Mandelbulb 3D Tutorial: Different 3D Fractal Formulas and Hybrids

... or "formulae", to be truly correct ;)

Up until now, you've been working with the formula that loads on default when you open Mandelbulb 3D, that is to say, the original one:

Formula Tab Window

But just like 2D fractals, the possibilities in the 3D realm are also infinite.

The challenge is to find formulae that are aesthetically pleasing as well. Of course, actually coming up with mathematical formulae is beyond me (for now, I tell you!). Lucky for us, Jesse built in a Formulae tab in which he has programmed many, many different formulae, from different sources:

As you can see, you can use up to 6 formulae. What this means is that you can combine different mathematical formulae to create a new, unique 3D fractal. But before I get to that, let me show you a few examples of single formulas. 

As is visible above, the default formula that loads for the default 3D Mandelbulb is "Integer Power". Notice that there is a little black dot inside the "Formula 1" tab. This means that it is active..

To choose a new formula, you simply drag the mouse over one of the buttons (3D, 3Da, 4D, 4Da or one of the adds) and choose an option. Note that any formula name that begins with an underscore (eg. _ptree_tess) is an add-on only, and won't do anything if you load it by itself. They are meant only as modifiers to actual formulas. Ok, here are a few examples of interesting looking formulae (in some cases I rotated them to show them off better, click to enlarge):

Beth1522

Riemann

GeneralQuat

MagVsXYZabs3

Ikenagabulb

benesi1pow2

ABoxVaryScale

Aexion1

There are of course many more. But these are just the beginning, for several reasons: First, as I've already said, you can make a hybrid fractal using two or more of these formulae. To do this, just go to the next Formula tab in line (eg. "Fo.2") and choose a different one, and then click on "Calculate 3D" to see what you get. In some cases you may have to zoom in or out. Here are some examples of hybridising formulae:

Mandelbulb with ABoxVaryScale

ABoxVaryScale with Beth1522

CommQuat with IQ-bulb

Beth1522 with CommQuat

The thing to remember is, just because you get a pixelated noisy mess when first trying a hybrid does not mean that it is worthless. Explore it a bit ;) ... Some really are just noise everywhere, but with some, if you play around in the navigation window, treasure mines of beautiful 3D fractals can be found. One such example, I have found, is combining "Bulbox" with the add-on "_AmazingBox" .. This is what you get:

But from here, I zoomed in. And I found what I call, 35th Century Earth ;) ... Apologies for the time it's taken to get to this Tut page. Lots happening in my life right now.

If my tutorial is helping you, please consider supporting me over at my Patreon page.
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Go to the next Tutorial page - Mandelbulb 3D Tutorial: Cutting

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: The Hidden Mandelbrot Set

Ok, so here's one you can spend hours on, despite its rough, untidy appearance upon first loading:

Firstly, it isn't called the "Hidden Mandelbrot" for nothing. If you start zooming into the more interesting (less noisy) areas, you'll find you'll discover a few familar shapes :) (And yes, some that are not so familiar)... Have a look:

Really nice, right? :) Now, while there is no Plug-in Setup option for the Hidden Mandelbrot, you can generate Julias with it. Here are some good examples, with x/y locations:

Fractal eXtreme: The Auto Quadratic Set

This set is strange and unique. Upon first loading, you get this stretchy design:

 

Interesting, no? A swirly, stretchy monster. And indeed, if you're looking for any kind of swirling galaxy fractal, zooming into this Auto Quadratic is ideal, as these examples can attest:

But now, what other options are there? Julias? No, unfortunately not. But if you got to Options > Plug-in Setup you will find a "randomise" button. And when have you ever encountered a randomise button and not thought, Ooooh, I wonder what this does... And indeed, upon clicking on it, I discovered a myriad of different shapes and forms. In fact, I had a hard time forcing myself to stop clicking.

Fractal eXtreme: The Complex Newton Set

I'm excited about the next few posts because the fractals are getting really interesting and beautiful, and so much more fun to blog about. You'll now see a few perfect examples of why I got so into this fractal art thing in the first place. Also, after posting the next three (or so) posts, I'm going to really start making my own artistic pieces, as well as make some changes to this blog in order to showcase my future creations :)

But for this post, let me stick to the title: The Complex Newton Set. As you can see here, upon loading the default, it looks very similar to the Classic Newton:

The Complex Newton Set

But it has two more "spokes", and thus all the smaller self-similar iterations are different as well. Ok, cool, but now, what's nice about the Complex Newton is that you can change it's exponent. It can be either any real number (essentially, the shape cycles between having many more spokes to being a very slow-loading circle/eye-shaped thingy... ) or any complex number (much more interesting, as can be seen here):

Fractal eXtreme: The Barnsley Sets

In my previous post I said I would now go into all the other kinds of fractal sets Fractal eXtreme has available.  (Auto Quadratic, the "Hidden Mandelbrot", Barnsley 1, 2 and 3, Classic and Complex Newton, and Nova/NovaM).

I've since played around with those various sets and believe me, some of them are mind-blowing. The thing is, I can't fit them all in, in one post. So I'll start with the less interesting Barnsleys. Actually, before I start with them, let me just cover one that really doesn't merit much attention: The Classic Newton. This is what it looks like when it loads:

Actually I think I changed the colours to make it a bit more exciting, hehe. And sure, it doesn't look so bad, you might say. But here's the thing: When you zoom in, there's only more of the same. Truly, what you see above you is all there is. I got hopeful when I noticed it had a "Duplicate as Julia" option (which you may have noticed some of the others don't have) but surprise!, no matter where you Alt-click on the original, the Julia looks exactly the same! And thus, these are the last few seconds I will spend on this "fractal". (Ok, just a few more to give it at least this small tribute: It is the first step towards the Complex Newton, a much more interesting cousin).

Ok, now for the Barnsleys! Once again, I'll start with the more boring option: Poor old Barnsley 2. Upon loading, this it what you get:

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.