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.
Comments
Awesome.
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...).
Something something Lie group...
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.
That's such a good idea! Sadly I am at work right now so I can't hack on it for a while, but I love it