Can we rotate 3D objects in 4D space?

[ramollari]

Suppose you have a piece of paper in the shape of a left shoe, on a flat surface. Suppose also that it can only stay in contact with the surface. No matter how you move it or rotate it in the 2D surface, it cannot become a right shoe. However, if you move it out of the 2D space, flip it and return it to the flat surface, it will now become a right shoe. By analogy, could we move a real left shoe into 4D space, 'rotate' it properly and obtain a right shoe for the other foot? Would this be possible with the curving of 3D space?

 

[MiGUi]

Yes, you can rotate 3D objects in a 4D space. You can represent it by a 4x4 matrix.

 

[hemmul]

Suppose you have a piece of paper in the shape of a left shoe, on a flat surface. Suppose also that it can only stay in contact with the surface. No matter how you move it or rotate it in the 2D surface, it cannot become a right shoe. However, if you move it out of the 2D space, flip it and return it to the flat surface, it will now become a right shoe. By analogy, could we move a real left shoe into 4D space, 'rotate' it properly and obtain a right shoe for the other foot? Would this be possible with the curving of 3D space?

huh! sure! if the 4th dimension, we consider, is time there is no problem: after several years of extreme usage, any fine shoe will finally wear out and it will equally be suitable for right as well as for left leg

 

[selfAdjoint]

The rotations in Euclidean four space are given by the group SO(4), but the transformations on Minkowski spacetime are given by the Poincare group SO(1,3), which is different.

 

[MiGUi]

We can imagine thousands of non-Minkowski 4D spaces, and there the rotation would be more easy.

 

[selfAdjoint]

Thousands of "flat" non-Minkowsi spaces? Non-euclidean?

 

[hemmul]

The rotations in Euclidean four space are given by the group SO(4), but the transformations on Minkowski spacetime are given by the Poincare group SO(1,3), which is different.

yep!

anyway, AFAIK, 4D rotations can be divided into: boosts (say rotation in xt plane) and usual spatial rotations (say in xy plane).
I don't think that spatial rotations would flip us the shoe (Q: why?), so it's up to the boosts.
Now, let's consider in details how do we rotate the 2D shoe:
1. introduce a new dimension
2. rotate the shoe in <new dimension><one of old dimensions> plane
3. project the shoe back to 2D.

Doing it analogiously for 3D->4D would look like this:
1. introduce 4th dimenion
2. BOOST e.i. rotate in <new dimension><one of all dimensions> plane
3. project back to 3D.

but what are boosts giving us? - they just resize the corresponding coordinates, so, the shoe will be slightly deformated, but not flipped... and the reason? - Minkowski geometry of space-time.

Note, that in case of "Euclidean 4D" we really can flip the shoe using the algorithm above...

right?

 

[MiGUi]

A 4D space with a metric tensor equal to identity. Minkowski's one has not this metric tensor.

 

[selfAdjoint]

A 4D space with a metric tensor equal to identity. Minkowski's one has not this metric tensor.

Yes, and you specify the metric tensor equal to the identity, i.e. diag(1,1,1,1), and you get ONE geometry; everything else flows from that. So what do you mean thousands? Just scaling factors won't give us a different geometry.

Now Minkowski 4D geometry has, I hear, thousands of distinct metric structures...

 

[MiGUi]

It was and expression... you can think in many 4D vectorial spaces. I don't know if they are 27, 683 or else.