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If I understood well, a 4D rotation is around a plane and not an axis, right?

My first question is: which 4D plane is perpendicular to XY, XZ and YZ planes?

And the second is: how I can accomplish a 4D rotation of the 3D space around this plane? (preferably in GLSL code)

Thanks in advance,

ZW, YW and XW respectively

The easiest way to do a rotation is to flip around two different normals. A flip is: point -= 2*normal*dot(normal, point);

In 3d you can rotate in a plane by choosing any two normals in that plane, the same goes for 4d and any larger number of dimensions.  :)

Sorry if I'm saying nonsense, my mind is not fully 4D-operational yet ;D, but... shouldn't exist a plane in 4D space that is perpendicular to all the mentioned planes at the same time?

My idea is the following: in 2D, as is commonly known, for any vector Z, we can define (Z*Z) by squaring the vector's length, and doing a rotation (doubling the angle). Besides is a 2D rotation, we can see it as a 3D rotation around the Z-axis (the rotation vector 0,0,1), so the result is always on the XY plane. What I was thinking of, is to translate this to 3D space and 4D rotation. My first thought was of taking W-axis as the rotation vector, as it is perpendicular to the other axis that defines 3D space, but don't knowing certainly which angle to measure for doubling it... and also then I read something about 4D rotations and they are around a plane, not only an axis, so which rotation plane makes the result of a 4D rotation always within XYZ 3D space? (W=0)

It was pure intuitive thinking and speculation, I don't fully undertand 4D rotations yet, but does this even make some sense at all?

No I don't think so, there exists an axis though which is perpendicular to all three... the W axis. (or whatever letter you want to give it)

It might be helpful to think of all rotations (including 2d and 3d ones) as rotating in a plane, rather than around an axis. That way the concept of rotation doesn't change as you increase the number of dimensions past 3d.

Excuse me, can someone show an image of this W axis and his relationship with the threedimensional space?
In mathematical terms I´m unable to imagine such an axis, but geometry is another thing, if I can draw it I can understand it.

Errmmm -  given that we live in 3D (spatially) there's no way to instantaneously visualize a 4th dimension.
I guess with 3D printers one could now compute a projected 3D version of a 4D object though ;)
Personally past 3D I think more in terms of matrices/arrays than in geometrical terms !!

Edit: An addendum to my first sentence is "unless you're an autistic savant".

A great site for discussing 4D visualization is: http://eusebeia.dyndns.org/4d/vis/vis

It also discusses 4D rotations: http://eusebeia.dyndns.org/4d/vis/10-rot-1 - as Tglad says, the important thing is to think of rotations in terms of planes, not axis.

Okay, I'm on shaky ground here, but I'm gonna put in my tuppence anyway...

Seems to me that there is an infinite amount of planes perpendicular to each of the planes you mention. Wouldn't it be the same as a line in 3D space, where a line has an infinite amount of lines perpendicular to it in a certain point on that line? On the other hand, there is just one plane perpendicular to that line in that same point. So if we extrapolate to 4D, I would say that for a certain line in a plane, there's only one space that's perpendicular to that plane, but an inifinite amount of planes.

I'll leave it to the experts to make mincemeat of my statement...  :sad1:

     Since I can´t follow the math reasonements, and have no idea about programming, I feel ready to talk,  :hmh:

     As I understand it, a plane has 2D and rotates in a 3D space, as far as we know, this is our world, and after taking a look at some of the links, I´ve only seen threedimensional images, drawings and ideas, the only 4D implied is the time used by the plane to rotate, time being the fourth dimension .
      
      I think an human observer would always perceive three dimensions, imagine a dimensional scale with hundreds of dimensions ,an observer, moving up and down in this scale would be like a man looking trough a narrow window in an elevator, only seeing three levels. no matter how high or down in the scale.(This scale exists, it´s the vibrational scale ranging from infrared to ultraviolet in our perception, and far more out of it, since infrared and ultraviolet are just our limits.).

      Limits would be in our perception, but without those limits we would be unable to understand a multidimensional fractal universe, just like when we recognize a pattern in a fractal exploration, and we see, "a garden", " a building" etc.

      I said this all to clarify my question, because to each human being, Geometry can be understood, but maths are just for some special people.