φ (phi), also known as the Golden Ratio, was thought by the ancient Pythagoreans to have mystical properties. Even the more level-headed Greeks ascribed it artistic ones, for instance considering the most eye-pleasing rectangle to be one where the ratio of the long side to the short side is φ. As shown above,
The Last Supper the Mona Lisa contains many golden rectangles.
They also have the nice property that if you divide one into a square and another, smaller rectangle, the new rectangle is also golden.
This gives us a clue for how to find φ: call the larger side of a golden rectangle a and the smaller side b. Once we carve out a square of size b, what’s left is a rectangle with large side b and small side a-b. Since that’s also golden, we know that:
(1) a/b = b/(a-b)
(2) b^2 = a^2 – ab
(3) 1 = (a/b)^2 – (a/b)
recognizing that a/b is φ and putting everything on one side,
(4) φ^2 – φ – 1 = 0
Applying the quadratic formula gives us
(5) φ = (1 +- √5)/2
At the moment we’re only interested in the positive solution, so
(6) φ = (1 + √5) / 2 ≈ 1.618
φ is interesting arithmetically too. Consider the Fibonacci sequence, which begins with two 1s, and thereafter makes new members by adding the previous two members:
1, 1, 2, 3, 5, 8, 13, 21, 34, …
The sequence of ratios of two adjacent Fibonacci numbers begins:
1, 2, 1.5, 1.666, 1.6, 1.625, 1.615, 1.619, …
It’s not hard to prove that this ratio converges to φ. It (almost) doesn’t matter what the first two numbers are: the ratios still converge to φ. Let’s do something outlandish like -17 and 12:
-17, 12, -5, 7, 2, 9, 11, 20, 31, 51, 82, 133…
The ratios are:
-.706, -.417, -1.4, .286, 4.5, 1.222, 1.55, 1.645, 1.608, 1.622
The ratios start out all over the place, but converge towards φ soon.
Since from (4), above, φ^2 = φ + 1, we could also start with the numbers 1 and φ to build an additive sequence whose ratios are all φ:
(A) 1, φ, φ + 1, 2φ + 1, 3φ + 2, 4φ + 3, …
which is identical to
1, φ, φ^2, φ^3, φ^4, φ^5, …
So, do the ratios of all additive sequences converge to φ? As I said above, almost. There’s one exception, based on the other solution to the quadratic (5): (1 – √5) / 2 ≈ -0.618 . For now, let’s call this ψ. It’s easy to verify a few things about ψ:
(7) ψ = -1/φ
(8) ψ = -(φ-1)
(9) ψ^2 = ψ + 1
The last of these means we’ve found an additive sequence whose ratio isn’t φ:
(B) 1, -1/φ, 1/φ^2, -1/φ^3, 1/φ^4, …
The ratio is always exactly -1/φ. In fact, that’s the only such sequence, and the numbers have to be exact. If we start even a little bit off, say with 1 and -.6, we get:
1, -.6, .4, -.2, .2, 0, .2, .2, .4, .6, 1.0, 1.6, …
At this point we have the regular Fibonacci sequence with each term multiplied by .2, so we know the ratios will go towards φ. In fact, if you calculate (B) on a computer and print out the ratios, you’ll find them going towards φ, because a computer doesn’t do exact arithmetic, and the little bits of roundoff error are sufficient to push it over the edge.
So we’ve found an important difference between φ and -1/φ:
- An additive sequence whose initial ratio is φ will keep the ratio φ
- An additive sequence whose initial ratio is -1/φ will keep the ratio -1/φ
- An additive sequence whose initial ratio is anything else will converge towards the ratio φ
That is, the sequence with ratio φ is stable (also called an attractor), while the sequence with ratio -1/φ is unstable. Another way to look at this is that any additive sequence is a linear combination of (A) and (B). Since the terms in A tend towards infinity while the terms in B tend towards 0, the only way for A not to dominate B is for the sequence to be all B with no A.
This doesn’t even scratch the surface of the things to be said about φ, but it’ll do for now.