My Mandelbrot Set Tattoo

I did it, at last :) And despite the misgivings of some, it came out beautifully. What do you think? :

Mandelbrot Set Tattoo

I'm extremely happy with it :) ... I intend on getting it worked on further, when I'm able to afford it. "The possibilities are infinite" :P ... Well, as infinite as the skill of the tattoo artist allows. I worked on an idea and showed it to the guy who did mine before he started, and he seemed quite apprehensive, frowning, and saying that it would be impossible to get 'that effect'.

Here's what I'm talking about:

I mean, yes, I understand it would be a difficult process, but surely it's not impossible?

Anyway, I'm reconsidering this even now, though, hehe. As cool as having a fractal spine could be, I'm now thinking of keeping things simpler... I'll work on a few designs and post them later.

So why did I get a tattoo of the Mandelbrot? ... Well, the way I see it, it represents the essential, underlying nature of the universe. That's why. ... Chaos. Infinity. The hidden language (and beauty) of mathematics that exists not only in our Euclidean man-made world of technology and architecture, but also (as fractal geometry is proving to us) everywhere in the natural world of trees, rivers, organic bodies and galaxies.

Lastly, my apologies for not posting anything at all for 6 weeks! ... Life, you know... I'll get to Mandelbulb 3D again soon. There has been quite a nice update recently, I noticed :) .. Cheers Jesse.

Fractal eXtreme: The NovaM Set

I cannot begin to tell you how awesome this fractal set is... "The possibilities are infinite", as they say on Fractal Forums, and in the case of NovaM, this is truly the case. As with all the other sets I've shown you since the original Mandelbrot, I won't go into the maths of this one because honestly, even though I am more than capable of understanding it all if I wanted to, I'm just too fascinated with the colourful results to care. From what I remember from reading about it though, I can tell you that it is based on the Newton Set formula, but it adds a constant (either a real or complex number) after each iteration. This constant is what Fractal eXtreme allows you to vary, through the Plug-in Setup option.

And not only does it have that variable constant, it also has a Julia form... Are you starting to imagine what possibilities we have here? Well I'm gonna show you. First, take a look at the fascinating creature that loads as the NovaM default:

NovaM x: 3

Isn't it gorgeous? As the caption states, if you go have a look at its Plug-in Setup, you'll find x is 3, and y, 0.

From here, you have a multitude of options. You could simply start zooming in. Or, using Plug-in Setup, you could start changing that x value. (Or the y value, or both!) Just with this feature alone you can yield thousands of amazing Fractals to explore, yet it's only the beginning... For every change you make to those x and y values, you are able to generate an array of Julias and of course you're able to zoom into those as well(!).

Do you see how I spent hours playing with this Set last night?

Fractal eXtreme: Creating different kinds of Mandelbrot Sets.

It's been a few days since my last post because honestly, after understanding the basics behind what generates a fractal, especially the Mandelbrot, the next inevitable step for me was to download as many different Fractal programs as I could and start experimenting :) ... It has been a virtual mushroom trip, to say the least.

For now though, let me stick to Fractal eXtreme. Such a nifty little program! So much more to it than one initially thinks... You've probably played around with it a bit yourself already but for the sake of being complete, I'll start at the beginning.


The first obvious thing is that you need to do is choose a Set when the program opens. It's default is the standard and much loved Mandelbrot set, but you can choose from many others.

Listed below the Mandelbrot are more Mandelbrots using different powers in their formulas. As it explains in the program, the higher the exponent, the more nodes the Mandelbrot has (always one less node than the power).

There's also an option called Mandelbrot Arbitrary Power, which is a lot of fun. You know that the normal Mandelbrot set has the function f(z)=z^2 + c behind it. Well, with the Arbitrary Set, you can set the exponent to any real number you want. The resulting fractals can be out of this world.

Then, just when you thought the Arbitrary Power was cool, along comes: The Mandelbrot Complex Power ... That's right: z^(some complex number) + c ... Instead of jading you to the adjectives 'incredible' and 'amazing', let me show you. Examples to follow of selected Mandelbrots of which I've spoken about so far.

Mandelbrot normal exponent changes :

Standard Mandelbrot Set

Mandelbrot^3 [ f(z)=z^3+c ]

Mandelbrot^8 [ f(z)=z^8 + c ]

Arbitrary Power changes:

Mandelbrot^3.5

Mandelbrot^2.5

Mandelbrot^1.7

Complex Power changes:

Mandelbrot^(8,1.73i)

Mandelbrot^(3.1,2.5i)

Mandelbrot^(2.08,0.36i)

One thing you'll notice with making changes to the exponent in these ways is that, the higher the exponent, the longer it takes for the program to render a good-looking image, especially the more you zoom in. But you don't need to zoom in very far to discover really beautiful fractals. Go ahead and try some of the different Mandelbrots, experiment with colours, etc. To change the Arbitrary and Complex powers once you've loaded the default, you need to go to Options > Plug-in Setup.

And there you have it :) Hope you're having fun :) ... Fractal eXtreme has a few other very interesting options for creating new Fractals (Auto Quadratic, the "Hidden Mandelbrot", Barnsley 1, 2 and 3, Classic and Complex Newton, and Nova/NovaM), but those I'll show you in the next post.

Fractal eXtreme: Exploring the Mandelbrot set and Changing Colours

Ok, from here on out, I'm going to keep the maths down to a minimum :)

For the past few days, I've been playing around with Fractal eXtreme, often getting lost for hours within its beautiful depths. Now it's time to share with you all that I've learnt about it and the Mandelbrot set.

First, for all who have landed on this post first, and don't wish to go back and read all the maths explanations, here's what the Mandelbrot set looks like in Fractal eXtreme:

Really beautiful colour variations, no? And it's very easy to play around with the colours: Just go to Options > Colour Mapping, or Colour Palette.



Colour Mapping is for big, general changes. You'll see that changing the 'offset' is most effective, and adjusting the 'speed' can make things very interesting within a certain range.

The Mandelbrot Set

The Mandelbrot Set

 Ok, on we go to the fractal design that probably led you here in the first place... The famous Mandelbrot set.

So you know now that there are as many Julia sets as there are complex numbers (an infinite amount), and each set is generated according to the results of iterating the function f(x) = x² + c ... So in other words, for each and every Julia set, there is a corresponding formula, each having a unique value of c (either a real, imaginary, or complex number).

Did you play around with Fractal eXtreme as I suggested in the previous post? Really cool, right? But I bet you felt a bit restricted in terms of which numbers generated the coolest Julia sets. You may have noticed that entering anything under -2 in the X field didn't do anything too interesting. And the same for above 1. And in the Y field, you're were even more restricted. Anything above 1.4 or so, and under -1.4, hardly produced anything worth calling a fractal. This is because, as I explained in the previous post, Julia sets are coloured according to how fast all the points on their complex plane map out or "escape" to infinity. So when your c value (real,imaginary or complex number) is too big, obviously every point mapped with our function f(x) = x² + c escapes to infinity very very quickly, and thus, the image is coloured mostly monochromatic, (blue in the case of Fractal eXtreme's default colour scheme).

Ok, to continue... Take a look at these Julia set examples:

f(z)= z² + (0.3,0.01i)

f(z)= z² + (0.18,0.7i)

f(z)= z² + (-1.36,-0.05i)

f(z)= z² + (0.3,0.25i)

f(z)= z² + (-1.3,0i)

f(z)= z² + (-0.71,-0.17i)