## Rotations in even and odd dimensions

If you don't know where to post something, put it here and an administrator or moderator will move it to the right place.

### Rotations in even and odd dimensions

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.
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
anderscolingustafson
Tetronian

Posts: 315
Joined: Mon Mar 22, 2010 6:39 pm

### Re: Rotations in even and odd dimensions

I'm confused
can you add some pics for illustration?
Secret
Trionian

Posts: 162
Joined: Tue Jul 06, 2010 12:03 pm

### Re: Rotations in even and odd dimensions

This is just an effect of parity.

Keiji

Posts: 1969
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

### Re: Rotations in even and odd dimensions

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.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger

wendy
Pentonian

Posts: 1987
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

### Re: Rotations in even and odd dimensions

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.
People may consider as God the beings of finite higher dimensions,
though in truth, God has infinite dimensions

Prashantkrishnan
Trionian

Posts: 114
Joined: Mon Jan 13, 2014 5:37 pm
Location: Kochi, Kerala, India