I have had some very helpful contributions by email explaining how a 4-cube can rotate (or, presumably, roll) at one and the same time about two separate planes. These two pivotal planes, as I understand it, must meet only at a point, not along a line. Thus once you have chosen one face from the 24 square faces of the 4-cube to be a pivotal plane, that leaves only one set of four other faces (all parallel to one another) to choose from for the other pivotal plane. I shall be able to explain this better (probably?) with diagrams in a later post.
John Baez very helpfully directed me to http://eusebeia.dyndns.org/4d/vis/09-rot-1.html which goes into 4D rotation in some detail, with helpful animations. Also to http://en.wikipedia.org/wiki/SO%284%29 which is altogether more advanced mathematically. Too advanced for me right now I am afraid, but I shall try to learn enough to follow it. Mark Newbold also gave that wikipedia reference. Plus he has, on http://dogfeathers.com/java/hypercube2.html, an extremely helpful animation of a rotating 4-cube, which you can stop mid-spin in order to probe into its subtleties. I am in the process of doing this, with great interest.
Once you have settled on a pair of pivotal planes, on both of which the 4-cube is required to pivot, it does seem to take a considerable feat of mental gymnastics to be able to follow the 4-cube as it executes its double rotation (or rolling). However – just possibly – and with the aid of the above sites, I am getting there. Slowly. Perhaps. I think.
I shall keep trying. Meanwhile, can anyone help me to get to grips with the “too advanced” website referred to above? Is there a site which could gradually work me through the math on which it is based, matrices and so on?
Some of the animations I have looked at of a rotating 4-cube show its various component 3-cubes distorting themselves into trapezoidal shapes during the rotation. I find this slightly distracting, for the same reason that I prefer the fig 9 view of the 4-cube (from my original blog) rather than the fig 3 view. In real life (if that means anything) the 8 component 3-cubes of a 4-cube will perfectly retain their 3-cubical shapes during rotation, won’t they?
They may well appear, what with perspective and whatnot, to distort themselves into trapezoids as the 4-cube spins. But, surely, this is just a consequence of the 2D paper on which we poor benighted 3D humans are forced to depict them? If only we could see properly into the fourth dimension, we would surely find that all 8 component 3-cubes retain their 3-cubical shapes throughout. Wouldn’t we?
