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#juliaset

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did some experiments tracing #ExternalRay in #InflectorGadget Julia morphing.

the #ExternalAngle (expressed as binary expansion) of the rays of the inflection nodes in the #JuliaSet have unchanging periodic part corresponding to the starting minibrot, while the length of the preperiodic part increases linearly with each morph (e.g. adding a constant number of bits each time), zoom depth is constant, but the stretching dynamics sometimes need precision to be increased.

compare with #MandelbrotSet morphs by deep zooming, where the length of the periodic part (and thus iteration count) increases exponentially with each morph (e.g. doubling each time), and zoom depth goes up typically by 150% (requiring 150% higher precision).

Just published: web version of #InflectorGadget inflection mapping gadget for #MandelbrotSet #JuliaSet #fractal #fractals #MathArt

mathr.co.uk/ig/

Pro-tip: keep the Add tool selected and use mouse wheel to zoom if available, otherwise (e.g. touch screen) you need to keep switching between tools

Click the hash button in the top left to update the address bar URL before sharing (otherwise it might not be up to date), here's one I did today (3 things in a rabbit: right-handed tree, left-handed tree, line):

mathr.co.uk/ig/#eyJ2IjoxLCJjZW

Suite du fil

added `{ preserveDrawingBuffer: true }` to the webgl2 context creation, now Firefox right click Save Image works properly (previously it saved a 100% transparent image).

apparently this can cause performance issues especially on mobile, so I will see about adding a different way to save later.

also made it update the URL hash automatically when adding inflection points (previously only on manual button click). this avoids losing work when accidentally refreshing the page etc. can't make it update on every action (zoom, pan, colours, ...) otherwise browser complains about too much history spam.

here are five things in a 2/5 Julia set: disk, line, three tree variants (distinguished by how they attach to the whole). not very refined (only 3 steps of each, 16 inflection points total including the initial Julia set location).

today I added a quantization option to #InflectorGadget, to snap inflection points to #MandelbrotSet nucleus (first one) and preimages of #JuliaSet attractor (later ones) - this makes all the nodes as circular as possible

this also makes the structure exactly reproducible (by a human, no automation yet) from instructions as there's no fuzzy variation coming from imprecise mouse click locations

the human touch adds a certain pleasant softness, so quantum mode is optional

10 years (exactly) on from my unanswered question

"Interior Distnace Estimate for Julia Sets - Getting Rid Of Spots"
math.stackexchange.com/questio

I think I found the answer: apply l'Hospital's rule to the limit for attracting (not super-attracting) interior distance, take the maximum of all such distance estimates.

en.wikipedia.org/wiki/L'H%C3%B

Not very principled (I haven't checked the necessary conditions for the rule to be valid), but seems to work ok for removing the spots at the preimages of the attractor.