The Marco Polo of Mathematics
While surfing through the Netflix movie guide the
other night, I hesitated on The Da Vinci Code and decided to watch this
controversial 2006 mystery-thriller once more.
Like National Treasure and Raiders of the Lost Ark, it’s
just the type of movie I love, full of nail-biting, scavenger-hunt
adventures. The more twists,
head-scratching clues, and puzzling codes the better. It had been a while since I’d last seen it —
boy, Tom Hanks looked a lot younger 15 years ago, but then so had I. Not long into the movie, we are in the Grand
Gallery of the Louvre (scene above). The
curator lies dead on the floor and Hanks, Professor Langdon, is being
interrogated by hot-headed Police Captain Bezu Fache. I just love the name “Bezu Fache.” Conjuring up a character name like that alone
could account for the large payday of $6 million author Dan Brown pocketed for
the film rights. All that aside, Bezu
then shows the professor our first clue.
It is a series of numbers followed by two phrases:
13-3-2-21-1-1-8-5
O, Draconian Devil!
Oh, Lame Saint!
Turns out “O, Draconian Devil” is an anagram for “Leonardo Da Vinci,”
another clue, while “Oh, Lame Saint” unfolds to “The Mona Lisa.” Fortunately, “The Mona Lisa” by Da Vinci is
just down the hall. Thus, with deathly
intrigue, two deciphered anagrams related to Leonardo Da Vinci, kick-off this
action filled adventure, The Da Vinci Code, and account for its
title. What intrigued me most, however,
was the series of numbers that we quickly learn are the first eight numbers of
the Fibonacci number series, although somewhat scrambled from their proper low
to high order for use later in the movie.
Today in mathematics, Fibonacci numbers form a
number line called the Fibonacci Sequence.
To form the sequence, each follow-on number in the string is the sum of
the two preceding numbers. You get the
idea. It’s really simple. The top of the order of the string is the
sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on forever. As in the movie, there is mystery here. The Fibonacci Sequence is significant because
of its off chute derivative, 1.618, and its inverse 0.618 (1/1.618), known as
the Golden Ratio.
This ratio becomes evident when you divide a higher number in the sequence by the preceding number (Ex: 2 ÷ 1 = 2.0). At first, the ratio, beginning at 2.0, wobbles around and is not constant. As
 |
| As the Terms of the
Number Sequence get Larger (34, 55, 89) there is Conver- gence to the Stead State Value 1.618 |
This Golden Ratio was known to ancient Greeks like
Euclid. The Fibonacci Sequence however
was apparently new, although it is uncertain whether its creator ever realized
its connection to the Golden Ratio. This
may be because the first 30 or so ratio values of successive numbers in the
sequence seem to have no consistent pattern to help identify the constant
multiplier pattern (1.618), but here I’m only guessing.
So big deal you may say,
Golden ratios, strings of numbers to infinity, even pleasing Golden
Rectangles … just amusing number parlor tricks.
Why should I care? More than a
playful waste of time, it is possible to find the ubiquity of the Fibonacci Sequence
and associated ratio throughout nature on into inanimate objects and
surprisingly even into a stock market buy/sell strategy called Fibonacci
Retracement.1 Some
golden numberists claim that nature's processes are "governed" by the
golden ratio. This magical sequence for
instance is said to underlie the number of petals of flowers. Lilies and irises have three, buttercups,
parnassia, and rose hips have five, some daisies have 8 or 13, chicory 21. The sequence can also be seen in the way tree
branches form or split. It is said to be
found in the logarithmic spiral of some seashells like a nautilus shell, and
even the cochlea of our inner ears. Not
surprisingly, spiral galaxies like our Milky Way are said to follow the
familiar Fibonacci pattern. Claims are
also made, some attributed to Leonardo da Vinci, that facial and body
proportions of humans also comply with this ratio.2
I won’t go into how it is believed to permeate throughout
the Great Pyramid of Giza, built around 2,560 BC, only to mention that today’s
ratio of its height to base is 0.636.
This is close to the Golden Ratio, varying by only 0.025%. This difference may be accounted for by
today’s measurement of the pyramid’s height and base absent its once smooth
outer covering. Yet there is no evidence
the Egyptians ever knew of the Golden ratio.
The mystery continues to this day, yet I wonder if we are looking for
associations like this a little too hard.
How had the Fibonacci Sequence and Golden Ratio, believed by some to be
a numbering convention of the universe, how had it been embedded throughout
nature’s biological systems, even into the creation of inanimate objects by man
like the pyramids? Has it been there as
a fundamental pattern all this time? If
so, that’s an incredible role for a single ratio to play.
Other than Fibonacci, who thinks of this stuff? Certainly not me. My guess, a fringe wants us to believe the
world is all about math. Here is a
YouTube clip from the film noir movie “Pi” where a math
type who meets a Jewish Kabbalist searching for meaning in Hebrew text would
have you think so. Emergent patterns
from the sequence can get really nerdy from here:
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 in the sequence equals 11 times the seventh one of the
ten.
Is it some hidden way for us to understand our world? They just may be right as we daily learn to
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| Pisa’s Piazza dei Miracoli |
For a guy like me, who can’t get through a 4-Star Sudoku puzzle, I wondered who conjured up this stuff. Was there actually a Fibonacci? Yes and no, for his actual name was Leonardo Pisa Bigollo or Leonardo of Pisa and as you can guess, he was Italian. Years ago, we visited Pisa on the vast plain of the Orno River. I recall parking our car just outside the walls of Piazza dei Miracoli (Square of Miracles) and walking through the gate onto its grassy field. Back then, we used cameras and just the right positioning to capture ourselves, arms extended, appearing to be holding up the leaning tower. It’s what we call a “selfie” today, but we needed help. To the left rose those iconic buildings: the Romanesque, striped-marble Cathedral of the Assumption of the Virgin Mary, its
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| Statue of Fibonacci
in Campo Santo |
Baptistry, the tilted 184 ft (56m) white-marble cylindrical bell tower, and the lesser-known Capo Santo Cemetery. Campo Santo literally translates to "Holy Field” because it is said to have been built around a shipload of sacred soil from Golgotha, in Jerusalem, brought back to Pisa from the Third Crusade in the 12th century. This is where we came upon a greater-than-life-size statue of Fibonacci over the spot where his body rests, off in one corner of this vast rectangular cloister-like building. While there we learned that in July of 1944 during a battle, a stray Allied shell started a fire in the Campo Santo. The fire burned for three days, causing its lead-covered timber roof to collapse. The destruction of the roof and the resultant heat caused severe damage, especially to the frescoes.3 Repairs, following the war, attempted to return it to its former majesty. As for the tower, its nearly four-degree slant (I always thought it more) is irreparable which of course plays to its attraction.
Leonardo’s father, Guglielmo Bigollo, was a wealthy
Italian merchant, trading post manager, and customs official in Béjaia, in what
we call Algeria today. Appropriately
named, his surname meant "traveler.”
As he traipsed throughout the Mediterranean and Near East, he took young
Leonardo along with him to places like Syria, Egypt, Greece, Sicily, North
Africa, and Provence. All the while he
encouraged Leonardo to learn mathematics which Leonardo absorbed from local
tutors. Leonardo described it this way:
"Having been
introduced there to this art with an amazing method of teaching by means of the
nine figures of the Indians, I loved the knowledge of such an art to such an
extent above all other arts and so much did I devote myself to it with my
intellect, that I learned with very earnest application and through the
technique of contradiction anything to be studied concerning it and its various
methods used in Egypt, in Syria, in Greece, in Sicily, and in Provence, places
I have later visited for the purpose of commerce." 4
On his visits to Africa, Leonardo was exposed to a new
numbering system. Instead of Roman
numerals, the people there used a system of numbers that they had learned from
the Hindu people farther east in India.
This alien mathematics proved far simpler to manipulate. In this system, XVIII became the much easier
18. In each area he visited, he learned
new mathematical concepts based on this Indian numerical approach. In Egypt, he learned about fractions. In Turkey and Syria, he discovered methods of
measurement. In Greece, he learned
geometry, and in Sicily he used division and subtraction. In this, his equivalent to a “Grand Tour,”
this Marco Polo of Mathematics amassed the mathematical genius of the
Hindu-Arabic number system based on the system of numbers zero through nine
(0-9).
It was evident to Leonardo that arithmetic
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| With a Slight Error
in This Addition Example (a C was Dropped) It’s Still Difficult to Imagine Operations Using Roman Numerals |
 |
| A Page from Liber
Abaci with Handwritten Fibonacci Sequence Scribbled in Right Margin |
While best known for the Fibonacci Sequence that bears his
name, his other most notable contribution to mathematics was his historic work,
Liber Abaci, (The Book of Calculation).
It is historic in that in this work he introduced Indian numerals (the Modus
Indorum) to the West. In it he
included riddles to demonstrate the utility of the eastern numbering
approach. It may not have been on par
with the works by another Leonardo named da Vinci or Isaac Newton’s Philosophiae
Naturalis Principia Mathematica, which came much later, but it proved to be
the brick and mortar foundation that underpinned their mathematical
expressions. As powerful as its impact
eventually became, it was a hard sell following its publication, when Leonardo
was 32, in 1202. Granted, Gutenberg had
yet to invent his printing press. That
laid over 200 years in the future, so understandably dissemination of its
message was slow. Comfortable with the
Roman system, people of his day also resisted the change. At first, the voice of his essay was small
but like yeast, it gradually grew to have a profound influence on Europe’s
adoption of the Hindu-Arabic numbering system along with its associated
operations.
We know Leonardo Pisano Bigollo today by his posthumous nickname, Fibonacci. The name was made up in 1838 by a Franco-Italian historian. This moniker
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| Leonardo Pisano Bigollo, Nicknamed Fibonacci |
came about because he was “The Son of Bonacci.” which in Latin translates to “Filius Bonacci” (Fi-Bonacci). However, he just may have been referenced even earlier. If indeed it was the same Leonardo, in 1506 a notary of the Holy Roman Empire, Perizolo, mentions a Leonardo as "Lionardo Fibonacci."7 Born in 1170 AD, this Pisian, properly called a “Pisano,” went on to discover the unusual properties of the numerical series. How it was derived may be a tall tale, certainly a “cotton tail,” for it is said that Fibonacci came up with the sequence when calculating the ideal expansion of pairs of rabbits. A puzzle Fibonacci posed in the Liber Abaci, the infamous rabbit problem, forms the basis for the sequence. It involved a hypothetical question concerning the number of pairs of rabbits there would be at the end of twelve months when restricted in a garden. He began with one pair which did not reproduce in the first month but produced one pair of offspring in each month thereafter, with all the pairs following the
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| Example Growth in
Rabbits per Fibonacci Sequence |
same reproductive pattern. Conveniently, rabbits can mate at the age of one month so that at the end of its second month, a female can produce another pair of rabbits. He assumed the rabbits never died and that the female always produced one new pair (one male, one female) every month from the second month on. A definite pattern emerged. “At the end of month two, there would be 1 grown-up pair of rabbits and 1 baby pair; at the end of month three there would be 2 grown-up pairs and 1 baby pair; at the end of month 4 there would be 3 grown-up pairs and 2 baby pairs.” Leonardo realized that by adding “any two consecutive numbers in the pattern,” you’d get the next correct number. He noted that after each monthly generation, the number of pairs of rabbits increased from 1 to 2 to 3 to 5 to 8 to 13, etc., and identified how the sequence progressed by adding the previous two terms resulting in a sequence which could, in theory, extend indefinitely. Reproduction like this may have happened where we live. While riding my Vespa recently, I just missed one cotton tail scooting across my path which would have certainly messed up Fibonacci’s sequence. While I occasionally dodge a rabbit, I leave it to my Calitri friends who dodge cinghiale (wild boar) on their roads.
Time has taken advantage of his achievements. His sequence, which by the way has never
directly led to the discovery of a fundamental law of nature, has become a
cottage industry. As I hinted at
earlier, some try too hard to force-fit its properties into all kinds of
measurements. Over the centuries his
number sequence and related Golden Ratio/Golden Rectangle are seen to
miraculously appear just about everywhere.
It is even found in our
calendar. It isn’t a prominent day like
some holiday or Saint’s Day, but November 23rd is celebrated as Fibonacci
Day. Why? The explanation lies in the numbers of
course. Someone noticed that when the
date is written in the mm/dd format (11/23), the digits in the date form a
Fibonacci sequence: 1,1,2,3. Of course,
the first numbers of the sequence! I
just came in from cutting the lawn and I swear I was seeing perfect Golden
Rectangles in the patterns I made all over the lawn as I trod along and Golden
Spirals approaching the logarithmic spiral of a nautilus shell in every turn. Would it be madness to take all this as
fact? Much of it lies in the eye of the
beholder, however, and I fear often backed-in to fit.
Sometimes nature does follow Fibonacci’s golden
idea. Nature is full of patterns and
regularity. But whether it is some
mystical governing principle underlying a deep mathematical basis of the
natural world, well, while there is some evidence, the jury is still out on
that. Yes, the number of petals on a
flower or arrangement of leaves on a sprouting stalk do keep to the sequence as
I’ve mentioned. But maybe to avoid
obscuration, their numbers (3, 5, 8, 13, 21 ...) might be arranged to optimize
exposure to the sun. Mother Nature
herself may be the greatest mathematician.
Had Fibonacci stumbled upon nature’s coded connection
to mathematics? Had he simply opened
the door to an advanced Indian number system?
Had he facilitated both? While
Fibonacci chose reproductive rhythm (recall the rabbits?), it is said that a
2nd century BCE work by Sanskrit grammarian Pingala (circa 450 to circa 200 BCE)
in his Chandahshastra (The Art of Prosody) developed a methodology based
on rhythms and the arrangement of tones most pleasing to the ear. His patterning concerned beats in music and
the rhythmic structure of Sanskrit poetry follow the initial numbers in the
Fibonacci Sequence.8
More recent evidence shows that Hemachandra, an Indian mathematician,
compiled a treatise concerning cadences of various lengths in 1150 AD, while
Leonardo Fibonacci presented a similar thesis (The Book of Calculation) to that
of Hemachandra’s commentary 52 years later in 1202 AD. For this reason, mathematicians and the
scientific community are beginning to highlight this Indian contribution. When mentioning Fibonacci numbers today, it
is referred to as “Hemachandra-Fibonacci Numbers.” With discretion, I’ll forgo this controversy
mired in the clouded memory of centuries.
I’d rather stick with Fibonacci having popularized the use of
Hindu-Arabic numerals in the West, replacing that unwieldy Roman number
alphabet. Come to think about it, it was
Leonardo who made it possible for me to maneuver, using this imported alien set
of numbers, … here a 7, there a 2, no, no, it’s a 4, everywhere row by column a
1-9 … throughout my Sudoku puzzles.
While Fibonacci played a cameo role in the opening
scenes of The Da Vinci Code, his work is largely responsible for the
dominance of the decimal system used throughout the world today. Like the bug that just whizzed by my screen
has no inkling of the Hindu ones and zeros whizzing through my computer, I
can’t fathom our world, this computer in fact, operating on Roman
numerals. To this I say, grazie e
bravo Leonardo Fibonacci, who like that fly, had no inkling of how his
actions would affect the world.
From
That Rogue Tourist,
Paolo
1
= https://www.mathnasium.com/examples-of-the-golden-ratio-in-nature
2
= https://www.lockhaven.edu/~dsimanek/pseudo/fibonacc.htm
3 =
https://dlf.uzh.ch/sites/camposanto/damage-destruction-and-restoration/
4 =
https://en.wikipedia.org/wiki/Fibonacci#cite_note-20
5 =
https://www.bnd.com/living/liv-columns-blogs/answer-man/article207337709.html
6 = https://en.wikipedia.org/wiki/Roman_numerals
7.=.https://en.wikipedia.org/wiki/Fibonacci#:~:text=The%20name%20he%20is%20commonly,Leonardo%20as%20%22Lionardo%20Fibonacci%22.
8 = https://www.lockhaven.edu/~dsimanek/pseudo/fibonacc.htm
