Zooming the Mandelbrot Set

[dazzledbylight]

The incredible beauty and wonder of the Mandelbrot Set

This is here on the off chance that some I-lander loves this stuff, maybe
had a chance to do their own zooming!

Seems the only active general LS thread is theStraya one...
so c'mon Aussies: any of you, or a friend ever wax poetic about this?

I'm approaching this from a Visual Artist POV, though I understand a bit
of the math. I've just been looking up how to assign colors to get the
beautiful color effects

OTOH if you've never heard of this but this peaks your curiosity just search for
it using the thread title.

 

[Kieran McConville]

Don't leave us hanging, just tell us what the Mandelbrot Set is! I know one could look it up, but one could also get you to tell us what it is!

 

[dazzledbylight]

Oh... ya think? Ooooops!!!

Commencing explanation...

Ok. Remember that graph paper in (?)middle school for plotting equations, or the equilavent on a computer, tablet screen?
Horizontal line usually called x, vertical axis called y. Usually looks like a plus sign - same amount of numbers set up on each line from center point. The numbers on the righthand and upward sides are your typical 1,2,3,4,5, etc.. But don't the leftward and downward sides they are the imaginary numbers -1,-2,-3,4,-5 etc.

The Mandelbrot set is so far the most conplex mathematical construction ever created.
It uses a repetitive (iteration) formula where the answer you get becomes a part of the next repetition of the equation.

It wasn't actually discovered until the mathematician Benoit Mandelbrot was able to plot it using massive amounts (then) of computing power in a digital graph. He was taught by a mathematician named Gaston Julia who discovered a series of smaller but still very complex sets (named after him "Julia Sets")

When the results fa

 

[dazzledbylight]

Con't...

While less complex, he was only able to make crude drawings of what he could completely express formulaically. Again it wasn't until powerful enough graphical computing power existed to created pictures the Julia Sets.
(It turns out these Julia Sets reside around this Mandelbrot Set)

The M-set (abbreviation) itself is at the center of the graph. It is a solid complex shape usually colored black. But what is around it (the boundary) is what people swoon over.
The m-set itself occupies the graph area from only 2 to -2.

So you are going through these iterations of this equation. When the answer stays small or falls to zero it is part of that solid black central figure. When the numbers get larger and larger the answer is said to diverge appearing in assigned colors out side of the set.

The bigger numbers can go to infinity. What people coloring it do is assign a color for how long it takes an iteration in the repetition at a certain point ( each pixel on the screen [goes through the iteration pricess of the formula] on the computer screen) zoom off towards infinity.

When I/we/they say "zooming the m-set it means zooming into the colorful area (the boundary) around the met-set.
I know that a bit to take in, but knowing this here's where the (visual) Magic starts because this small math constructed image at first glance "hides" a whole "world".

got to run an errand... Be back for the rest! Sorry! in case anyone is reading this now.
( oh, the crowd of onlookers, not! )

 

[dazzledbylight]

Con't 3 (hopefully finale)

You're probably familiar with the word "fractal". It's a shape that is complex, detailed in the same way at any scale. If it's exactly the same it's called " self-similar", if rather close but not exactly the same then it's partly self-similar. Look up (if you want to) Koch snowflake a full fractal. A partial one (early description) would be looking at he coast of Britain. You see an irregular coastline. You zoom in and see a irregular beach line. Further still you see the irregular line the pebbles at the edge of the waterline. Detail at every scale, instead of smoothing out at a closer and closer range.

In fact CGI is based upon fractals. It was a way to render the natural world more realistically, digitally. We all have seen it get better and better in the past 25 years.

The M-set is more horizontal. Imagine a circle on it's right "side" with a deep cleft at the 3 o'clock position. Then on it's left side just touching is a circle about 50% smaller, it's center is on the same horizontal line/plane at the clefts "v" point. Each of these 2 circles are studded with irregularly circle shaped "buds". The leftward most one one the smaller left circle from that on the same horizontal plane is a thine line with tiny projections.

On the big circle in 12 & 6 o'clock positions are the same (a bit bigger) buds as on the left side - except that there is a wavvy " y" (one line shorter than the other) instead of a straight line. It also has little projections.
Finally you will see descending and ascending on either side of these buds spaced apaet similar buds getting smaller and smaller.

Picking a set of colors we'll go from white to turquoise to blue to midnight navy.
Now we will start to zoom in. We zoom in on that 12 o'clock bud with the "y" on the longer part. Hmm... you see there appeares to be a black shape inside the brighter colors. Intrigued you zoom in towards that and.....


there embedded in the brighter colors is that same shape, oh a bit different but very recognizable- only (about) 150% smaller han he big circle.

 

[dazzledbylight]

Con't 4
A complex subject deserves a complex answer! This should be it!
(Maybe it's 90% smaller, I don't have my proportion wheel to figure it)

Then you begin to notice that in these wavy lines extending out from these even tinier buds of this mini-brot, are those MORE little blacks "spots" embedded in those lines? You zoom closer and yes.. yeah, there are. The more you zoom in the more you see of them. Only now the lines around them begin to form more complex shapes and repetitions. Ornate lace- like spirals, wild triangles some built out of intricate interior patterns, studded curves like scythes, Julia Sets like tethered floating islands, winding complex ribbons, similar and different shapes appearing multiplied and multiplied and on an and on...

At a certain depth the original Mandelbrot Set would be the size of our solar system, zooming in more... the size of our galaxy, to our...
Universe...

They say we can't truely see the complete m-set because it would take infinite computing power, but like π (pi) we can get closet and closer. All the while being astonished at the unfolding beauty that touches the edge of Infinity.

Addendum 1: the straight leftward line comes out of the leftward most bud, the irregular "y" from the 12 & 3 o'clock buds. Didn't seem quite clear on re-reading.

Addendum 2:
The way I described zooming into the M-Set ( though not as deep as I described it towards the end- that's what I saw years later on the Web) that's is what I experienced back in (?) 1986 at a special IBM computer art exhibition where I got to explore it with my own eyes.

This wonder awaits you on-line, and if so inclined you can get the programs to set it up on your own computer ( but you do need to wade through some math in certain ones).

 

[Mack_Again]

i probably should study complex analysis at some point.

 

[dazzledbylight]

I did the best I could

 

[Kieran McConville]

Thank you for the explanation, I've been having a busy week, but will take it on board.

See, I was afraid the Mandelbrot Set was going to turn out to be a particularly decadent upper-class social clique from the interwar period. As in 'Tom Ripley, you simply must meet the Mandelbrot Set. They're a riot!"

 

[dazzledbylight]

Or like "after I finish my polo game, I'll catch up with the Mandelbrot Set this eeeeeveng!"

Glad you got through the explaination.
( actually turned out to be a good brain exercise for me )