51–59 of 59 posts
I love the brutalist simplicity of the site, and the extremely high density of quality information. For those two young to know — this is a taste of what made the early web so exciting.
I believe it's like a generalisation of the well-known 'bifurcation diagram' of the logistic map such that it's got more dimensions, and these images are like slices of it 'through the page'.
It makes gorgeous soft and subtle details! Unfortunately since it's essentially a histogram it's tricky to produce high res renderings. Paul Bourke is a total hero of this realm! Great stuff!
Marty Golubitsky's Homepage https://www.asc.ohio-state.edu/golubitsky.4/ Books by same: https://www.asc.ohio-state.edu/golubitsky.4/mgbooks.html https://en.wikipedia.org/wiki/Marty_Golubitsky Fearful Symmetry: Is God a Geometer? Ian Stewart + Martin Golubitsky is a good read, Ian Stewart is well known to UK recreational math fans and frequently hosts STEM material on th BBC. https://www.goodreads.com/en/book/show/165…
Earlier quoted context omitted.
Yes, the high level rotational symmetry. There's some gross level structure that's preserved while the individual points are (presumably) completely aperiodic/chaotic. What's some insight into the gross level symmetric features appearing? How do you convert something that's completely periodic to an aperiodic/chaotic system with gross level periodic structure? What are the operations that allow the gross level period…
They iterate f where f(z) = (a₀ + a₁ z ź + a₂ Re(zⁿ) + a₃ i) z + a₄ źⁿ⁻¹. You can easily check that f(z exp(i 2 π / n)) = exp(i 2 π / n) f(z).
These are awesome, but it would be nice to understand, even at some heuristic level, why they're periodic with the period they do have. Alternatively, what is a change to a completely periodic orbit to chaos with the same "periodic symmetry" that gives some enlightenment of where the chaos is being inserted and why it isn't destroying the periodic orbit. Does anyone have an idea of whats' going on, or references that…
I'm also not sure if the book answers your question(s) on this apparent aymmetry aspect, but Chaos and Fractals by David Feldman is a fantastic introduction to these topics with lots of good pictures. It is not a proof-based book, but it does include mathematics at manageable level. The book gives a great introduction to strange attractors, namely the Henon map, Rossler attractor, and the Lorenz attractor.
Also check out Jim "Chaos" Crutchfield's work, and the mesmerizing video feedback he made with his analog video processing computer that he built at the University of California, Santa Cruz for his Ph.D. in physics in 1984. Jim Crutchfield https://en.wikipedia.org/wiki/James_P._Crutchfield https://csc.ucdavis.edu/~chaos/index.html Space-Time Dynamics in Video Feedback https://www.youtube.com/watch?v=B4Kn3djJMCE A fil…
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