i tried using float instead of double to speed it up, negligible change
but it overflowed much sooner, so i used floatexp (float with separate int exponent for more range).
after adding some extra operators, it compiled including Eigen C++ matrix stuff
now it is even more very slow...
...and i think it broke because it seems to infinite loop after outputting a single blank frame...
do eigen decomposition for jacobian J product at escape (first normalize J by dividing by max abs element to avoid overflow)
J* J = V D V*
then scale+stretch of normalized J is
V sqrt(D) V*
separate into overall scale (product(sqrt(D))) and divide the diagonal by scale^(1/n) where n is the dimension (usually 3 or 4).
then compute the (weighted) sum of all the stretch matrices (currently evaluating J at 70% of the way to the surface at each pixel), normalize to scale 1 again, and smooth over time to avoid stroboscopic oscillations
can't really tell if it works properly because my offline renderer is way too slow when running in termux on my phone (~30mins for 30secs of 36x64px video depending on location) and i don't have interactive one yet; so i'm not really sure what i'm looking at...
i suspect a lot of the time bz+1 will evaluate numerically to 1 because z is small, in which case the moebius reduces to linear with extra steps. linear in that case would be fine as the z^2 term would be lost in rounding noise anyway...
concretely (Z+z)^2 - Z^2 = 2 Z z + z^2
moebius of the form
m : z -> a z / (b z + 1), for |z| < r
for single step:
a = 2 Z
b = - 1 / (2 Z)
r = sqrt(eps) |a|
becomes
z -> 2 Z z + z^2 + z^3 / (2Z - z)
with the last term less than rounding error
combining adjacent steps
m_2 . m_1 : z -> (a_2 a_1) z / ((b_2 a_1 + b_1) z + 1)
r = min(r_1, r_2 / (|b_1| r_2 + |a_1))
(not 100% sure on my derivation of combined r...)
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...
given self-adjoint matrix M* M,
the stretch part of the polar decomposition of M can be found quite easily:
do eigen decomposition
M* M = V D V*
then stretch of M is
V sqrt(D) V*
I don't need the rotation part, but that could be achieved by combining with
M M* = U D U*
with full polar decompositions
M = (U sqrt(D) U*) (U V*)
M = (U V*) (V sqrt(D) V*)
hoping that this might do something useful...
working on autoskew for 3D fractals
more complicated than 2D where root-mean-square of complex directional distance estimates works well.
in nD, singular value decomposition gives
A = W D V*
where W, V* unitary and D diagonal
from which two polar decompositions are
A = (W D W*) (W V*)
A = (W V*) (V D V*)
I think a volume integral of some function of the polar decomposition from SVD of the jacobian at escape might be useful, but there are a lot of fiddly variations and I have not got anything satisfactory yet.
or maybe root mean square of matrices could work, who knows...
AlgoRest tonight is CANCELLED due to extreme weather.
edit: CANCELLED due to extreme weather :(
event in London next week:
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://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).
I added implementation notes to my page about my piece Golden Ratio Maximus (published in Taper#12 "Tools"):
https://mathr.co.uk/web/golden-ratio-maximus.html#Implementation
I added some incomplete implementation notes to my page about my piece Seize It With Both Hands (published in Taper#14 "Sonnets") https://mathr.co.uk/web/seize-it-with-both-hands.html#Implementation
I need to get better at writing implementation notes at the time of implementation rather than trying to reconstruct my thinking years later...
I wrote up some even briefer notes on the implementation of my piece Tarn Generative Map Algorithm #1 (published in Taper#15 "Crossroads"): https://mathr.co.uk/web/tarn-map.html#Implementation
I wrote up some brief notes on the implementation of my piece Hyperloop (published in Taper#16 "For Good Measure"):
5000th toot! #introduction time.
I'm originally from London, UK, now based in Alt Empordà. I make #CreativeCoding #MathArt. I use Linux since 2004 or so, currently happy with Debian though I need to find something else to keep my remaining 32bit hardware up to date #permacomputing. Most of my code is published under #copyleft licenses.
I started making music on classic #Amiga, using #OctaMED SoundStudio until 2003. Then I used #PureData until 2011, but mousing patch cables was increasingly not fun so I wrote pdlua to embed Lua in Pd and later fixed some things to let libpd be compiled with emscripten to run in web browsers. In 2012 I started #LiveCoding music in the #C programming language, which I still do.
Always interested in #fractals, I wrote some libraries to support my investigation of the #MandelbrotSet. I took over maintenance of Kalle's Fraktaler 2, but eventually I was fed up with its code and wrote Fraktaler-3 from scratch, to support deep zooming into hybrid escape time fractals. A companion program Zoomasm is for video output.
I have a lot of projects, most sleeping until I feel inspired to work on them some more. Currently working on Mandeltron, a thing that combines live-coding with the Mandelbrot set.
More on my website, including blog going back to 2005: https://mathr.co.uk
My laptop has an AMD 780M integrated GPU, by default assigned 4GB VRAM from the total 32GB in the machine.
I tried increasing it in the BIOS to 8GB, but the system (Debian Linux with X.org) became unstable so I switched it back.
When I have time I may see if I can report kernel / firmware bugs so it can get fixed.
https://m.youtube.com/watch?v=K3GHE4sC2jg Gigh Zack (with assistance from LLM...) applied perturbation techniques (popularized in the last years for deep 2d fractal zooms) to the 3d mandelbulb.
working on a submission these past days. think the third rewrite today (earlier versions had various problems) and subsequent minification pass means it is about done now.
hoping that forbidden "regex eval functions" means things like `new Regex("some string")` and not `/some regex/`.
have an idea for another one too, will see how that goes. it's maybe on the "wrong" end of the math / literary spectrum though.
https://taper.badquar.to/16/about.html#submit deadline for Taper#17 "Prime" submissions is August 17th 2026.
Taper#16 "For Good Measure" is out now!
Featuring my mathematical piece "Hyperloop" among many others.
https://taper.badquar.to/16/hyperloop.html
Taper is an online zine for computational poems in less than 2KiB.
Is my piece a poem? I might argue that the way mathematical proofs fit together is poetic, or more spaciously, along the lines of "everything is political", that "everything is poetry".