Welcome to Fractal Forums

I joined FF some years back... I hadn't really gotten into it yet when one day I got a message that things were changing. When the email announcements from this group quit coming, I looked in to see what was happening and my impression was that it had gone dormant. Now, with Mandelbulbs flashing across the Web, I check in to discover that FF is alive and thriving... cool.

I didn't know the search for 3D sets was still on... I found two ways to generalize the M-set to 3-space some years ago
http://www.ixitol.com/3D%20Msets.jpg
If someone wants to explore these things, I'll post the formulas...

Ciao, Russell
http://www.ixitol.com/eCard.jpg

This is a trial attempt to attach an image...

hi there, this looks like extrusion, and lathing :angel1:

That's what you get when you apply a single rotational axis in 3D space (so your 2D "complex multiplication" happens in a flat plane).

Assume your point to iterate is called Z. I suppose in the extruded objects you must have chosen the same rotation axis for all points Z (e.g. a coordinate axis), in the lathed objects you chose an axis such that the rotation rotated Z within a plane defined by the origin, Z and the orthogonal projection of Z onto one coordinate axis.

When you add another rotation on a different axis, you will get more interesting objects. The Mandelbulb uses two rotations in a spherical coordinate system (one to rotate angle phi, one to rotate theta), followed by raising vector's length to a given power.

I've been experimenting with non-perpendicular axes recently, slightly different from the spherical coordinate system approach. The resulting shapes look like anything from living plancton to Donald Duck ;)

As it happened, my approach to generating 3D M-sets was from a different direction and it doesn't really involve rotations. In trying to find a way to rotate the M-set on it's axis of symmetry, I thought to build on the equation z = z^2 + c. Since the complex numbers are isomorphic to the cyclical group C4, I sought to construct a number system on C6, and after a while I was successful. I called these ordered triplets 'triternions' (T). Given real numbers a,b,c, I can write T = a + bj + ck, where j^3 = k^3 = jk = kj = 1. A surprise that sort of snuck up on me was to discover that j and k are neither real nor imaginary.

I typically rule the X axis with the real numbers, then Y,Z with j,k respectively and I call points on these axes x,y,z. Then, as I eventually figured out, the formula
a = b = c = 0
a1 = a^2 + 2*b*c + x
b1 = c^2 + 2*a*b + y
c1 = b^2 + 2*a*c + z
a = a1, b = b1, c = c1
where iteration continues while a^2 + b^2 + c^2 < 8

This generates (surprise again) the extruded set!

Not at all what I'd expected, but an interesting result nonetheless. Here are some of my early 2D zooms on this 'T-set' (below). Maybe certain of your techniques can be utilized to provide new looks at this object.

Greetings, and a belated Welcome to this particular Forum !!!      :D

These look very similar to some I created several years ago, but I was using a program called QuaSZ to render 3-D fractal objects.