Show HN: Beyond Z²+C, Plot Any Fractal

#1

Post by akunzler » Tue, Jul 15, 2025, 6:24 PM UTC

I've always been dissatisfied that simple Mandelbrot explorers proport themselves as a Fractal Graphing Calculator. In summer break between semesters, I started making a real graphing calculator, parsing LaTeX to WebGL to let you graph most any combination of z and c.

Fun ones to try include - sin(z^2+c) - c^z - z^{1.7}+c

Also supports animation, just enter any other letter and turn it into a variable. Supports Mandelbrot or Julia Set style calculation.

Use with a graphics card or integrated graphics

Show HN: Beyond Z²+C, Plot Any Fractal
juliascope.com

Re: Show HN: Beyond Z²+C, Plot Any Fractal

#2

Post by mg » Tue, Jul 15, 2025, 6:39 PM UTC

I'm not sure if every fractal can be expressed as an iterative formula f(z,c).

In 2012 I found a fractal by using a fundamentally different approach. It arises when you colorize the complex plane by giving each pixel a grey value that corresponds to the percentage of gaussian integers that it can divide:

https://www.gibney.org/does_anybody_know_this_fractal

Re: Show HN: Beyond Z²+C, Plot Any Fractal

#3

Post by OgsyedIE » Tue, Jul 15, 2025, 6:42 PM UTC

This was a great nostalgia trip to my days on fractalforums, before the web got much denser. I tried playing around with the settings but I was unable to reproduce the two-dimensional version of Tom Lowe's Mandelbox map, discovered in 2010:

https://sites.google.com/site/mandelbox/what-is-a-mandelbox

There are galleries on the other pages of the site, if anybody is interested.

Re: Show HN: Beyond Z²+C, Plot Any Fractal

#4

Post by akunzler » Tue, Jul 15, 2025, 6:54 PM UTC

mg wrote: post #2

I'm not sure if every fractal can be expressed as an iterative formula f(z,c). In 2012 I found a fractal by using a fundamentally different approach. It arises when you colorize the complex plane by giving each pixel a grey value that corresponds to the percentage of gaussian integers that it can divide: https://www.gibney.org/does_anybody_know_this_fractal

Good point, this site then supports every (as far as I know) fractal you make with iterations of complex numbers and constant cutoff values, mandelbrot style.

There are surely infinitely many more ways to generate other families of fractals though

Re: Show HN: Beyond Z²+C, Plot Any Fractal

#5

Post by akomtu » Tue, Jul 15, 2025, 7:13 PM UTC

mg wrote: post #2

I'm not sure if every fractal can be expressed as an iterative formula f(z,c). In 2012 I found a fractal by using a fundamentally different approach. It arises when you colorize the complex plane by giving each pixel a grey value that corresponds to the percentage of gaussian integers that it can divide: https://www.gibney.org/does_anybody_know_this_fractal

What's the heck is gaussian integers? I've tried to parse your article, but still confused.

Re: Show HN: Beyond Z²+C, Plot Any Fractal

#6

Post by Sharlin » Tue, Jul 15, 2025, 7:16 PM UTC

akomtu wrote: post #5

mg wrote: post #2

I'm not sure if every fractal can be expressed as an iterative formula f(z,c). In 2012 I found a fractal by using a fundamentally different approach. It arises when you colorize the complex plane by giving each pixel a grey value that corresponds to the percentage of gaussian integers that it can divide: https://www.gibney.org/does_anybody_know_this_fractal

What's the heck is gaussian integers? I've tried to parse your article, but still confused.

Simply the complex numbers where the real and imaginary parts are both integers. Eg. 0, 3+i, 123-45i, -7+8i. Same as the 2D grid of integer Cartesian coordinates.

Re: Show HN: Beyond Z²+C, Plot Any Fractal

#7

Post by ttoinou » Tue, Jul 15, 2025, 7:28 PM UTC

My favorite alternative to Mandelbrot is the Monkelbrot, I made this 13 years ago (probably I discovered this formula on the old fractalforums.com)

https://www.deviantart.com/titoinou/art/The-42-MonkelBrot-29...

  f(z) = ( (z*c-1)^2 - 1 )^2 - 1

It features Classic Quadratic Mandelbrots z^2 and also Quartic Brots z^4 in one set, that is apparently connected (I didn't prove this yet...). Also, it doesn't go crazy like others alternative, it stays nicely behaved like the original Mandelbrot set. You can copy paste "( (z*c-1)^2 - 1 )^2 - 1" without the quotes on this site to explore the fractal

It's really fascinating when navigating the fractal to try to understand where would a z^2 minibrot appear vs. where would a z^4 minibrot appear

Re: Show HN: Beyond Z²+C, Plot Any Fractal

#8

Post by mg » Tue, Jul 15, 2025, 7:30 PM UTC

akomtu wrote: post #5

mg wrote: post #2

I'm not sure if every fractal can be expressed as an iterative formula f(z,c). In 2012 I found a fractal by using a fundamentally different approach. It arises when you colorize the complex plane by giving each pixel a grey value that corresponds to the percentage of gaussian integers that it can divide: https://www.gibney.org/does_anybody_know_this_fractal

What's the heck is gaussian integers? I've tried to parse your article, but still confused.

You can think of them as the complex equivalents to normal integers.

Complex numbers have two components. If both are integers, the complex number is a Gaussian integer.

Re: Show HN: Beyond Z²+C, Plot Any Fractal

#9

Post by bmacho » Tue, Jul 15, 2025, 7:40 PM UTC

akomtu wrote: post #5

mg wrote: post #2

I'm not sure if every fractal can be expressed as an iterative formula f(z,c). In 2012 I found a fractal by using a fundamentally different approach. It arises when you colorize the complex plane by giving each pixel a grey value that corresponds to the percentage of gaussian integers that it can divide: https://www.gibney.org/does_anybody_know_this_fractal

What's the heck is gaussian integers? I've tried to parse your article, but still confused.

https://www.google.com/search?q=gaussian+integer

Re: Show HN: Beyond Z²+C, Plot Any Fractal

#10

Post by OgsyedIE » Tue, Jul 15, 2025, 7:46 PM UTC

ttoinou wrote: post #7

My favorite alternative to Mandelbrot is the Monkelbrot, I made this 13 years ago (probably I discovered this formula on the old fractalforums.com) https://www.deviantart.com/titoinou/art/The-42-MonkelBrot-29... f(z) = ( (z*c-1)^2 - 1 )^2 - 1 It features Classic Quadratic Mandelbrots z^2 and also Quartic Brots z^4 in one set, that is apparently connected (I didn't prove this yet...). Also, it doesn't go crazy like ot…

It is tragically the case that most of the archives of fractalforums are irretrievable and lost media. The archive.org copies are very incomplete and the database dumps, as far as my research last year could figure out, are locked behind a group of moderators of an inadequately programmed successor site who don't want to share them, considering the dumps to be a status moat for themselves.