Faster Mandelbrot Set Rendering with BLA: Bivariate Linear Approximation

The latest and greatest (as of the time of writing, in May 2023) method for computing Mandelbrot set images is a form of bivariate linear approximation, which is referred to as BLA. Discussion of this method began in late 2021 on the fractalforums.org site (from the first post on page 3 of this thread). I made some failed attempts to implement BLA in 2022, but over the last few months I've finally got an experimental BLA working in my JavaScript Mandelbrot set viewer, Very Plotter.

In this post I'll show how to use BLA in the Very Plotter v0.10.0 release. Then, I'll give some background on the theory and implementation of BLA, and show the performance I'm seeing at some example locations.

Arbitrary Precision Square Root in JavaScript

I've recently been working on a square root function for my arbitrary precision floating-point numbers used in Very Plotter, my JavaScript explorer for the Mandelbrot set among other things.

JavaScript has a built-in BigInt type. This allows performing math operations on integers of arbitrary size, but there is no built-in BigInt square root function.

First I'll show my floating-point numbers that are built using of this BigInt type, and then I'll explain my square root algorithm.

Stripe Average Coloring

Very Plotter has been updated to version 0.9.0!

This release includes a few smaller nice things but the biggest new feature is Stripe Average Coloring (demonstrated in the image here).

Stripe Average Coloring is a completely different way to color the pixels of the Mandelbrot set than traditional iteration count coloring. It does a terrific job of adding beautiful detail to the areas that look empty, smooth, and featureless in iteration count images.

In this post we'll look at this coloring method and some example images created with it.

Cosmology Theory 2023

Papers published in February 2023 make the case for black holes potentially being the source of dark energy, where "dark energy" is the mysterious "force" responsible for the observed rapid expansion of the Universe. I will note right off the bat here that as of March 2023, Sean Carroll (and as he pointed out, the silence of other prominent physicists on this idea indicates they may share the same opinion) does not think this theory has a good chance of "being on the right track." With that being said, it's still fun to think about.

But if indeed black holes are the (only) source of dark energy, then, in the distant future, once all black holes evaporate, there may not be any influence of dark energy remaining in the Universe. At that point, the Universe may collapse.

Taking that possibility, and combining it with some other theories, chief among them being Roger Penrose's Conformal cyclic cosmology theory, creates a sensible overall theory for a cyclic Universe.

Orion Nebula and Siril

While I was out photographing comet C/2022 E3 (ZTF) I also took some untracked shots of the Orion Nebula. I've never had any success in photographing a deep sky object before so I'm glad with how it turned out.