system
Given the following post, respond as if you're commenting on Reddit:
human
If I take a two dimensional object and rotate it through the third dimension, it appears that the object has become a mirror image of itself to a two dimensional observer. Is there an equivalent to reflect a 3D object via rotation in the fourth dimension?
gpt
Yes. In every 3d rotation, there is a fixed plane (the rotation plane) and a fixed line, the axis of rotation. This decomposition in rotation is required for galaxys, solar systems and atoms exist.

In 4d we don't have this asymmetric decomposition. Instead, every 4d rotation has two fixed planes. Solar systems and atoms can't exist in 4d because that fixed line in 3D becomes a fixed plane. We can try find a 4d rotation that produces a 3d reflection keeping this decomposition in mind.

So let's try to reflect an object across the YZ plane and through the X axis. Let's have W be a fourth orthogonal axis. In a reflection, the plane of reflection is fixed, so that means that the YZ plane is one of our planes of rotation. But nothing happens to points on the reflection plane, so in 4d this has to be a zero degree rotation. But we still have the XW plane to work with. Since an object is moved to the other side of the origin, this suggests that we do a rotation through the angle pi (a 180 degree rotation) in the XW plane. It turns out that when you look at where a 3d object lands after a 180 degree rotation through the XW plane, while having a zero degree rotation about the YZ plane, you do get a reflection.

In fact, any N-dimensional reflection can be seen as a N+1 dimensional rotation.