So yesterday I tried out something I wondered for quite a while...

I took the Singular Value Decompoition of an aribitary 2D vector M={{x},{y}}, squared the results in a nearly trivial manner and multiplied it all together.

What I discovered, was exactly what I hoped for:

M²=U²Sigma²transpose(V)²,

where U and V in general are rotation-style matrices and Sigma is the Matrix of Singular values. It behaves a lot like a radius.

In this special case, V = 1 and Sigma = {sqrt(a^2+b^2),0,0}.

squaring a square matrix, which is the case for both U and V, is always defined. Sigma will always be as close to a diagonal matrix as possible, which also is defined: Simply square all the numbers.

Well, doing so for said 2D vector above, you actually get {{x²-y²},{2xy}}, which perfectly corresponds to complex number multiplication.

Now, what would stop anybody to do the very same thing for a 3D vector?

Nothing, luckily.

So I have a more or less natural extension of the Mset to 3D or to nD in general

What I got for 3D after svd, squaring and simplifying, is this:

x->x²-(y*z*(x+sqrt(x²+y²+z²)))/sqrt(x²+z²)

y->x*y+z*sqrt(x²+z²)

z->x*z+(y*(z²+x*sqrt(x²+y²+z²)))/sqrt(x²+z²)

now, just add the constants a,b,c and you should, in theory, get an Mset as close as possible to a "true" 3D version.

At least as far as vector stuff goes.

The Singular Value Decomposition is so to speak the "ideal" decomposition of any Matrix where all the information can be found: rotation+translation, scaling, back-rotation+translation

Using it should make it possible for a wide range of problems to be "propely" extended, as far as this even exists.

I already experimented a bit with this SV3D MSet and did an xy-slice buddhabrot.

Evident from that is, that this is NOT a extension in the sense of simply adding a dimension to the Mset: the slice has the same basic body vs. head structure but besides that doesn't look anywhere close to the basic Mset usually seen.

It's rather a port to 3D, based on directly perform a "double angle" operation in a 3D-vector sense.

How exactly this angular conversion works in this case, I'm not sure about.

I'll soon post the xy plane and after that start to render the xz plane. What I can see so far is, that it is symmetric to the y-axis....

I'm also curious what happens when looking at the full 3D object... How much whipped cream vs how much "actual detail"...

By now I'm kinda certain that the "true" nD Mset with n>2 will always include whipped cream. I guess, it's partially due to the whole thing being a kind of rotation. So I wouldn't be surprised if the above transform results in that. Still I'd love to see the results or how it works together with other variants. - The various hybrids look awesome and adding another "template" to play with and cross-section with other sets could give new interesting patterns...

xy plane (edited and updated)

xz plane