Another way of showing 4D rotations is to sweep out a surface. Just as a point can trace a line as it moves, we can create a surface from the path traced out by a line rotating in 4D. A huge range of different surfaces can be generated in this way, depending on the lines chosen, the projection and the axes of rotation. The following are just a few examples of such surfaces, all using a circle as the initial curve (it is interesting to note that if you start with a circle, it remains a circle throughout the rotation, this is because stereographic projection is conformal) :
The two videos above show the non-orientable surface known as a Klein bottle. It might not look like the usual representation of a Klein bottle but it is topologically the same surface, and I think this embedding of it (known as a Lawson klein bottle) is actually much more elegant.
An unusual embedding of the Mobius strip can also be generated in this way:
This is the a mobius band with a circular edge, also known as a Sudanese mobius band (I have no idea where the name comes from).
April 2, 2009 at 4:43 pm
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April 4, 2009 at 11:38 pm
Perhaps if the atomic structures of the material that forms some hypothetical newly evolved cortex of the brain – if this atomic material were to contain an axis lying in a fourth direction; let’s say a .0000006 seconds into the future and .0000005 seconds into the past, and with particles at these distances, then information of four space might be able to be processed into perceptual interpretation mediated through the 4-D neurons of this cortex.
March 27, 2010 at 5:29 am
there is no backwards.