The Fibonacci fib

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.


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