I just made my #RaspberryPi Pico render the #MandelbrotSet and return the data so that I could create an image file on my PC!

Not a world changing achievement, but I am happy that it works and it's just the first step towards more computation on the Pico!

Here are a few notes on what I did!

1/X

https://mathr.co.uk/web/z2cz5.html
I wrote up some #math #maths #mathematics about properties of filled-in #JuliaSet #fractal for the #polynomial mapping $f_c(z) = z^2 + c z^5$, including visualization of its #MandelbrotSet analogue.

Decorating the workplace.
#complexfunction #juliaset #mandelbrotset #misiurewiczpoint #tanlei

@stecks Lastly, (apologies for the overload!) here are some ideas/requests for other people who might be interested:

- some classics: #PythagoreanTheorem , #MandelbrotSet , or #E8

- emoji for when you're thinking hard (maybe the orb from the orb-ponderer meme?)

- emoji for when something you thought was hard was actually easy!

- emoji for when something you thought was easy was actually hard!

someone emailed me about one of my #haskell projects from 2010 so I managed to get it working again.

https://hackage.haskell.org/package/mandulia-0.8.0.1

ghc-9.4.2 needs --allow-newer=text for OpenGLRaw to build, but apart from that it should work fine with ghc-8.0.2+ (and maybe earlier versions too). tested on debian testing (bookworm, not final) with freeglut3-dev installed.

quick start:
run `mandulia main`
press "a" to enter attract mode
press "F11" to toggle full screen
press "ESC" to quit

#fractals #JuliaSet #MandelbrotSet

Kalles Fraktaler 2 + development is mostly stopped from my side.

Here are some #AlternativeTo KF2+:

- FractalShades (Python)
- Fractal Zoomer (Java)
- Fraktaler 3 (C++)

For more details see:
https://mathr.co.uk/kf/kf.html#future

Important KF bug fixes may still be released from time to time, but I do not have enough time to implement the #BilinearApproximation (#BLA) optimisation in this project.

BLA is much better than #SeriesApproximation and #BivariateSeriesApproximation (which is called NanoMB1 in KF), in terms of efficiency of calculations, generality of applicability, simplicity of implementation, and robustness of results.

#KF #KF2 #KallesFraktaler #KallesFraktaler2 #Kalles #Fraktaler #fast #deep #zoom #DeepZoom #EscapeTime #fractal #Mandelbrot #MandelbrotSet #BurningShip

kf-2.15.5 released!

full announce: https://mathr.co.uk/blog/2021-12-05_kf-2.15.5_released.html

highlights include single reference implementation for glitch avoidance (as suggested by Zhuoran on fractal forums), and lots of new formulas (mostly from FractalAlex)

this is probably the last kf release from me for the foreseeable future...

#KF #KF2 #KallesFraktaler #KallesFraktaler2 #fractal #fractals #MandelbrotSet #BurningShip #EscapeTime #DeepZoom

https://mathr.co.uk/blog/2021-06-04_old_wood_dish.html I wrote a blog post with analysis and variations on a theme by jwm-art from 2010

#MandelbrotSet #DeepZoom #Fractal #Art

New bugfix release: https://mathr.co.uk/kf/kf.html#kf-2.15.2.3

- fix SIMD implementation of sgn() (test location by FractalAlex)
- fix opengl pack alignment for non-multiple-of-4 width bitmaps
- fix palette From Image not updating number of colours (reported by Microfractal)
- fix palette From Image to use maximum of width and height
- add PNG to supported images for palette From Image (reported by Microfractal)
- fix argmin|z| glitch correction for OpenCL
- report detailed timing information in command line mode
- update dependencies
- boost to 1.76.0
- tiff to 4.3.0
- openexr to 2.5.6
- glfw to 3.3.4

#KF #KF2 #KallesFraktaler #KallesFraktaler2 #Deep #Zoom #DeepZoom #EscapeTime #Fractal #Fractals #MandelbrotSet #BurningShip

New blog post "deep zoom theory and practice"

https://mathr.co.uk/blog/2021-05-14_deep_zoom_theory_and_practice.html

#deep #zoom #DeepZoom #fractal #fractals #mandelbrot #MandelbrotSet #BurningShip #graphics #math #maths #mathematics #FloatingPoint

Figured out how to plot wakes implicitly.

Given a wake with parameter ray angles $s_-, s_+$, for each pixel $c$ in the image trace the dynamic rays at those angles towards the Julia set: if and only if they land together, then $c$ is in the wake.

An application of Theorem 2.5 from https://arxiv.org/abs/1709.09869 "A survey on MLC, Rigidity and related topics" by Anna Miriam Benini.

Previously I had been tracing the two parameter rays into a polygonal boundary and filling that using rasterization, to do: benchmark and compare the two methods in various scenarios.

#MandelbrotSet #JuliaSet #fractal #maths #ComplexDynamics #ExternalRays

https://mathr.co.uk/kf/kf.html#kf-2.15.1.6
https://mathr.co.uk/kf/kf.html#kf-2.14.10.7

Two bugfix releases of #KF in current branch and stable branch: turns out switching #MinGW from #Win32 to #POSIX #threading model had some unforeseen consequences (aka bugs, one of which was "zooming out sometimes resets zoom to infinity or zero" which broke interactive use), so I reverted those changes.

#KallesFraktaler2 is fast #fractal #DeepZoom #FreeSoftware for #MandelbrotSet #BurningShip etc

https://mathr.co.uk/kf/kf.html#kf-2.15.1.5
kf-2.15.1.5 released!

- fix long double rendering for hybrid formulas with abs
- fix typos breaking OpenCL floatexp bailout checks
- fix guessing for OpenCL floatexp
- fix endless reference loop in OpenCL
- merge changes from kf-2.14.10.6

https://mathr.co.uk/kf/kf.html#kf-2.14.10.6
kf-2.14.10.6 released!

- fix long double rendering for formulas with abs
- fix floatexp rendering for fractional bailout norm power
- fix some float conversion warnings and potential problems
- make prepare script modular (second argument determines libraries to build, default is all libraries)
- patch OpenEXR to try to fix issues with non-ASCII paths on Windows

#kf aka #KallesFraktaler #KallesFraktaler2 is #fast #DeepZoom #FreeSoftware for #EscapeTime #fractal like #MandelbrotSet #BurningShip etc. For Windows but works in Wine and is cross-compiled from Linux using MinGW.

kf-2.15.1 released!
with OpenCL acceleration, hybrid formula designer, exponential map, and more
get it from https://mathr.co.uk/kf/kf.html#kf-2.15.1

- #OpenCL support for perturbation iterations (requires double precision support on device: #CPU should work, some #GPU might not)
- #HybridFormula editor (design your own fractal formula)
- #ExponentialMap coordinate transformation (useful for export to #zoomasm for efficient #video assembly)
- rotation and skew transformations are rewritten to be more flexible (but old skewed/rotated KFR locations will not load correctly)
- kf-tile.exe tool supports the new rotation and skew transformations
- the bitrotten skew animation feature is removed
- a few speed changes in built in formulas (one example, RedShiftRider 4 with derivatives is almost 2x faster due to using complex analytic derivatives instead of 2x2 Jacobian matrix derivatives)
- flip imaginary part of Buffalo power 2 (to match other powers; derivative was flipped already)
- slope implementation rewritten (appearance is different but it is now independent of zoom level and iteration count)
- smooth (log) iteration count is offset so dwell bands match up with the phase channel

#deep #zoom #DeepZoom #fractal #fractals #free #software #FreeSoftware #mandelbrot #MandelbrotSet #BurningShip #KallesFraktaler #KF

"Charred Bard"

https://archive.org/details/charred-bard
https://youtu.be/NMKBBk-yf_4

#Zoom into a #hybrid #fractal, in black on white with rainbow fringes.

59 iterations of #MandelbrotSet formula, then 1 iteration of #BurningShip formula, then repeat, until escape.

137bpm #techno soundtrack is composed in #Clive, using tempo-synced feedback delays with 1/2-octave bandpass filter bank, where each band's gain depends on the level of the bands below.

Rendered as an exponential map with work-in-progress KF-next, using a custom zoom assembler to flatten it into video frames.

https://archive.org/details/special-branch

Short #zoom into a #hybrid #fractal, in black on white with rainbow fringes.

59 iterations of #MandelbrotSet formula, then 1 iteration of #BurningShip formula, then repeat, until escape.

Soundtrack is an experiment in extreme timestretching.

Rendered as an exponential map with work-in-progress KF-next, using a custom zoom assembler to flatten it into video frames.

Noticed that the filaments connecting the core of embedded Julia sets go different ways on opposite sides of a valley. While superficially similar, the biggest spiral implodes from many branches with a layer of details offset on each branch that resembles a two-fold spiral, to a truly two-fold spiral.

More information in image descriptions.

#EmbeddedJuliaSet #MandelbrotSet

https://archive.org/download/zoom-interpolator-tests/mathr-exr-zoom-interpolator-gl4-test-3.mp4 950MB 1920x1080p60 3m35s

#MandelbrotSet #fractal #zoom #video #ColourCycling #Dynamo

raw iteration data keyframes from #KF in #EXR container colourized using #OpenGL 4 with per-output-frame processing. 32 samples per pixel from 4x4 sized keyframes for #AntiAliasing

https://arxiv.org/abs/1911.03796 #amreading #maths #MandelbrotSet #ExternalRays

A formula to transform rays from hyperbolic components to the real axis.

New to me terminology:
- vein (not exactly sure what this is, paper needs more diagrams and examples in binary instead of decimal)
- pseudocentre (tip of ray pair or vein)
- complexity (length of binary expansion of dyadic rational / pseudocentre / vein)
- kokopelli component (upper period 4 island)
- maximally diverse sequence