If you landed here because you read about Luca, Leonardo, Fibonacci, and the Golden Ratio and you wanted to know more, then you came to the right place! We’re going back in time 2,200 years to the days when the great mathematician, Euclid, discovered the Golden Ratio, 1.618….
OK – Let’s Follow in Euclid’s Footsteps
Euclid wanted to figure out how to divide a line into two pieces so that the ratio of the whole line to the longer piece is equal to the ratio of the longer piece to the shorter piece. It’s easier to see with a picture.
Let’s say we have a line made up of two segments, Segment “A” of length “A”, and Segment “B”, of length “B”. The length of the whole line is A + B.
Let’s say that the point, “P” separates the two segments. Euclid wanted to find where to put point “P” so that the ratio of the lengths of “A” to “B” will be exactly the same as the ratio of the length of (A + B) to the longer of the two segments, which is “A”.
Why did he want to know that? That’s a great question. I don’t know the answer. (I have a guess or two, but perhaps another time… ). Anyway…
In algebra, it looks like the equation on the left, and graphically, it looks like the picture on the right:
To make things a tiny bit easier for ourselves, instead of writing “A/B” all the time, let’s say that whenever an “A/B” turns up, we’re going write “φ” instead.
This isn’t the same sign used by the guy formerly known as Prince. It’s the 21st letter of the Greek alphabet, “phi.” (We’ll pronounce it, “fee”). So now we’re going to write “φ” in place of “A/B”, OK? In other words…
A/B = φ
When we find the right place for the point “P” to go, we said that A/B will be equal to (A+ B)/A, so if A/B = φ, we can also say this…
(A + B)/A = φ
Right? Nothing new here. Everything is on the up and up.
So far, So Good
It doesn’t matter what the lengths of the lines are. The ratios of the lengths are all we care about. Hey — we can rewrite the stuff on the left-hand side like this:
But A/A = 1, so now we can say…
Remember the rule about the reciprocal of a fraction…
If we use the reciprocal rule, since A/B = φ, then 1/ φ = B/A.
If we put in our φ’s for the A’s and B’s, our little A/B equation can be written like this:
1 + 1/φ = φ
Do you buy that? I hope so. Now let’s multiply each side by φ. No harm in that. Perfectly legal. We just have to remember to multiply each term on each side by it. Let’s just write the whole thing out…
1 x φ + (1/φ) x φ = φ x φ, which gives us…
φ + 1 = φ2
OK, just one more step… Let’s subtract φ and 1 from each side to get all of non-zero terms on one side of the equation and zero on the other. That gives us this:
φ2 – φ – 1 = 0
Mathematicians love equations like this. So clean. So simple. To a mathematician, this equation is elegant. Beautiful, even!
Time to Plug in some Numbers
Equations of this form are called “quadratic” equations. The solution to any quadratic equation has already been figured out, so from here we’re home free. I’m going to go out on a limb and assume you’ll be OK with us not going through, step by step, to see where the solution comes from.
So here it is… For any quadratic equation written in the form,
the solution is …
OK, stay with me here… I know this looks like a math accident waiting to happen, but we’re almost home. For us, “x” is “φ”, “a” is 1, “b” is “-1”, and so is “c”. When we plug those into the solution, here’s what we get:
Remember that negative one, (-1), times negative one equals one? Well, it does, so here’s what we’ve got…
My calculator says that the square root of 5 is 2.23607… where the dots mean that the numbers go on forever. Let’s plug it in to get our solution…
Guess what 3.23607… divided by 2 turns out to be? Did you already guess what it is? It’s our old friend,
1.618….
Now Wait Just a Minute!
As Jack Benny might have said at this point, “Now wait just a minute!” This is the EXACT same number, right down to the dot-dot-dot, that we got if we divided subsequent numbers in the Fibonacci series. Remember?
It’s been a while now, but remember where we started? We said that the point, “P” separates two segments of a line. Euclid wanted to find where to put point “P” so that the ratio of the lengths of “A” to “B” will be exactly the same as the ratio of the length of (A + B) to the longer of the two segments, which is “A”.
He found that the answer to his question is to put “P” in the spot where the ratio of A to B is 1.618…
But what do Euclid’s lines have to do with Fibonacci’s bunny rabbits? That’s the thing about this number. It has a way of turning up in the strangest places. But the FIRST place it turned up, or the first place anyone saw it written as a number, was on Euclid’s desk.
Back to Luca, Leonard, and the Divine Ratio
