claude

what if... one could construct a low precision #MandelbrotSet orbit corresponding to a specific high precision orbit, without doing any high precision calculations?

I'm experimenting with constructing "tuned" periodic orbits by a simple algorithm (cycle one orbit, replacing the periodic 0 with successive values from a second orbit; scaled by Julia size of the first orbit, then iterate backwards with Newton's method to fix it up to be less approximate)

seems to work in some simple cases, in other cases the refinement pushes it to a wrong orbit*, so I think my initialization is wrong. plotting the images after each step of Newton refinement shows it takes some bonkers path instead of moving smoothly from the approximate to the accurate....

• e.g. in attached video I specified to go to the same side relative to the mini each time, but it doesn't do that...

#maths

edit: CANCELLED due to extreme weather :(

event in London next week:

https://luma.com/london-evub

Algorest
Wednesday 24th June 2026 7pm
Home Deptford, London SE8 4AD

I'll be doing visuals early in the night using Mandeltron, my work in progress thing for live-coding the Mandelbrot Set.

https://mathr.co.uk/web/mandeltron.html

Guillaume Collombet (with assistance from LLM...) on fractal chats discord suggested using #MoebiusTransformation instead of #LinearApproximation to accelerate #MandelbrotSet #DeepZoom .

The advantage is that the quadratic term of the perturbation formula is now exact, with a cubic remainder term. for single step, assuming that z < sqrt(eps) 2Z makes this remainder be lost in rounding noise, while for the linear approximation the corresponding condition is z < eps 2Z.

given two moebius with validity radii, the validity radius for their composition can presumably be calculated without much difficulty.

eps is typically 1e-16 to 1e-8, depending on how much precision loss you think you can get away with.

Guillaume claims moebius gives a practical computational efficiency improvement over linear. I haven't tried it myself yet.

However, while I think it should work well for #JuliaSet, where composition of two moebius in z gives another moebius in z, the nonzero c term for mandelbrot set ends up with something with zc and c^2 terms when composing the two functions, which Guillaume handwaves away reasoning that c is usually tiny and z is also small most of the time. I'm not 100% convinced that iteration (composition coefficients getting larger) wont eventually make them significant...

managed to get tidal-mondo building with MicroHs. had to workaround some differences from GHC that might or might not be bugs.

haven't used it yet so I don't know if it still works after my changes.

wip pull request at codeberg

#Haskell #uzulang #TidalCycles

https://uzu.lurk.org/t/challenge-non-linear-time-transform/6171 an interesting challenge was posted, I did a #TidalCycles thing with #MoebiusTransformation in response, but I don't have tidal fully running (only tidal-core and tidal-mondo, nothing hooked up to making sound) so i haven't been able to listen...

the idea is that for time outside the arc (0, 2) time advances linearly as normal, but inside it smoothly speeds up and slows down, fitting in 4 cycles in the usual time for two.

I used the computer algebra system #Maxima to solve for the coefficients given values/slopes (3 per transform).