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Ìf the complex plane is a plane - is there also a complex space? or a complex 4d-'space'?

The quaternions :D

...and in 8d are the octonions. Both are complete fields, fields means they have a * and + operator between points. Complete means there are no points missing from the space.

However neither have a squared operation that is conformal.

You can also create a space that is 4d because it is just two complex planes, but there isn't anything special about this, and it isn't conformal either, in squaring.

It really depends on what properties you are hoping for. Another extension would be Geometric Algebra. If you want a LOT of accumulated information about that, you can check out this thread:
http://www.fractalforums.com/new-theories-and-research/geometric-algebra-geometric-calculus/
or alternatively google it.

uh.. this is becoming really interesting. :)
I didn't realize that this is what quaternions are and only knew them as 'just another formula in mandelbulb3d'

thanks for introducing us ;)

if anyone else is interested:
https://www.youtube.com/watch?v=3BR8tK-LuB0

not sure if I misunderstood the concept, but in the numberphie video it is said that you need quaternions to rotate in 3 dimensions. and to rotate in 4 dimensions you need octonions.
the usage of the word dimension seems to confuse me here a bit..

Interesting question. Yes, quaternions are used for rotating in 3 dimensions. So the orientation of some object like your tea cup can be written as a quaternion. e.g. written as w,x,y,z: 1,0,0,0 is aligned to whatever world coordinate axes you are using, and 0,1,0,0 is rotated 180 degrees around the x axis. 0.707,0.707,0,0 is rotated 90 degrees around the x axis.
Even though there are 4 numbers, they represent 3D rotations because you only use unit-length quaternions... otherwise the quaternion represents an orientation and a scale.
Also, for every quaternion its negative (negative in each component) represents the same orientation. The reason for this is profound and requires quite some explanation.

Regarding octonions, I have never heard of any real physical quantity that they represent... they don't represent 4D rotations as you can do that with two complex numbers. The best I have found is a physical interpretation of split-octonions, which relates to balls rotating on balls (J Baez).

I've read about octonions, describing 8 dimensional super symmetry in string theories. So if you see that as physics (I'm not sure, if I do), octonions have a physical representation or interpretation.

Just to mention quaternion fractals are much longer known, as those new generation fractals leading to the development of mandelbulb3d and others. Mostly, their 3d display where not seen as very interesting. the m-set is a simple rotation body and the j-sets do not show much "fractality" and they lack of the diversity, we know from other fractal types.

I think that this is only due our limited perspective. We can only see a single layer of the onion at a time.
Stardust explained this well in a comment here (https://www.fractal.institute/buddhabrot/):
"Deep zooms create intricate patterns in a 2D plane which is visible to the eye. A deep zoom of the 3D Mandelbulb or BS fractal will have similar layers, but our perspective in 3D lies within the fractal rather than outside it. The mandelbulbs generally use distance estimation rather than iteration depth, but we are only scraping the outer surface of the fractal. It is like looking at an intricate 2D render with fixed bailout and no coloring"

Apparently you can associate Octonions with two balls, one with trice the radius of the other, the smaller ball rolling on the surface of the larger one. Kind of obscure a connection, but interesting nonetheless:
http://math.ucr.edu/home/baez/octonions/

Quarternions are kind of like the bi-vector part plus the scalar part of 3D Geometric Algebra. (kind of, because I think the sign convention is a little different, but the whole thing behaves essentially the same.)

In Geometric Algebra, you basically get all the combinatoric variations for sections in your space. For 3D space you get a total of 2³=8:

• 1   0D component   (scalar)
• 3   1D components   (the spacial directions)
• 3   2D components   (the three perpendicular planes that can be built from them)
• 1   3D component   (representing the volume of the space, also acting like an imaginary unit and a "dual" operator - it turns directions int planes and vice-versa, for instance

The three 2D plane-components are signed and can be put together to describe an arbitrary directed sine of a 3D spacial rotation. (This is also true for Quaternions: the ijk components are equivalents of the sine but for 3D spacial rotations.)
The scalar component, meanwhile, stores the rotation's cosine (also true for Quaternions)

(I am neglecting the size of the four components here: It's only true for Quaternions and Multivectors that are normalized. If they aren't normalized, you'll additionally scale up/down your points with 0 at the origin.)

You can see that these things are correct from how you can find the sine and cosine of the angle between two vectors in your usual tried and true vector algebra:




The difference between the Quarternion- and the GA- convention are simply whether it's that formula for the sine or there should be an extra negative sign. (Cross products anti-commute so changing the order of arguments changes the sign. This is a generalization of the fact that the sine is an anti-symmetric function: Plugging in a negative angle is the same as plugging in the positive one and changing the sign.)
The reason the cosine is unaffected by this is simply because the dot-product commutes - changing the order of arguments changes nothing at all (this, in turn, is a generalization of the cosine being symmetric: plugging in a negative angle changes nothing at all)
The end-effect this has on a given rotation simply is that one turns things clock-wise, the other counter-clockwise. - I can never remember which does which though. It's a trivial change anyway.

I personally really like Geometric Algebra for its clarity. I never got these facts about Quaternions before I learned them through a Geometric Algebra lens. Quaternions are by far not as obscure as they seem once you understand this.

A 4D rotation can be expressed in quaternions:  P' = APB  (see wikipedia article: https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space)

In complex numbers you have exactly one plane, for quaternions you have a plane for every direction in 3D space.  So historic notation of a complex number:

z = a + bi

where 'i' is the only possible plane direction (historically i² = -1) translates into quaternions as:

Q = q + bu

where 'u' is any of the possible set of unit bivector directions in 3D space (u²=-1).  In complex number bi can be represented by a single scalar, but a quaternion bu requires three and so a quaternion requires four scalars.  The subset of quaternions that can be expressed as above (scalars a, b and unit bivector u) are a commutative sub-algebra isomorphic to complex numbers (or simple "a complex plane).

I don't have a working knowledge of GA, but yeah quaternions are a scalar plus a bivector.