See http://eusebeia.dyndns.org/4d/vis/09-rot-1.html. It bothers me a bit to find this splendid and excellent website (which I shall call eusebeia for short) speaking of a 4-cube (or indeed anything which exists in 4D) executing “3D-like rotation”.
Why? Well, maybe I’ve misunderstood something but this is how I see it. Consider life, and rotation, in 2D. All you need is a single point – the centre (or center if you insist) of rotation – and (given the speed, the sense and the radius) the rotation is 100% defined.
Now what of 3D? You might think that rotation in 3D, e.g. of a weight whirled round on a piece of string, could also be about a fixed, pivotal point. But no! Take a given weight, string length and speed of whirling and assume that all you know is a pivotal point. In that case you find that you’ve no idea which plane the whirling rotation is taking place in. It could be in a vertical North-South plane, a vertical East-West plane, or a horizontal plane. Or indeed in any one of an infinity of planes in between. You find that you must know the axis of rotation in order to define it.
Now go into 4D. Any simple, non-composite (i.e. non-Clifford) rotation is now about a plane. Hence the name, planar rotation. So if all you know is the axis of rotation, say the North-South axis, you don’t know enough. Any rotation which seems to be about an N-S axis could in fact, in 4D, be about the NSEW plane, or the North-South-up-down plane, or about the North-South-fourwaurds-backwaurds plane. Or indeed about any one of an infinity of planes which include the N-S axis and lie in between the planes just mentioned. (I use the words “fourwaurds” and “backwaurds” – or F and B for short – to mean plus and minus in the direction of the fourth dimension. See the original post.)
In the light of all this, therefore, I suggest that it is imprecise – in fact inappropriate – to speak of “3D-like” rotation in a 4D hyperspace world. Now, I admit that there are some varieties of rotation which do truly take place in a 4D world and which might truly, at first sight and to our human eyes, appear to be 3D-rotation-like. But this appearance, I suggest, is misleading.
If we limit ourselves to orthogonals, there are 6 types of rotation in 4D. Let us use NSEW to denote compass points, UD for up and down, and F and B for fourwaurds and backwaurds as explained above. So the 6 types of 4D rotation are about the following 6 planes: NSEW; NSUD; NSFB; EWUD; EWFB; and UDFB.
Consider an ordinary 3D 3-cube existing in our familiar 3D space, with its edges running NS, EW and UD. Now rotate it, say, about the NSFB plane. This will appear (to a 3D person living in 3D who can’t see into the 4th dimension) to be a “3D-like” rotation. Please note however that this rotation clearly cannot be 100% 3D-like (at least as far as the all-seeing mathematician is concerned – and (s)he of course is the one who set up this rotation initially and caused us all these headaches). Why not? Because the plane about which the rotation takes place includes the mysterious FB – fourwaurds and backwaurds – axis.
Exactly the same applies if we take that same 3D 3-cube and rotate it about the EWFB plane. Ditto for rotation of it about the UDFB plane. The rotations may appear to us humans to be 3D-like. But they’re not really. At least I think not anyway.
