# Can we rotate 3D objects in 4D space?

1. Oct 22, 2004

### 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?

2. Oct 22, 2004

### MiGUi

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

3. Oct 23, 2004

### hemmul

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

4. Oct 23, 2004

Staff Emeritus
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.

5. Oct 23, 2004

### MiGUi

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

6. Oct 23, 2004

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

7. Oct 24, 2004

### hemmul

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, lets 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?

8. Oct 24, 2004

### MiGUi

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

9. Oct 24, 2004