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He talks about 120 degree rotations of triangles that leave them invariant. Then he proceeds to talk about flipping them with an interesting (at least to me) remark - "there's nothing that says I'm confined to moving (the triangle) in the plane".

A natural scenario that comes to mind following this is: if there was an observer that's purely confined to the 2D plane, and if there were a triangle on that plane, then the observer could completely specify and map out the triangle by roaming around it and measuring its lengths and angles.

Also, for that observer, the notion of 120 degree rotations would be a totally natural symmetry

to think about. On the other hand, a flip would be a not-so-natural, "higher dimensional" symmetry that's not confined to the 2D plane.

I'm currently studying special relativity and in that context we usually talk about homogeneity (translational symmetry) and isotropy (rotational symmetry) as far as I know, which are symmetries in our usual 4D spacetime. So I'm curious as to whether there are "higher dimensional" symmetries in our case as well. I'm sure fields like group theory address such questions, but I'm curious about how prevalent/useful this "higher dimensional" symmetry (there's probably a proper technical term for this which I don't know, hence the quotes) idea is in Physics.

Are there any fields (in Physics) where the notion of such kind of symmetries is useful?