Six ways to planar-rotate a 4-cube

In an earlier post I implied that I could show the “skeleton” of a rotary 4-cube with just four still pictures of it with its corners lettered A to R (leaving out I and O). This is wrong of course, or incomplete anyway, because  there are six different ways in which a 4-cube can execute planar rotation and ought not each to have its own set of pictures?

Thus there are six different sets, each of four parallel, plane faces, about which the 4-cube can pivot. So rather than my measly 4 pictures, actually it would seem to take 6x4 = 24 of them to show the full range of a 4-cube’s rotations from the start through ¼ turn, ½ turn and ¾ turn positions.

By chance, a 4-cube has 24 square faces. Coincidence?

Well, no, not really. Because it wouldn’t need 24 pictures. It would need 6 sets of 4 of them. But the openers of each set would all be the same, namely the starting position. That leaves 24 – 5 = 19 different pictures.

Am I right?

John Scott
johnscott.hyperspace@gmail.com

4D rotation without distortion (more or less)

I still never cease to be impressed by Mark Newbold’s animations of a rotating 4-cube.   See http://dogfeathers.com/java/hypercube2.html.

Question:  (to Mark or anyone skilled enough to organise animations of this kind):  is there some way to start from my “fig 9 view” of a 4-cube (see my original blog) and show an animation of it as it rotates in 4D?   Preferably without distorting it out of its original fig 9 shape?

Reverting to simple, planar (i.e. non-composite, or non-Clifford) rotation, one can readily draw still pictures of the fig 9 4-cube in its start position and its positions after a ¼ turn, ½ turn and ¾ turn.   I may add these later.   If we identify the 4-cube’s corners by the letters ABCDEFGHJKLMNPQR, the four pictures will all be the same with only the corner-lettering changed.   (I choose to leave out I and O;  it’s what I was taught to do.)
For my part, I think these 4 pictures would give a good (albeit limited) account of what happens to the 4-cube as it spins or rolls.   At least you could see exactly where all 16 corners were at each quarter-turn.

Perhaps someone may be skilled enough to animate this view of a 4-cube as it spins or rolls?

By comparison, I have trouble following or analysing the animated rotary 4-cube at the top of the blog.   Partly this is because as it spins it keeps distorting its component 3-cubes into trapezoids.   Mainly though my problem is that one can’t shout “STOP” and see just where all its 16 corners have got to.   Mark Newbold’s Dogfeathers animation is brilliant in that respect however and I must analyse it further.

The two pivotal planes of a composite-rotating 4-cube

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?

Rolling a cube in multi-dimensions

Squares (2-cubes), cubes (3-cubes) and hypercubes (4-cubes) and how they roll

The simplest form of a cube is a square, or 2-cube, in 2 dimensions. When you roll such a square, along a line within its 2D world (fig 1), it pivots on its corners.

As the square rolls, from West to East say, each corner in turn hits the line and acts as the pivot. In effect the 4 corners form a circuit, with 4 “stations” on the circuit. As the square rolls, the “business” corner, that is to say the one on which it pivots, moves round the circuit from station to station. So the business corner moves one step, or one station, round this circuit each time the square executes a quarter-turn.

Now consider rolling an ordinary 3D cube, or 3-cube, from West to East. A 3-cube pivots on its edges. It has 12 edges to choose from. We’ll start off our 3-cube flat on a table somewhere with 4 of these edges pointing North-South, 4 pointing East-West and 4 pointing up-down. See fig 2.

If we now roll it from West to East, just 4 of these edges will act as pivots, namely the ones pointing North-South. Once again, just as with the square, each of these 4 North-South edges in turn hits the table and acts as the pivot. They form a circuit with 4 stations on it. As the 3-cube rolls, the business edge, the one on which it pivots, moves round the circuit from station to station, moving by one station at each quarter-turn. The four pivotal edges, for a West/East or East/West roll, are shown in bold in fig 2.

What about a hypercube, or 4-cube? As we have seen, a rolling 2-cube, or square, pivots on its corners, whereas a rolling 3-cube pivots on its edges. We would expect, therefore, that a rolling 4-cube would pivot on its surfaces, its square faces. At least for planar rotation, which is what I am considering here, that turns out to be the case. The rolling 4-cube does indeed pivot on its square faces, of which it has 24 to choose from. Once again though, only 4 of them will act as pivots for a given variety of roll (according to my understanding at least). Which 4?

There are, no doubt, many ways of drawing a 4-cube. Probably most often seen is the view shown in fig 3. What I prefer, however, is to derive a 4-cube as follows. Start with a 0-dimensional point (fig 4). ve this point through unit distance to form a 1D line (fig 5). Now move the line through unit distance at right angles to itself (fig 6). This creates a 2-cube, or square, which exists in 2D. Next, move the square through unit distance at right angles to itself. This creates an ordinary 3D cube - a 3-cube (fig 7).

Now move the 3-cube, again through unit distance at right angles to itself - what on earth does that mean? Well, in our 3D world, nothing. But we can draw it on paper and imagine it easily enough by analogy.

See fig 8, which shows the result of moving the 3-cube in an imaginary direction at right angles to itself. What this movement creates is a 4-cube or hypercube (fig 9). And, at least when considering rolling and rotation of the hypercube, I find these motions easier to visualise with the fig 9 view than with the fig 3 view (though both views are equally valid in their different ways).

(It is not, perhaps, immediately evident that figs 3 and 9 show the same object. But consider: both show two “obvious” cubes (i.e. cubes drawn in the way we expect to see them). And in each case there are 8 straight lines joining up corresponding corners of these two “obvious” cubes. The two views, fig 3 and fig 9, are, in fact, equivalent. The two look different because they are distorted in two different ways in order to show a 4D object on 2D paper.

Note that the 2D square (2-cube) is bounded by its 4 component straight-line edges. The regular cube (3-cube) is bounded by its 6 component square faces. We would expect, therefore, that the hypercube (4-cube) would be bounded by 8 components in the form of 3-cubes.

Fig 10 shows a hypercube plus compass points identifying North, South, East, West, up, down, and, in addition, plus and minus in the 4D direction. I call these "fourwaurds" and "backwaurds" (F and B for short); the unconventional spelling is to remind us that these are unconventional directions in the unfamiliar fourth dimension.

As you can verify from fig 9 or fig 10, or indeed from fig 3, the hypercube has 16 corners, 32 edges, and 24 square faces. In addition, it is indeed bounded by 8 cubical "volumes”, just as we would expect. These 8 volumes comprise the 8 component 3-cubes which make up the 4-cube or hypercube. 2 of these 8 3-cubes are separated from each other along lines running North and South. This pair of 3-cubes is shown in bold in fig 10, and the N-S-running edges which join them are shown dotted in fig 10.

Another pair of 3-cubes (shown bold in fig 11) are separated along lines (dotted in fig 11) running East and West. A further pair of 3-cubes (bold in fig 12) are separated along lines (dotted in fig 12) running up and down. The final pair of component 3-cubes (bold in fig 13) are separated along lines (dotted in fig 13) running fourwaurds and backwaurds in the assembly.

Of course, many of these eight 3-cubes, or eight “volumes”, which we have been looking at do not, at first sight, look like ordinary 3-cubes. But how could they, on 2D paper? Compare for instance the six “square” faces of the 3-cube in fig 2. Only two of these look truly square in the picture, again because a 3-cube is 3D and paper is 2D. We take this distortion in our stride, because we are so used to it. With practice, we can also take in our stride the distortions in figs 9–13.

What about rolling of the 4-cube and how does it rotate as it rolls? There are in fact, I think, 6 different ways in which (that is to say 6 different circuits of 4 stations around which) a 4-cube can roll as it executes so-called planar rotation. Each circuit uses one set of 4 of the 4-cube's square faces (there being 24 square faces of the 4-cube in all) as the faces on which to pivot. Just as a rolling 2-cube (or square) pivots on its 4 corners and a rolling 3-cube pivots on 4 of its 12 edges, so a rolling 4-cube pivots on 4 of its 24 faces.

Let's consider a 4-cube executing planar rotation as it rolls from West to East. In this instance, let us suppose that the 4 pivotal surfaces, on which it rolls are the ones shown in bold in fig 14. These 4 pivotal surfaces are parallel to one another and they form four stations on a circuit. As the 4-cube rolls, each of these 4 pivotal surfaces in turn acts as the pivot; and the “business” surface, the one on which pivoting takes places, moves round the circuit from station to station, one station at each quarter turn. All this is analogous to the behaviour of a rolling square or a rolling 3-cube.

It is amusing, if you have the time, to follow the progress of the rolling 4-cube round all 4 stations on the circuit. And, of course, fig 14 shows in bold just 4 pivotal surfaces on which the 4-cube pivots as it rolls. There are, in all, 6 such sets of 4 surfaces, 6 different circuits around which the 4-cube can roll. So, if you have time, follow the 4-cube’s progress round all 6 of them. Study, if you like, the 6 different ways a 4-cube can roll.

If you have even more time, try playing with 5-cubes and 6-cubes (or as high-cubes as you like). A 5-cube, for example, will have 32 corners, 80 edges, 80 faces, 40 volumes and 10 hyper-volumes. When it rolls, it will pivot on just 4 of the 40 volumes, which will form the 4 stations on a circuit. So there will be 10 different circuits around which it can roll. Similarly a rolling 6-cube will pivot on 4 out of its 60 hypervolumes, arranged in a circuit. There will be 15 different circuits around which a 6-cube can roll. And so on.

If the above is correct, it is interesting, is it not, that all “cubes”, from the 2-cube (square) through the 3-cube, 4-cube and so on, roll in a similar way. All have one or more circuits, each circuit having 4 stations on it, around which the rolling takes them. These 4 stations on a circuit are the 4 pivots about which the n-cube rolls on that particular circuit. During rolling, the “business” pivot – the one about which the n-cube swings during one particular quarter-turn – moves round the circuit of 4, step by step, from station to station. And the number of different circuits around which an n-cube can roll will always, I think, be ½ of n ( n - 1 ).

At least, I hope all that is right. Please correct me if it is not.

The above deals with planar rotation, as defined in the Wikipedia article I mentioned at the beginning. But what about the composite rotation to which that article refers? I await with total fascination the explanation, from some kind person, of this composite rotation. I shall say no more on the matter for now. But I strongly suspect that I shall be able to show that, contrary to popular belief, turning a left boot over in 4D hyperspace does not always transform it to a right boot.

It does sometimes. But not always. I think.

"Composite rotation"

I thought I understood rotation. I have explained above  how squares (2-cubes), cubes (3-cubes) and hypercubes (4-cubes) roll. But there is something I don't understand.

See http://simple.wikipedia.org/wiki/Hypercube for a good article on hypercube rotation [the relevant extract is on the "Wikipedia article" tab above]. It's good but I don't understand it fully. It speaks of planar rotation of a 4-cube, which I think I do understand and which I have (I hope) explained correctly below, and composite rotation - which I don't and I haven't.

See the animation here from the Wikipedia article. This does not seem to be of the planar type. I presume it's of the composite type. Whoever did the animation clearly understands it much better than I do.

Help, please.