Fascinating. Thanks for sharing! Sometime back I had run into a related experiment where the author setup a simple 1 layer NN with a shift-register feedback and explored the state space of neuron activations over large iterations. The observation was beautiful in that the state space maps traced out attractors. See here if you are curious - https://towardsdatascience.com/attractors-in-neural-network-...
I learned something fascinating this week! (Unrelated to OP). The algebraic computation of the Hausdorff dimension of a fractal set is *exactly the same* as the Master Equation of the asymptotic analysis of algorithms! I.e.: where a Sierpinski triangle has dimension log(3)/log(2) ~ 1.58—intermediate in between that of a line and of a plane—that's the very same log(3)/log(2) as in the O(n^(log(3)/log(2)) of, say, the Karatsuba algorithm. In Karatsuba, when you recursively split a digit representation of a number in half (2), you get three new multiplications (3). In a Sierpinski triangle, rescaling by a factor of 2 increases the number of non-empty triangles at that scale by 3. And it really is the *same* manipulation: the Hausdorff dimension of the fractal is a critical exponent of an asymptotic function growth rate, a function of the diameters in a covering set in the ε→0 limit.
The first program I spent money on was a fractal app on the Amiga that would allow me to navigate and drive around the fractal using an Atari joystick. So yeah, fractals are core part of compute experience for me
Are there any useful generalizations of how complex attractors like this typically work? Some systems will have multiple attractors - stable points of the system's dynamics, but what about features/regions of a complex attractor like this - e.g. the circular regions looking like Jupiter's red spot - are they typically/ever sub-attractors in their own right (enter once and never exit)?
Fascinating. Thanks for sharing! Sometime back I had run into a related experiment where the author setup a simple 1 layer NN with a shift-register feedback and explored the state space of neuron activations over large iterations. The observation was beautiful in that the state space maps traced out attractors. See here if you are curious - https://towardsdatascience.com/attractors-in-neural-network-...
Also beautiful. Thank you for sharing it on HN.
I see Hénon and I think of Tarbell, and yep, there it is.
Fractals are due a resurgence, as I recall the Internet used to be about 5% fractals by volume.
I learned something fascinating this week! (Unrelated to OP). The algebraic computation of the Hausdorff dimension of a fractal set is *exactly the same* as the Master Equation of the asymptotic analysis of algorithms! I.e.: where a Sierpinski triangle has dimension log(3)/log(2) ~ 1.58—intermediate in between that of a line and of a plane—that's the very same log(3)/log(2) as in the O(n^(log(3)/log(2)) of, say, the Karatsuba algorithm. In Karatsuba, when you recursively split a digit representation of a number in half (2), you get three new multiplications (3). In a Sierpinski triangle, rescaling by a factor of 2 increases the number of non-empty triangles at that scale by 3. And it really is the *same* manipulation: the Hausdorff dimension of the fractal is a critical exponent of an asymptotic function growth rate, a function of the diameters in a covering set in the ε→0 limit.
https://en.wikipedia.org/wiki/Hausdorff_dimension
The first program I spent money on was a fractal app on the Amiga that would allow me to navigate and drive around the fractal using an Atari joystick. So yeah, fractals are core part of compute experience for me
We in the fractal art community have been having a great time making new software, fractal types and doing international meetups for decades :)
Are there any useful generalizations of how complex attractors like this typically work? Some systems will have multiple attractors - stable points of the system's dynamics, but what about features/regions of a complex attractor like this - e.g. the circular regions looking like Jupiter's red spot - are they typically/ever sub-attractors in their own right (enter once and never exit)?
Beautiful.
I always find it a bit shocking that there is so much structure in the states of chaotic systems as they evolve over time.
Thank you for sharing this on HN.
Balancing right on the knife's edge between simplicity and randomness.