The crux of the matter is that rotations are things that happen in planes; they are inherently two-dimensional phenomena. A single rotation can only swap pairs of coordinate axes, or reverse the orientation of both (or do both together, or nothing). What is essential is that it’s messing with two axes, any time it messes with one. It’s 0 or 2. No 1.
But to swap left and right, we only want to mess with one axis.
We just need that extra dimension of freedom to couple a “trash” axis (the fourth dimensional one) with the one we want to flip, in order to keep the other original two untouched. I’m this way we can preserve 2/3 dimensions and reverse the third, provided we can have a fourth that also changes
To be convinced that you can’t do get it done with rotations in 3D (why you can’t use the z axis, then “un use” it cleverly) probably takes abstract algebra; permutations and stuff (like that one episode of futurama). I can’t tell if it’s an even vs odd thing, or if simply having an extra thing is enough.
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Elaboration of how adding a dimension helps, going from 2 to 3 dimensions:
Picture the letters d vs b.
If you are constrained to the two dimensional plane, you can’t turn one into the other with only rotations. We have to invent the mental image of “reflections”, which are somehow qualitatively different from rotations (it took a long time to flesh this stuff out, when people finally looked beyond 3 dimensions)
But if you have a big cardboard cutout of a d, you can just rotate it along a new axis—one that runs through the plane (through the stem of the d, say). The axis itself isn’t new, but the plane containing our d would be moving through a new dimension as it rotates
In this case, as we swap left with right, the x axis moves to the -x axis, lining up with the z-axis half way through
If the fingers of the glove are pointing in the +y direction (left thumb toward +x), then we simply need the same kind of transformation: a way to swap +z and -z (palm directions), without affecting x or y
In 4D, there’s a plane containing the z- and w-axes (w being our newest direction). We simply rotate the +z axis toward the -z axis within this plane, hitting the w axis of at the half way point, exactly as in our 2 vs 3D version
The crux of the matter is that rotations are things that happen in planes; they are inherently two-dimensional phenomena. A single rotation can only swap pairs of coordinate axes, or reverse the orientation of both (or do both together, or nothing). What is essential is that it’s messing with two axes, any time it messes with one. It’s 0 or 2. No 1.
But to swap left and right, we only want to mess with one axis.
We just need that extra dimension of freedom to couple a “trash” axis (the fourth dimensional one) with the one we want to flip, in order to keep the other original two untouched. I’m this way we can preserve 2/3 dimensions and reverse the third, provided we can have a fourth that also changes
To be convinced that you can’t do get it done with rotations in 3D (why you can’t use the z axis, then “un use” it cleverly) probably takes abstract algebra; permutations and stuff (like that one episode of futurama). I can’t tell if it’s an even vs odd thing, or if simply having an extra thing is enough.
———————-
Elaboration of how adding a dimension helps, going from 2 to 3 dimensions:
Picture the letters d vs b.
If you are constrained to the two dimensional plane, you can’t turn one into the other with only rotations. We have to invent the mental image of “reflections”, which are somehow qualitatively different from rotations (it took a long time to flesh this stuff out, when people finally looked beyond 3 dimensions)
But if you have a big cardboard cutout of a d, you can just rotate it along a new axis—one that runs through the plane (through the stem of the d, say). The axis itself isn’t new, but the plane containing our d would be moving through a new dimension as it rotates
In this case, as we swap left with right, the x axis moves to the -x axis, lining up with the z-axis half way through
If the fingers of the glove are pointing in the +y direction (left thumb toward +x), then we simply need the same kind of transformation: a way to swap +z and -z (palm directions), without affecting x or y
In 4D, there’s a plane containing the z- and w-axes (w being our newest direction). We simply rotate the +z axis toward the -z axis within this plane, hitting the w axis of at the half way point, exactly as in our 2 vs 3D version