I did the Brot

[Ags]

I was toying with the HTML <canvas> element and finally got round to drawing a Mandlebrot set. I wanted to make it infinitely zoomable, but I discovered after zooming a million times or so, Javascript runs out of precision. I need a bigger number!

[badhex]

 

[Gfamily]

Very neat.

As you may know, the Mandelbrot set is the map of all connected Julia Sets.

Back when I started BASIC programming, I developed a MB generator, and then developed a secondary function to plot the Julia Set that corresponded to any particular point on the MB set.

It was interesting to see how the overall shape of the JS depended on where the starting point was on the MB set. After a while you could tell where to start if you wanted JSs that had 2 main branches, or 3 branches, or 4 or more.

Some interesting findings.

If I was a born mathematician I'd probably have been able to make something of it - though more likely, I'd have found it was a well known thing in the field.

Now I'm retired I must try to recreate the programme. 

Edited June 18 by Gfamily

[Ags]

9 minutes ago, Gfamily said:

As you may know, the Mandelbrot set is the map of all connected Julia Sets.

All I know is its maths, its infinitely complex, and it's on my screen 🫠

This is actually stage zero of a web based image processing idea I have. Now that I can draw Mandelbrot, I can draw the contents of a FITS file...

 

[Ags]

15 minutes ago, Gfamily said:

As you may know, the Mandelbrot set is the map of all connected Julia Sets.

Thinking about it, how can sets be "connected"? If two sets are connected aren't they just one set?

[Gfamily]

2 minutes ago, Ags said:

Thinking about it, how can sets be "connected"? If two sets are connected aren't they just one set?

A Julia Set is a set of points on a plane surface. If every point in the set  is adjacent to another point in the set, so they form a single area on the surface, rather than having separate islands, it's said to be connected.

[Ags]

How can this be mathematics? It seems organic.

[vlaiv]

Mandelbrot set is best understood as:

Set of all points c in complex plane that have following property:

Let z_sub_n = z_sub_n-1^2 + c

which is iterative formula. If we start with z_0 = 0 - we never leave unit circle - or module of any such iteration stays below 1 (up to infinity).

In Mandelbrot set - c has coordinates of point we want to examine - if it belongs to set or not.

Julia set on the other hand has the same exact iterative formula

z_sub_n = z_sub_n-1^2 + c

and it is used to define a set of points on complex plane by the same criteria - except, here c is constant so we can have say Julia set for c=0.2 or c=0.5-0.3i (any complex number), but starting z or z_0 is coordinate of point we examine.

These sets are related and have similar fractal properties - but different thing is used to determine if point belongs to set or not. There is single Mandelbrot set, but there are infinitely many Julia sets - one for each constant c.

[Ags]

It's life Jim, but not as we know it.