. That is why the spirals are imperfect. The plant is responding to physical constraints, not to a mathematical rule."
Nature is using the best option available to that plant or snail or whatever within the general location and allotted time. Imagine that, Nature can do math without ever having sharpened a pencil. It is also interesting to note that Nature has also solved the same problem in other ways in other species without a Ratio in sight!
But those of the school of Mystical Mathematics see the Ratio as something from Above or Beyond or whatever, and ride off into the sunset on their Golden Spiral Nautilus.
The first question that has to be asked is what evidence is there that Nature uses Arabic Numbers in our traditional format? Are we sure that nature doesn't do its computations in Binary or Hexadecimal or Base 8 or something we don't even have a name for that would completely change the sequence, and thusly the Ratio, once you're past the first few numbers in the sequence.
[Sidenote: In Binary (ones and zeros) the Golden Ratio- φ, becomes the Golden String in an unbelievably long series of numbers with some interesting attributes of its own even though it is all ones and zeros that actually stretches to Infinity! IE: 1011010110110101101011011010..... ]
Let's state it clearly and simply. Yes, the Ratio, and the Sequence, and the Spiral, and the String, and the Rectangle, and the Proportions and all the rest are Interesting.
Yes they are- Interesting.
The Human Face does NOT routinely conform to the proportions suggested by the Ratio no matter what da Vinci said in his various works and famously outlined on the 'sketch of an old man'. Again, go through the line at a fast food restaurant and check out the profiles of the people against the squares of the Rectangle. If you don't get arrested, you'll discover that those that answer to it without bending the grid or the person's face are a rarity.
Some naturally occurring plants and animals do seem to at least be aiming that way as a natural solution to a natural problem. However, there are at least equal numbers that do not, having found another solution to the same problem, such as the placement of seeds in a seed-head or the growth of a shell or the way rain bands circulate around the eye of a tropical cyclone (as we observed earlier with Ivan and Charley). Some appear to answer to another mathematical equation, others may indeed be random.
And while the mathematical extrapolation of the Ratio and its relatives could lead one to transience or some other religious enlightenment, the same can be said with many other numerical expressions including the most simple computation- 'add one', which will lead to infinity as well.
Let's wrap this one up.
The Fibonacci Sequence and its derivatives is a fascinating bit of mathematics.
The Rectangle and Spiral and Ratio and Quadrant and so on are more or less useful computational aids and geometric tools.
They are NOT a religious device.
The only real mystery here is why some people would choose to assign to them mystical properties or insist on seeing them expounded where another option is equally applicable.
It comes to this, humans like to see patterns even where none exist. Such as seeing faces in clouds, 'pareidolia' if you want to attach a fancy word to it. It is the way we're put together. And for the most part, it is OK, there's nothing inherently wrong with that.
We also like to assign meaning to those patterns. For instance, if there IS a face on Mars, it is there for a reason. Somebody put it there. And so on.
You can see where the problems begin. Perhaps it is coincidence and chance and odd happenstance, perhaps it isn't. But who is to decide?
Whether or not there is meaning behind the pattern is part of the discussion of Metaphysics being carried on elsewhere on the Desk.
The pattern is there. We as a species have discovered it. We'll leave it at that.
There you have it. Fibonacci and his numbers and all they have brought us.
-30- (or in this case... 1, 1, 3, 5, 8, 13, 21, 34, 55...)
A basic introductory animation into all this. textism.com/bucket/fib.html
Something of a support group for mathematicians: www.mscs.dal.ca/Fibonacci
The 'golden section' and other fun with Fibonacci www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci
Phi is everywhere at goldennumber.net
Phi 2000 as mentioned in the article.
www.World-Mysteries.com looks at the numbers.
On-Line Encyclopedia of Integer Sequences index of Fibonacci links, with some out to... how many do you want? www.research.att.com/~njas/sequences
Part of the site- Myths and Mysteries of Science http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
The Platonic Solids Math Academy Online / Platonic Realms ™ www.mathacademy.com
www.angio.net Pi out to ... yeah.
A page for the Perrin Numbers in the Planetmath.org/encyclopedia
This one compares Padovan's Spiral to Fibonacci's
To coin a term "Fascinating"
A page detailing the Nazcan lines http://www.unmuseum.org/nazca.htm with a picture of the hummingbird as mentioned in the quote.
Two links explaining the 'face in the clouds' - pareidol
skepdic.com The Skeptic's Dictionary
Lenin on a Shower Curtain at www.badastronomy.com
MEDIA DESK LINKS
Other articles in the Mystery Series include the Coral Castle, The Voynich Manuscript, King Arthur and the Philadelphia Experiment.
Why the Desk didn't even TRY to work out the Ratio....
To the main page http://themediadesk.com
[NOTE: The Desk is not affiliated with any of the above outside sites or entities. Mention of a particular document, equation or theory is not to be taken as an endorsement by or of the Desk. No infringement or disrespect is intended or is to be assumed. thank you ]