The True Meaning Of The Fibonacci Sequence

Ok, this one's got nothing to do with future fire

I wrote the following for a previous blog of mine*. It was a very silly blog (even sillier than this substack), populated with nonsense, sport and occasional references to science. The blog’s tagline: Where we leave our reserves of potential in the ground, untapped, for the benefit of future generations. The first post dates back to 2006 - future fire was but a twinkle in my eye at that point - and the last post is from May 2021, three months after I wrote this, now lightly edited for your enhanced enjoyment.

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I have been asking my daughter about math. She doesn't particularly like it, somewhat to my disappointment but of course also much to my fatherly understanding. But I do (like maths). And she's just started high school, which to my eyes represents a six year long chance for me to relive my glory days of math. Sure, I peaked in my first or second year of university, but high school is where math came together in my head. So of course I am keen to find out what she's being taught, and see if I can jog my memory, help her out and reconnect with maths. 

Well, the first bit of homework was not too bad. Some inexplicably finely printed (okay, explicable - budgets are tight these days) questions to do with order of operators (x and / before + and -), add the brackets to make the equation correct, fill in the missing number to make the equation correct etc. I found a mistake in an otherwise flawless set of answers. And was even able to help her with a question from another set that had been stumping her. 

Monumental cemetery of Pisa - Leonardo Fibonacci (Stefano Stabile, via Wikimedia Commons)

But then this evening, just before bedtime, she informed me (well, I may have asked her first) that her latest assignment was to find out the significance of the Fibonacci sequence. An ultra-quick aside, for those unfamiliar. The Fibonacci sequence goes:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

Get it? To get the next number in the sequence, just add the previous two numbers. The first two numbers just are. The rest can be deduced from that simple rule. 

Well, I knew this sequence. I've read about it many times, it is quite a pleasing pattern and it also relates to the spiral shape of shells. Or something. I told her this, including the "or something" part, and said we should be able to help her. I was sure I could find something on my bookshelf that would include a little description of the Fibonacci sequence and its significance. 

The first book I tried was Morris Kline's Mathematics in Western Culture. I've yet to read it but now that I've stumbled upon it, I might just give it ago. I checked the index but found no entry for Fibonacci. The next book was Douglas Hofstadter's Le Ton Beau de Marot. Hofstadter's indices (indexes?) are tours de force and I knew that if he'd discussed it then I'd find it in the index. Sadly, no. Undeterred I moved five spots to the left and picked up Gödel, Escher, Bach. Goddamn jackpot [h/t Carl Weathers].

There were multiple references to the Fibonacci sequence and an entirely separate reference to Leonardo of Pisa. Satisfied that the mission was accomplished, I handed the book to my daughter and encouraged her to inquire within.

Some minutes later, after putting one of my other daughters to bed, I sought the eldest out only to be told in no uncertain terms that the answer was not within. Yes the Fibonacci sequence was in there, but there was nothing about its meaning, its significance. Really? Surely that couldn't be. My daughter assured me she had checked each and every instance listed in the index, and had even skimmed the preceding and following pages to be sure. 

Alas, I had to send her to bed without an answer.  Of course, I would not be suffering the same fate, so I made my way back to the bookshelf and set out to find a book that dished the dirt on the true meaning of Fibonacci's numbers. A couple of other Hofstadter books? Nope. Dennett's Darwin's Dangerous Idea? Dawkins' The Ancestor's Tale? Nope, nope. Increasingly desperate, I turned to Barrow's Impossibility and Book of Nothing. Nothing, impossible! What about John Allen Paulos' Innumeracy? Uh-uh. My prized copy of Feynman's Lectures on Physics (Vol. 1) ? Why would it be in there, of course no. I kept trying, because there had to be something on my shelf that went into this famous little sequence, right? I did not hold much hope when I picked up H.G. Wells' A Short History of the World and I was not disappointed (because I didn't have hope, not because it was in there). I must have tried a few more, but my heart wasn't really in it any more by that point. If only I still had Silver's beautiful The Ascent of Science. Why did I get rid of that? Surely it would have been in there. [Ed. This is the book I borrowed from the library the other week!]

Of course the sadness was that what had once been beautiful - a search through well worn pages of some old favourites on my bookshelf, in service of the next generation of mathematicians - was on a path to becoming an abomination. A google search. I could already see the millions of results, all to be ignored in favour of a pithy summary placed handily at the top (or is it top right?) of the search results. Not even using Duck Duck Go could right this wrong.  Wikipedia wouldn't be so bad, I told myself. It's one of the success stories of the internet, seemingly. If we had a real encyclopaedia (both my parents did when I was a child), we'd have gone to it. What's so different? 

But it felt like a real blow. Like there's no knowledge and no progress without giving in to that damn machine, without staring at that godforsaken screen, without mindlessly consuming those first few algorithm-generated crumbs. Defeat. Ah, maybe there's no shame in just looking it up. Bah! I'm not defeated yet!

I read through Hofstadter's entries on the sequence, in both GEB and I Am a Strange Loop. He talks about the significance of the sequences, alright, but in rather different terms. He speaks of the meaning they had for a young Kurt Gödel, and the derivation (algorithm) of the sequence, and its complement. He talks about the fact that after considerable struggle, mathematicians were able to show that there was a reason (possibly even a darn good one) that 8 and 144 were the only perfect cube and perfect square in the entire sequence.

All of that was significant. All of that had plenty of meaning. But it wasn't going to cut it for my daughter. She needed to have some string of text from some random [modern usage here, my apologies] website that said 'Fibonacci's number is famous for appearing in many natural forms such as the Nautilus shell and ' ... well, I can't think of the other examples. 

Hang on, what if I try it in R, the open source free programming language? Why yes, I can create a variable containing the Fibonacci sequence and plot it and the meaning will reveal itself to me! I don't have to be defeated by the internet (forget for a moment that R was obtained via the internet). Dammit, I can't see any patterns. What am I doing wrong? Should I be plotting alternating numbers? Rotating the graph? Using squares. This is hopeless.

My mind turned to graph paper. Perhaps I could sneak into my daughter's room without waking her, find her maths workbook, and start making my own plots. Surely I'd eventually stumble on the secret of the Fibonacci sequence that way? No, that was too farfetched. There was no way around it. I would have to 'search it up'.

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Graphical representation of the first few terms in the Fibonacci sequence.

The true meaning of the Fibonacci sequence is defeat. Machine over 'Mo Sapiens. Screen over page. School over Dad. Instant gratification over effort. A perfectly average diamond in a jewelry store over a gem in a trashcan (h/t Lin Yutang via Raymond Smullyan via our old friends Hofstadter and Dennett). Homogenisation over individual difference. 

But was all really lost? I had tried, hadn't I? I had resisted. I had fought. Is it not better to have searched (without a search machine) and not found than to have never searched at all? Couldn't this small striving be a seed, a harbinger of things to come? A role model even, to my digital native daughters, that tech wasn’t always the answer? That might be going too far. Nevertheless, I enjoyed the process. It may not have been of consequence, but it meant something to me to seek whence.

Grazie, Leonardo!

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Postscript. My vain but glorious efforts to avoid screens and googling take on additional meaning with the arrival of the wave of large language models (like ChatGPT) currently crashing over our heads. It seems quainter than ever to rage against the machine; de rigeur now to merge with the machine.

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*The blog still exists but good luck finding it. I tried googling the exact name using quotes and there were no results. This is testament to its pristine isolation from anything linking to it anywhere on the internet.