Date: 30 DEC 1980 0144-EST
From: DCP at MIT-MC (David C. Plummer)
Subject:  The higher order cubes (just the 4x4x4 for now) [93 lines]
To: CUBE-LOVERS at MIT-MC


And you thought the 3x3x3 was a complicated beastie...

I have plans for the 4x4x4 which I think can work. A first order
approximation is to take the 3x3x3 and split the edges in two.
This doesn't take care of the centers or axis, but those are
perhaps the most complicated anyway, and will get considerable
thought. 

WARNING: If you really want to see this, get out the graph paper
and correct the aspect ratio that the characters here will have.

The way I have labeled the cubies is as follows:
        Corners are labelled A (there are 8 of these)
        Edges are labeled both B and D (this defines the
                orientation) (there are 24 of these)
        Centers are labeled C (there are 24 of these)
So a face would look like:
        ABDA
        DCCB
        BCCD
        ADBA

Take a slice down the center of one of the planes and open it up.
It should look something like:

        .....&&&&&&&&++++++++
        .....&&&&&&&&++++++++           where & and @ are pieces of
        ..xxxxxx&&&++++++++++            the C type of cubie
        ..xxxxxx&&&++++++++++           + is a B/D type
        ....xx&&&&&++++++++++           x is the central axis
        ....xx&&&&&++++++++++
        ....xx&&&&&++++++++++
        ....xx&&&&&++++++++++
        ....xx&&&&&++++++++@@
        ....xx&&&&&++++++++@@
        ....xx&&&&/@@@@@@@@@@
        \...xx&&&/@@@@@@@@@@@
        .\..xx&&/@@@@@@@@@@@@
        ..\.xx&/@@@@@@@@@xx@@
        ...\xx/@@@@@@@@@@xx@@
        xxxxxxxxxxxxxxxxxxx@@
        xxxxxxxxxxxxxxxxxxx..
        .../xx\..........xx..
        ../.xx.\.........xx..
        ./..xx..\............

Gross, isn't it? There are a few things going on here that are
hard to show. That central cross must be able to rotate, so the
innermost parts of C and B/D are carved in somewhat. There is
another hairy constraint: the central axis MUST BE RIGIDLY
CONNECTED TO ONE AND ONLY ONE OF THE C TYPE CUBIES. The reason is
hard to describe. Hint: suppose you rotate a half cube portion.
It is possible that when the rotation is finished, the central
axis is misaligned. Connecting it to one face cubie forces it to
win. This may also make the central cross somewhat more fragile.

Forgetting about the central axis and the corner cubies (they
shouldn't be too able to work into the picture) we have the
following diagrams for the B/D and C type cubies. The numbers
indicate elevations:

                B/D                             C

        44444444                        4444
        44444444        ACTUALLY THEY   4444
        4443333333      ARE CURVED,     4443333333
        4443333333      BUT CHARS       4443333333.
        4433333333      HAVE POOR       4433333333.2
        4433333333      RESOLUTION      4433333333.2.
        4433333333                      4433333333.2.1
        4433333333                      4433333333.2.1.

I seriously think these will work. I hope to find somebody in
a material science type of lab that can make plastic molds so I
can actually try and build one of these. I hope to use a soft
plastic that is machinable, carvable, sandpaperable, etc, and
when I have a good one, make several pieces out of harder plastic
and see what happens. Perhaps I'll have something by the end of
January. 

I think that the 5x5x5 cube (though the tolerances might be
tighter) may actually be easier to construct. It may also be a
more interesting cube to work with.

Next: Notes on transforms on the 4x4x4 and 5x5x5 cubes.

PS: A mind blower: a DODECAHEDRON frob (can't call it a cube). I
thought of this one coming back on the bus. I haven't put
anything down on paper, but my minds eye tells me it has a
chance.

Comments welcome.