Tag Archives: fibonacci

“I’ve been thinking about numbers.” Stuart poked his fork into a pile of pasta in front of him and twirled. 

“Mostly, we think of numbers in terms of mathematics,” he said. “Or arithmetic. All very abstract. But I’m looking at them in terms of the humanities.”

Genevieve had spent the afternoon cooking up a Pasta all’Assassina, something she had just learned from YouTube. It piled up on our plates in small pyramids of spaghetti. 

“You know how there are these sequences of numbers in math? Like 2, 4, 8, 16, 32, 64, 128, 256 and so on. Or the Fibonacci Series: 1, 2, 3, 5, 8, 13 — you know. Logical sequences that seem to have some meaning outside mere math.”

“You mean like the spiral in a seashell or the arrangement of seeds in a sunflower…”

“Exactly. Well, these patterns, as patterns, are purely mathematical, in other words, they only exist in the Platonic ideal of mathematical thinking. I was looking for a sequence that made sense without arithmetic, that English majors could grasp at a gander.”

“And did you find one?” I asked.

“Absolutely. I considered the mythic or symbolic punch of numbers and came up with a sequence something like: 1, 2, 3, 4, 7, 9, 12, 21, 40, 42 — that’s kind of a ringer in this — 101, 1,001, one million, and finally, ‘billions and billions.’ That last, thanks to Carl Sagan.”

“If I understand you, these numbers carry significant weight in folktales, mythology, religion and popular culture. But don’t all numbers carry some kind of baggage? I mean, you left out eight from your list, but there is the Noble Eightfold Path of Buddhism.”

“You are right, most numbers have something, but my list considers only the big boys in the number-myth world. The ones that carry the heaviest weight. 

“One, of course, is the unity. It is the prime singularity out of which all else evolves or explodes, like in the Big Bang, or the One-True-God. Two is the duality of yin and yang, of pairs of opposites, of Yoruba twins, of Castor and Pollux, of the Navajo twins Monster-Slayer and Born-of-Water. Or of the salt and pepper shaker or even the right and left hands. Two is big in the Sequentia Stuartii. Yes, that’s what I’m calling it.” 

“Three is a quantum jump, though, I think,” I said. “Three is everywhere, from the Three Little Pigs to the Holy Trinity. Goldilocks and the Three Bears, in Greek mythology, the three Furies, Graces, and in Norse, the three Norns. In joke telling, there is the rule of threes, and in photography, we hear of the rule of thirds. Omni Gallia en tres partes divisi est.

“Yes, three is big,” said Stuart. “Four is a little smaller in the mix, but still, there are the four seasons and the four directions, the four Classic humors and elements, the Four Noble Truths of Buddhism…”

“The Four Horsemen of the Apocalypse. Let’s not forget my favorite. But I notice five and six don’t make your list.”

“Again, there are some references, but they thin out with five and six, and thus fly under my significance bar. Five has the pentagram, but most other associations are a bit more arcane, and therefore don’t have the currency of the big-number power. Six has what? Six sides to a die. Or if you want to go really esoteric, the Zoroastrian god Ahura Mazda has Six Immortal Holy Ones to attend him. I had to look that one up.

“Ah, but seven. Seven is king. It is the big kahuna of numbers. If I make a graph of number significance, seven is off the charts. The home run of numbers. Seven days in a week; seven seas; seven continents; seven hills of Rome; Seven Wonders of the Ancient World.”

“Seven Deadly Sins, again, let’s not forget my favorites.”

“And the Seven Cardinal Virtues,” said Stuart. “Seven planets in the Ptolemaic universe, seven notes in the diatonic scale. The seven liberal arts. Break a mirror and get seven years of bad luck.”

“And the number is so persuasive, someone decided there were seven colors in the rainbow. What is ‘indigo,’ after all, but just another blue. They added it so they could have seven colors.”

“God created the world in seven days. Six plus a day of rest. The Bible is full of sevens. Seven years of fat and seven of lean in the Pharaoh’s dream. Seven days of Passover. Seven year Jubilee cycle. Jericho was conquered on the seventh day after seven priests with seven trumpets marched around the city seven times. King David had seven elder brothers. After Elisha raised the child from the dead, the kid sneezed seven times. There are seven pillars in the House of Wisdom. 

“In the New Testament, seven demons are cast out of Mary Magdalene, seven loaves to feed the multitude in Mark and Matthew, the seven last words of Christ on the Cross. And Revelations is filthy with sevens. Seven golden lampstands, seven stars, seven torches of fire, seven seals, seven angels and their trumpets, seven last plagues, seven golden bowls, seven thunders, horns and eyes, diadems and kings.”

Genevieve had been mostly silent through all this, letting the boys blow their steam, but she joined in. “I was raised Catholic,” she said. “And I was brought up with seven sacraments, the Seven Joys of the Virgin and the Seven Sorrows of the Virgin, the Seven Gifts of the Holy Spirit. There were Seven Corporal and Seven Spiritual Acts of Mercy.”

“Sevens out the wazoo,” Stuart said. “And it isn’t just Christianity. There are seven chakras in Hinduism. And seven great saints, seven worlds in the universe and seven gurus. Agni, the fire god, has seven wives, seven mothers and seven sisters and can produce seven flames. The sun god has seven horses to pull his chariot. In the Rig Veda, there are seven parts of the world, seven seasons and seven heavenly fortresses. I could go on.”

And he does. 

“In Islam, there are seven heavens and seven hells. You make seven trips around the Kaaba. A baby is named on the seventh day of life. The seven sins of polytheism. 

“In the Baha’i faith the text is The Seven Valleys the soul traverses. It is the number of islands in Atlantis, the Seven Cities of Gold that the conquistadors searched for.”

“Gandhi had his list of the Seven Blunders of the World that cause violence,” said Genevieve. “Wealth without work, pleasure without conscience, knowledge without character, commerce without morality, science without humanity, religion without sacrifice, and — “

“I know this one,” said Stuart. “It’s in the news daily: politics without principle.” (And “men without shutting up,” thought Genevieve, but who was too polite to say it out loud.)


 “In China, there are the Seven Sages of the Bamboo Grove,” Stuart went on. “There are the seven lucky gods in Japanese mythology, and the seven-branched sword. The Buddha supposedly took seven steps at his birth. And believe, me, we’ve only scratched the surface of seven.”

“There are seven holes in the human head,” said Genevieve.

“And my favorite,” I said. I have a lot of favorites. “The seven directions. Some American Indian mythologies recognize seven: the four normal ones, plus up and down, and most importantly, the seventh direction — in. The inner world is one of the directions.”

“Let’s not forget The Seven Samurai and Bergman’s Seventh Seal,” I said. “Or Se7en.”

“Or Lina Wertmuller’s Seven Beauties, or Seven Brides for Seven Brothers. Or the 7 Faces of Dr. Lao.”

“Or The Magnificent Seven,” I said. “Or Seven Years in Tibet or Six Days and Seven Nights. Or The Seven Year Itch.”

It was becoming a contest. 

Return of the Secaucus Seven,” Stuart added. “And Snow White and the Seven Dwarfs. Seven Days in May. House of Seven Gables…”

The Seven Little Foys. The Seven Percent Solution. Seven Up! from Michael Apted’s Up series…”

“Stop it. Stop it now,” Genevieve said. “Boys,” she spit out, as a short summing up of an entire gender. 

“Eight pretty well disappears,” said Stuart, “but nine makes up for it. A stitch in time saves nine. Cloud nine is the ultimate in happiness. A cat has nine lives. Possession is nine points of the law. There are nine muses. The Norse god Odin hanged himself on the World Ash Tree for nine days to gain wisdom. In Christianity, there are nine fruits of the Holy Spirit. The Aztec and Mayan underworlds both had nine levels. Nine justices on the Supreme Court. Nine circles in Dante’s Hell. In J.R.R. Tolkien’s Middle-earth, there are nine rings of power given to nine ring-wraiths. Buddha had nine virtues.”

“There used to be nine planets,” I said. 

“Pregnancy lasts nine months,” said Genevieve.

“Nine players on a baseball team and nine innings in a game,” said Stuart. “And classical composers often had a superstition that their ninth symphony would be their last. Yes, nine is a pretty full number, contrasted with ten — the most bureaucratic of numbers. Yes, there are the Ten Commandments, but the number 10 is the least charismatic of numbers. It is the basis of the decimalization of the world. And you both know how I feel about that. Base 10 — Pfui.”

We both knew the antipathy Stuart harbored for metrification and the inhuman procrustification of the division of things into tens. Everyone familiar with Stuart knew. The next number in his sequence is 12.

“Twelve is a dozen,” he said. “It is 12 signs of the Zodiac, 12 hours of day and 12 of night. There were 12 disciples of Jesus and 12 tribes of Israel. Hercules had 12 labors. There are 12 Olympian Gods, 12 months in a year. Twelve notes in a dodecaphonic tone series. Twelve days of Christmas. Twelve jurors in a panel. Twelve knights of the Round Table. Twelve steps for Alcoholics Anonymous. Paradise Lost is in 12 books.”

“I can think of a bunch of movies with 12 in the titles: 12 Angry Men; Ocean’s  Twelve; 12 Monkeys; The Dirty Dozen; Cheaper by the Dozen; Twelve Years a Slave.”

“And Shakespeare’s Twelfth Night,” said Genevieve.

“Thirteen has plenty of mojo,” said Stuart, “but it is negative. Bad luck. There are at least three numbers with bad magic. Thirteen is one everyone knows. And 666 is the “mark of the beast.” And poor number 17 used to have no juice at all — one of the emptiest numbers, but now has quickly dropped from null to negative with the advent of Q. Nutjobs are going bananas every time they hear a number 17. 

“Then, there is 21, the points of blackjack and the minimum age to enter a casino and play blackjack. In many places, it’s the minimum age for a lot of things. The 21st Amendment ended prohibition, and when it did, you had to be 21 to buy hooch. Which you could do at the 21 Club in New York.”

“There are 21 guns in a 21-gun salute,” I said. “And the TV game show that was rigged, Twenty-One. There have been four movies with that name. Add up the dots on a die and you get 21. It’s the number of shillings in a guinea. Or was. 

“And according to Duncan McDougall in 1907, the human soul weighs 21 grams. I didn’t know they used metric back then.”

“Next is 40,” said Stuart. “It rained for 40 days and 40 nights. Christ spent 40 days in the wilderness. There are 40 Norse Valkyries and ‘Life begins at 40.’ Muhammad was 40 years old when he received his revelation. For Russians, the ghosts of the dead remain at the site of their deaths for 40 days. We listen to ‘Top 40’ radio stations. A short nap is ’40 winks.’ The number of Ali Baba’s thieves. ’40 acres and a mule.’ And the average work week, in hours.”

“I take it that 42 is in your list as a nod to Douglas Adams.”

“Absolutely. It is also Jackie Robinson’s uniform number, the number of lines in a page of the Gutenberg Bible, and the number on the back of the spider who bit Miles Morales, turning him into Spider-Man. It is also the number of the third most famous street in Manhattan (I give primacy to Wall Street and Broadway).

“A hundred is a useful number, but doesn’t carry much mythological weight,” Stuart continued. “But a hundred and one rises in power. It is the college course number of introduction. It is a book title more popular by far than ‘100.’ There are 101 Dalmations and the 101st Airborne Division is ‘the tip of the spear.’ There was a sappy recordings of the 101 Strings. Simon Bond’s 101 Uses for a Dead Cat. The George Orwell’s truly horrifying Room 101.” 

“Doesn’t that mean that 1984 is a number in your sequence?”

“Yes, I guess so. I hadn’t thought of it. My next number is 1,001. From the Thousand and One Nights of Scheherazade.  It is one of the 1001 Books You Must Read Before You Die. Buckminster Fuller called it a Scheherazade number in his book, Synergetics, meaning it is palindromic and a factor of any other Scheherazade number. But that is math, and my galoshes get stuck in the mud of math. Don’t ask me about it.” 

The number, million, said Stuart, used to have more meaning, when it was the largest number most people thought in terms of. Being a millionaire meant something back then. 

“But it still has some cache. In popular usage, a million means a lot. ‘You’re one in a million,’ ‘a million-dollar smile,’ ‘not in a million years,’ ‘a million-dollar question.’ And ‘I’d walk a million miles for one of your smiles.’ 

“The million has largely been replaced the the billion, ‘with a B.’ To be rich nowadays requires being a billionaire. Everett Dirksen reportedly observed, ‘A billion here, a billion there, and pretty soon you’re talking real money.’ Although he denied ever saying that, it was quoted in The New York Times in 1938. Also, Carl Sagan never actually said ‘Billions and billions,’ he did use it for the title of his last book. It was actually coined as a catchphrase on the Johnny Carson show. But the concept has now become a  name for a vague but large number, the Sagan. It joins other ‘indefinite hyperbolic numerals’ such as ‘gazillion’ ‘bazillion,’ and ‘umpteen.’ Since it is ‘billions’ plural and ‘billions’ plural, logically that would require that ‘billions and billions’ be at the very least four billion whatevers.”

At this point, Genevieve brought out the Ile Flottante and after we ate it, we sat at the table for a moment and looked at all the empty bowls and dishes. And the wine glasses calling out for refills. 

“We forgot the sequel, Guns of the Magnificent Seven,” said Stuart.

fibonacci in blue

Too often, we take what we hear at face value. Facts turn out not to be facts. No one changed your family’s name at Ellis Island. Didn’t happen.

These are not just myths, they are just things that sound like they could be true and so become embedded in our midden of common knowledge. No, Eskimos do not have 30 or 43, or 90 words for “snow.” Human beings do not use merely 10 percent of their brains.

This is all stuff for the Cliff Clavins of the world.

Sometimes this stuff gets caught in our mental wheel spokes because we simply don’t look closely enough.

Take the Fibonacci series. We are told that this interesting pattern of numbers governs much of what appears in nature, including the spiral patters we see everywhere from whelk shells to spiral galaxies. The problem is, observation does not support this idea, at least not as it is usually presented.

The series is created by starting with a zero and a 1 and adding them together, and continues by adding each new number with the previous, making the series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. The series has many interesting properties, one of which is the generation of the so-called “Golden Section.”

To the Greeks, the golden section was the ratio ”AB is to BC as BC is to AC.” It also generates the Fibonacci series and is said to define how nature makes spirals.

golden overlap

Look at the end of a whelk shell, they say, or the longitudinal section of a nautilus shell, and you will see the Fibonacci series in action.


Yet it is not actually true. When you look at whelks, you find spirals and the Fibonacci series creates a spiral, but the two spirals are quite different: the mathematical spiral opens up much more rapidly. The shellfish has a tighter coil. The whelk’s spiral makes roughly two turns for every turn the Fibonacci spiral makes. Math is precise, but nature is various.

fibonacci whelk

What I am most interested in here is not just the agon of conflicting beliefs, but rather the faith in mathematics, and the sense that math describes, or rather, underpins the organization of the world.

I cannot help thinking, in contrast, that these patterns are something not so much inherent in Creation, as cast out from our brains like a fishing net over the many fish in the universe.

Take any large string of events, items or tendencies, and the brain will organize them and throw a story around them, creating order even where none exists.

Consider the night sky, for instance, a rattling jostle of burning pinpoints. We find in that chaos the images of bears and serpents, lions and bulls. Even those who no longer can find the shape of a great bear can spot the Big Dipper. The outline seems drawn in the sky with stars, yet the constellations have no actual existence outside the order-creating human mind.

Ursa major

Our own lives — which are a complex tangle of events, conflicting emotions and motives — are too prodigal to fit into a single coherent narrative, even the size of a Russian novel. Yet we do so all the time, creating a sense of self as if we were writing autobiographies and giving our lives a narrative shape that makes them meaningful to us.

We usually believe the narrative version of our lives actually exists. Yet all of us could write an entirely different story by stringing events together with a different emphasis.

The question always arises: Are the patterns actually there in life and nature, or do we create them in our heads and cast them like a net over reality?

The issue is central to a brilliant movie made in 1998 by filmmaker Darren Aronofsky called Pi. In the film, a misfit math genius is searching for the mathematical organizing principle of the cosmos.

His working hypotheses are simple:

”One: Mathematics is the language of nature.

”Two: Everything around us can be represented and understood through numbers.

”Three: If you graph the numbers of any system, patterns emerge.

”Therefore: There are patterns everywhere in nature.”

Pi movie scene 3

The movie’s protagonist nearly drives himself nuts with his search until he cannot bear his own obsession anymore.

But the film also questions in a roundabout way whether the patterns exist or not.

When different number series — each 216 digits long — seem to be important, an older colleague warns our hero that, once you begin looking for a pattern, it seems to be everywhere.

It’s like when you buy a yellow Volkswagen and suddenly every other car on the road is immediately a yellow VW. Nothing has changed but your perception.

Mathematicians find patterns in nature, yet math itself is purely self-referential. It can only describe itself.

As mathematician/philosopher Bertrand Russell put it: ”Mathematics may be defined as the subject in which we never know what we are talking about nor whether what we are saying is true.”

In other words, ”one plus one equals two” is no different from saying ”a whale is not a fish.” You have only spoken within a closed system. ”A whale is not a fish” tells us nothing about whales but a lot about our language.

It is a description of linguistic categories, rather less an observational statement about existence. Biology can be organized as a system of knowledge to make the sentence false — indeed, at other times in history a whale was a fish.

Before Carl Linne, who created the modern biological nomenclatural system, there were many ways of organizing biology. In his popular History of the Earth and Animated Nature, from 1774 and reprinted well into the 19th century, Oliver Goldsmith divided the fish into “spinous fishes,” “cartilaginous fishes,” “testaceous and crustaceous fishes” and “cetaceous fishes.” A mackerel, a sand dollar and Moby Dick were all kinds of fish.

Plate from Goldsmith's "Animated Nature"

Plate from Goldsmith’s “Animated Nature”

Let’s face it, although the Linnaean system is useful, it is kind of arbitrary to organize nature not by its shapes, or where it lives, but rather how it gives birth or breathes.

”One plus one” likewise describes the system in which the equation is true.

It is possible to cast other patterns over reality. For instance, artists understand perfectly well how ”one plus one equals three.”

That is, there is the one thing, the other thing and then the two together: one sock, the other sock, and the pair of socks. That is three things.

Three things

Three things


In art, we constantly put one object up against another object and observe the interaction between them. In that sense, one plus one can equal three.

When mathematicians say that numbers describe the world, they are speaking metaphorically. Numbers do not, in fact, describe the world. The patterns of numbers seem to mimic the patterns we discern in nature and bear an analogical relation to them.

The fact that this seems to happen so often may be little more than the yellow VW effect.

For experience is large and contains multitudes, even infinities. In any very large set, patterns can be found.

That is the trick behind numerology. If the name Ronald Wilson Reagan can be turned numerologically into the symbol for Satan because each of his names has six letters, making the “666” or “mark of the beast” from the book of Revelations, well, looked at another way, it can be turned into a recipe for Cobb salad. All it takes is a system ingenious enough to do it.

Our hero in Pi believes in the Fibonacci spiral: ”My new hypothesis: If we’re built from spirals while living in a giant spiral, then is it possible that everything we put our hands to is infused with the spiral?”

He begins to sound more and more paranoid.

And paranoia has been defined as a belief in an invisible order behind the visible world.

Paranoia and idealism thus are siblings.

There seems to be hard wiring in the human brain that makes us cast patterns over the world. That hard wiring seems to bring forth what Carl Jung called archetypes, that is, the narrative patterns our brains spin out and the shape we then jigger all of actual experience into.

And when forced to choose between the coherent pattern and the incoherent reality, we always choose the pattern.

Perhaps we could not live otherwise. But it makes me mistrust idealism just as I mistrust mathematics.