The Fibonacci Matrix
ianthehenry.com
ianthehenry.com
Something I noticed in the plot where you can add points is that it's not actually using the continuous version of the transformation to interpolate the paths. It looks like the points are being linearly interpolated between integer powers of the transition matrix. Once you've got the eigenvalues and eigenvectors, you can easily raise the transition matrix to fractional powers to get things like square roots and show halfway points. I think if you interpolated that way then you'd get smooth spirals towards the golden ratio line, instead of bounces (keeping in mind that because one of the eigenvalues is negative you'd end up with complex numbers requiring a projection down from 4d to 2d...).
Of course you can also just have a good approximation of phi and multiply your coordinates by arbitrary powers of that. That will just slide you along the diagonal, but it becomes very accurate very quickly.
[1]: https://www.math.purdue.edu/~yipn/543/matrixExp19-I.pdf
One funny trick that brought some realism to the lecture: If a and b are the golden ratio and its conjugate, then f_n = a^n + b^n. But since |b| < 1, you can just do f_n = nearest_integer(a^n).
Well, almost. You need to multiply by a factor of 1/sqrt(5) before rounding.
the denominator is important.
also another interesting relation is a^n ~ f_(n+1) + f_(n-1) = f_(n+2) - f_(n-2)
Nice quip at the end as well :)
Even if we start with two numbers that are completely unrelated in the Fibonacci sequence – say, 8 and 41 – the simple way that we pick the next number of the Fibonacci sequence will cause us to approximate the golden ratio after only a few iterations:
8 / 41 = 0.1951219
(8 + 41 = 49) / 8 = 6.125
(49 + 8 = 57) / 49 = 1.16326531
(57 + 49 = 106) / 57 = 1.85964912
(106 + 57 = 163) / 106 = 1.53773585
He mis-adds in the third step 8+41 ought to be 41+49..
But that's not all. He says "if we start with [any] two numbers...in the Fibonacci sequence" but in fact you can start with ANY two numbers WHETHER OR NOT they are fibonacci numbers.. and perform the Fibonacci operation and divide adjacent numbers and it will converge to the golden ratio. E.g.
8
10 1.25
18 1.8
28 1.555555556
46 1.642857143
74 1.608695652
120 1.621621622
194 1.616666667
314 1.618556701Good call on the wording there, though -- changed it from "completely unrelated in the Fibonacci sequence" to "completely unrelated to the Fibonacci sequence" (41 is not a Fibonacci number).
Your sequence is
8
41
49
57
106
163
8
41
49
90
139
229Using the Binet closed form for Fibonacci with floating point numbers is problematic because of finite precision. Naive exponentiations of the golden ratio are not accurate. Part of the problem is the infinite precision needed to deal with sqrt(5).
One way to deal with that is to represent numbers in the numerator or denominator as (a + b √5) with a and b as integers. Whenever the √5 gets multiplied you separate it out as 5.
This is way is not faster. One ends up with the logarithmic complexity of exponentiation by squaring. Nonetheless its quite fun and accurate.
https://artofproblemsolving.com/wiki/index.php/Binet%27s_For...
You kinda have to use WebGL for this to look good, though, because the vanilla 2D canvas has no way to disable anti-aliasing, so you can't really get those crisp pixely lines.