Digging into understanding chaos, dynamics, and fractals. There are a lot of intriguing ideas present in this domain. First idea to think about are irrational and transcendental numbers (√2, phi, √5, e, pi), they are aperiodic meaning their decimal representation don't repeat.
Now, a kind of map to make sense of chaotic motion and fractals is the Logistic map:
An interactive version by @rickyreusser here: rreusser.github.io/logistic-map/
Original paper introducing the idea here: pdfs.semanticscholar.org/752e/0468e5e2e…
An interactive version by @rickyreusser here: rreusser.github.io/logistic-map/
Original paper introducing the idea here: pdfs.semanticscholar.org/752e/0468e5e2e…
@rickyreusser An interesting constant that comes out of this map is the Feigenbaum's constant (~=4.6692). It can be thought of as the fixed point reached on successively dividing the lengths between adjacent bifurcations in the logistic map.
@rickyreusser Now what can one do with this value? If we pan and zoom to approximate this ratio on the Mandelbrot set, it will create an infinitely repeating GIF. This is an example from Wikipedia, I have to verify if a continuous loop can be achieved in other configurations of zoom and pan.
@rickyreusser A visual intuition can be gained by thinking of the logistic map as existing in the Mandelbrot set as its cross section on the x axis. Source: en.wikipedia.org/wiki/Buddhabrot
I highly recommend going through this thread by @BuildSoil to get to know different contexts where this pattern appears. To name a few: Alan Turing's morphogenic patterns, prey-predator populations, autocatalytic feedback systems:
@BuildSoil Mandelbrot set can be thought of as a meta pattern / map of Julia sets. Here's a video by @hdfractals that gives a big picture idea — Mandelbrot is set of Julia Sets:
And here's @unconed's visualization of the same: acko.net/blog/how-to-fo…
And here's @unconed's visualization of the same: acko.net/blog/how-to-fo…
So Julia sets are motifs inside the Mandelbrot set and one can find the logistic map present inside them. Here's @bernatree visualizing this:
I guess at this point it might be valid to say the logistic map is a leitmotif present inside the Mandelbrot.
I guess at this point it might be valid to say the logistic map is a leitmotif present inside the Mandelbrot.
@bernatree Apart from this @bernatree has also created visualizations where H-Tree is derived by walking on the Mandelbrot set: and tongues of the Julia fractal:
His @ComplexTrees account is also well worth checking out!
His @ComplexTrees account is also well worth checking out!
@bernatree @ComplexTrees To get a feel for chaotic dynamical activity, one needs to familiarize themselves with the state space and phase potraits of dynamics. @robertghrist is doing really great work in visualizing these:
@bernatree @ComplexTrees @robertghrist And if you love these, I highly recommend playing with MIT mathlets. Truly the education space on the Internet is a gold mine for the interested layman to educate themselves with high falutin' (but ultimately comprehensible) ideas in an accessible format: mathlets.org/mathlets/linea…
