Rotations in even and odd dimensions

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Rotations in even and odd dimensions

Postby anderscolingustafson » Sun Jan 23, 2011 2:56 am

One interesting thing about rotating things in different numbers of dimensions is that reversing an objects dimensions in all the spacial dimensions that exist by is possible in all even numbers of spacial dimensions but is impossible in all odd numbers of dimensions.

Try for instance rotating an object one hundred eighty degrees once along its up and down front and back sides. When you do this the object will be upside down and its front will be were its back side was. It now has its dimensions reversed in two dimensions. Now try rotating the object one hundred eighty degrees again but this time leaving the top and bottom in the same place. When you do this the object will still be upside down and the left and right side will be in the opposite places from were they originally were but the second rotation has caused the front and back side to revert back to their original locations. The object still only has its dimensions reversed in two dimensions but its dimensions cannot get reversed in all three dimensions. If we could theoretically rotate the object the object one hundred eighty degrees keeping the top and bottom and left and right sides in the same place that they are but in a way that would cause the ana and kata sides to be in opposite places then we would have undone the part of the second rotation that undid the part of the first rotation were the front and back were moved to opposite places so then four of its dimensions would be reversed from the way they originally were.

So basically it is possible to reverse all the dimensions of a 2d object but not a 3d object and it is possible to reverse all the dimensions of a 4d object but not a 5d object and so on.
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Re: Rotations in even and odd dimensions

Postby Secret » Sun Jan 23, 2011 6:43 am

I'm confused
can you add some pics for illustration?
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Re: Rotations in even and odd dimensions

Postby Keiji » Sun Jan 23, 2011 9:28 am

This is just an effect of parity.
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Re: Rotations in even and odd dimensions

Postby wendy » Mon Jan 24, 2011 8:10 am

It's also possible, in every even dimension, for the reversal to happen by rotation around the origin. For those who are familiar with complex numbers, consider 2n dimensions as n complex, the rotation is done by x' = x cis(w.t), where w is an angular velocity.

In odd dimensions, the complete reversal, or reflection in a point, makes a mirror-image, and so x,y,z... can not be rotated onto -x,-y,-z,... Also, the 'hairy ball' situation applies, it is not possible on a sphere in an odd dimension for the wind to be blowing everywhere. It is possible in even dimensions, though.
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Re: Rotations in even and odd dimensions

Postby Prashantkrishnan » Sun Oct 02, 2016 6:29 pm

This follows directly from the fact that rotation is a two dimensional phenomenon. Rotation in n dimensions corresponds to dividing n by 2 and taking the remainder.
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