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The golden ratio. It has the property that
[1] A/B = (A+B)/A
Opening the brackets, we have
A/B = A/A + B/A
Since A/A=1, we have
A/B = B/A + 1
If we assume B=1, then
A/1 = 1/A + 1
or, more simply,
[2] A = 1/A + 1
Multiplying both sides by A,
A^2 = A/A + A
Replace A/A with 1:
A^2 = A + 1
Subtract one from the other:
A^2 - A - 1 = 0
Applying the usual quadratic solution formula, we obtain two solutions:
A = (1/2) (1 + Sqrt 5) ≈ 1.618 033 988 749 895...
A = (1/2) (1 - Sqrt 5) ≈ -0.618 033 988 749 895...
The first solution is "the golden ratio", commonly denoted by a
lowercase phi (φ).
As required by [2], 1/φ is 0.618... If you add 1 to that, you get φ
again - exactly as [2] demands. (Apparently this latter quantity, 1/φ =
φ - 1, is sometimes denoted by an uppercase phi, Φ. Notice that -Φ is
the other solution to our equation.)
Many people have noticed that the ratios of successive Fibonacci numbers
approach φ. For those that don't know, the Fibonacci numbers are defined as:
Fib[0] = 1
Fib[1] = 1
Fib[n] = Fib[n-1] + Fib[n-2]
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34
21 + 34 = 55
34 + 55 = 89
55 + 89 = 144
1 / 1 = 1
2 / 1 = 2
3 / 2 = 1.5
5 / 3 = 1.666 666 666 666...
8 / 5 = 1.6
13 / 8 = 1.625
21 / 13 = 1.615 384 615 384...
34 / 21 = 1.619 047 619 047...
55 / 34 = 1.617 647 058 823...
89 / 55 = 1.618 181 818 181...
144 / 89 = 1.617 977 528 089...
φ ≈ 1.618 033 988 749...
Note that φ itself is irrational, so no fraction can ever exactly equal
it. And yet, this sequence approximates it with ever increasing precision.
To many people, this is a seemingly magical property. BUT WAIT! It turns
out that if you apply the same rules but starting with ANY NUMBERS YOU
WANT, the same thing happens!
Pic[0] = 31
Pic[1] = 42
Pic[n] = Pic[n-1] + Pic[n-2]
31 + 42 = 73
42 + 73 = 115
73 + 115 = 188
115 + 188 = 303
188 + 303 = 491
303 + 491 = 794
42 / 31 = 1.354 838 709 677...
73 / 42 = 1.738 095 238 095...
115 / 73 = 1.575 342 465 753...
188 / 115 = 1.634 782 608 695...
303 / 188 = 1.611 702 127 659...
303 / 491 = 1.620 462 046 204...
φ ≈ 1.618 033 988 749...
The initial two terms are randomly chosen. Their ratio is nowhere near φ
(other than that they're roughly the same size, so their ratio is
roughly 1). But a few terms later, and they already agree with φ to
three significant figures.
Even if we start with widely different terms, this happens:
1 + 10 = 11
10 + 11 = 21
11 + 21 = 32
21 + 32 = 53
32 + 53 = 85
53 + 85 = 138
85 + 138 = 223
10 / 1 = 10
11 / 10 = 1.1
21 / 11 = 1.909 090 909 090...
32 / 21 = 1.523 809 523 809...
53 / 32 = 1.656 25
85 / 53 = 1.603 773 584 905...
138 / 85 = 1.623 529 411 764...
223 / 138 = 1.615 942 028 985...
φ ≈ 1.618 033 988 749...
Inexorably, the ratio approaches φ. But why?
Thinking about this intuitively, it becomes (slightly) less surprising.
Instead of looking at the symbols, think about what [1] actually
/means/. We have three quantities, {B, A, A+B}, and the ratio between
successive terms should be the same. We have the Fibonacci numbers,
which build each new term by adding together the previous two. Notice a
similarity there?
With the golden mean, we ask for a ratio such that successive steps
produced by adding always have the same ratio. With the Fibonacci
numbers, we /force/ each successive step to be produced by adding, and
the ratio then approaches the golden mean - seemingly regardless of the
starting point!
It's not dissimilar to the iterative blind deconvolution algorithm; you
start with random data, transform it, apply known constraints to the
result, inverse transform it, apply known constraints to that result,
transform, constrain, inverse transform, constrain... Eventually, the
system converges to a solution that satisfies all the constraints -
hopefully your original, unblurred image. ;-)
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