Leonardo of Pisa (Fibonacci) · Mathematics
The real mathematics linking the Fibonacci sequence to the golden ratio - the ratio of consecutive terms converging on phi (about 1.618) - done honestly, together with a clear-eyed debunking of the exaggerated ‘golden ratio in everything’ mysticism, most of which has nothing to do with Fibonacci at all.
Take the Fibonacci sequence - 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... - and do something Fibonacci never did: divide each term by the one before it. You get 1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615, 34/21 = 1.619, 55/34 = 1.618. The answers bounce up and down, but by ever smaller amounts, and they close in relentlessly on a single number: 1.6180339887..., an irrational number the Greeks knew and modern writers call the golden ratio, written with the Greek letter phi. The further out you go in the sequence, the more precisely the ratio of neighbours pins down phi. So hidden inside the additive rule of the rabbits is a fixed proportion that the sequence forever approaches but never quite reaches.
The golden ratio is genuinely a remarkable number, for reasons that have nothing to do with paintings or seashells. It is the unique positive number whose reciprocal is itself minus one: 1/phi = phi - 1 = 0.618..., so phi and its reciprocal differ by exactly 1. Equivalently, phi is the positive root of x squared = x + 1, which is why the Fibonacci ratios converge to it. It has the strangest continued-fraction expansion of any number - all 1s - which, by a precise theorem, makes it the most irrational number, the hardest of all to approximate well by fractions. That extreme quality is not decorative; it is exactly why phi appears in plant growth. A growing shoot that places each new leaf at a turn of phi of a full circle achieves the least overlap and the best exposure to light, precisely because phi resists being closely approximated by any simple fraction, so the leaves never line up into wasteful rows.
Here is a fact that surprises almost everyone: Fibonacci himself never connected his sequence to the golden ratio. In the Liber Abaci he presents the rabbit numbers as the answer to a breeding puzzle and notes the additive rule, and that is all - not a word about ratios, limits, or the ‘extreme and mean ratio’ he certainly knew from Euclid. The two ideas that we now think of as inseparable sat in entirely separate compartments of medieval mathematics: the golden ratio was a topic in Euclidean geometry, studied via lines and pentagons, while the sequence was a piece of commercial recreation. Joining them required someone to look at the sequence and ask a geometer’s question about its proportions - and the first person known to have done so, clearly, was Johannes Kepler, around 1608, nearly four centuries after the Liber Abaci. When you hear ‘Fibonacci and the golden ratio’ spoken in one breath, remember that Fibonacci would not have recognized the pairing.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that the ratio of consecutive Fibonacci numbers converges to the golden ratio phi (about 1.618), the positive solution of x squared = x + 1; that Euclid had defined this same ratio as dividing a line in ‘extreme and mean ratio’; and that Fibonacci himself never linked his sequence to it. Explain both the g…
Leads to Euclid.
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