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i've read on the wikipedia article that in 4 dimensions the hypercube, the Clifford torus, and a bunch of other 4-d figures that all seem to rotate oddly. the hypercube rotating in 4 dimentions is rotating like a vector feild on a torus (thanks to physics.stackexange, the picture is from that site)

i am wondering: why is it that is the fourth dimentional objects rotate oddly? my assumption is that it happens so the outer and inner cubes can swap positions.(a hypercube can be made by making 2 cubes, and connecting the two cubes with "edges") Is that the case? if not, does anyone know?

Alexander Day
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  • You read on Wikipedia that various 4D figures rotate oddly? That's a weird statement for an Wikipedia article to make. –  Apr 25 '17 at 22:25
  • A different point of view is that it’s rotations in two and three dimensions that are the oddities. Once you move beyond three dimensions, things aren’t as “locked together” under rotation as they are in two and three. – amd Apr 25 '17 at 23:05

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It's not that they rotate oddly. It's that when you choose a specific projection of their rotation to 3-space, the picture looks odd.

But that's nothing special. Imagine the cube in 3-space. Let it rotate. While it rotates, project it to the 2-plane. It looks kind of odd, right?

Or, how about the square in 2-space. Let it rotate. While it rotates, project it to the line. Even that looks kind of odd.

Lee Mosher
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