Date: Mon, 13 May 91 14:46:06 -0400 (EDT)
From: Dan Hoey <hoey@aic.nrl.navy.mil >
~~~ ~~~ Subject: Very silly ways of building very large cubes (was Re: 5by cubes)

Andy Latto <latto@lucid.com> wrote:

You can't make an order 21 cube, or any cube of order 7 or higher.
When you turn the top layer of such a cube by 45 degrees, the corner
cubie will not touch the other layers at all, so there's
no way to keep it attached, and it will fall off.

Then "Dale I. Newfield" <dn1l+@andrew.cmu.edu> responded:

There is no law that says that the cubes have to be the same size.

and showed that by making the outer layers thicker, we can increase
the size of the cube. There is another way around Andy Latto's con-
cern, and that is that we can--at least in theory--design a physical
cube that lets pieces overhang, such as corners that touch only two
surfaces, and yet still holds the pieces so they cannot be removed.

This idea (which came up in talks with Jim Saxe about a decade ago) is
to slice up the cube with a fresnel saw. A fresnel saw is used to cut
a piece of glass into two fresnel lenses out of pieces of glass, and
you find them in the same stores that sell plaid paint and jelly-
doughnut cookie cutters. (In case you don't know what a fresnel lens
(pronounced freh-NEL) is, for this note it's sufficient to think of it
as a surface with small concentric circular grooves in it. Kind of
like those old vinyl recordings people used to listen to, except that
the grooves are circular instead of spiral, and the grooves don't
wiggle back and forth.)

Now if you have two surfaces with mating grooves--each one has a ridge
that fits in each of the other's grooves--when you put them together
you can twist one with respect to the other, but you can't slide one
across the other, because the grooves are locked together. There is
one thing you can do that we don't want: you can lift one slab away
from the other.

The solution now is to get a *very* *sharp* fresnel saw, that cuts
hooked grooves that interlock with each other. You get surfaces with
cross sections that look somewhat like

```                            hook-in surface
_________       _______________________       _________
\     /         \     /           .           \     /         \     /
|   |           |   |            |            |   |           |   |
|   |  __       |   |  __        .        __  |   |       __  |   |
|   |    \      |   |    \       |       /    |   |      /    |   |
\   \___/|      \   \___/|      .      |\___/   /      |\___/   /
\       |       \       |      |      |       /       |       /
\_____/         \_____/       .       \_____/         \_____/
_____           _____________________           _____
/     \         /          |          \         /     \
| ___   \       | ___       .       ___ |       /   ___ |
|/   \   \      |/   \      |      /   \|      /   /   \|
\__  |   |      \__  |   axis    |  __/      |   |  __/
|   |           |    of     |           |   |
|   |           | rotation  |           |   |
___________/     \_________/             \_________/     \_____
```

hook-out surface

except that the surfaces are closer, so the hooked grooves are engaged
with each other. (Now we see why we need a fresnel saw, so that we
can cut the two mating surfaces in one cut, and avoid the problem of
trying to assemble two separated pieces (though we could get around
that difficulty messily with glue)).

So we may cut up a 2n+1 x 2n+1 x 2n+1 cube with a fresnel saw, to make
a large Rubik's cube. The only really touchy point is the need to
make the ``direction'' of each cut match the direction of the other
cuts at that ``depth.'' Here, direction refers to whether the hook-in
surface faces toward the nearest parallel side or away from that side,
and ``depth'' refers to the distance from the nearest parallel side.
This ensures that when we permute cubies around the directions of the
groove hooks will not change, so the grooves will always mate.

If n is large, then pieces of one slab will overhang at each turn, so
you can see the grooves on a whole surface of a corner, or on two
surfaces of an edge piece. But you can't pull the piece off, because
it won't move straight with respect to the rest of the cube, only in
curved trajectories. We have to keep the fresnelling small with
respect to the size of the cubies, and the tolerances are pretty
tight, but that's the regime we theoretical engineers are working in.
(I'd like to mention that cubes made with this method also have the
nice feature that there's a 2n-1 x 2n-1 x 2n-1 Rubik's cube on the
inside, so you can play with the theoretical invisible group while
you're at it.)

Now what about cubes of even side? The fresnel saw cuts two surfaces
that mate to each other but not to themselves. How can we get a
surface that mates to itself? I think the answer is that we can't.
But this doesn't mean we are out of luck, as there are several ways of
fixing up the center cuts of these cubes. Perhaps easiest way is to
embed a 2x2x2 cube in the center of the original solid cube, and use
it to hold the octants together. Unfortunately, this method requires
an appeal to the existence of even-sided cubes, rather than teaching
us how to build them.

The other ways of finessing the center cut involve the thin-center-
slab approach. You know you can simulate a 2x2x2 pocket cube with the
corners of a regular 3x3x3 Rubik's cube, and similarly you can
simulate any even cube with a larger odd cube. Also, we can make that
center slab very thin, so it becomes part of the supporting structure
rather than a significant part of the cube. We also remove the cor-
ners from the center slab, so it does not protrude from the cube. We
may even make covers for the cubies slabs adjacent to the center, to
cover up the crack where the center slab lives. We are ready to cube!

Or are we? The thin-center-slab suffers from the partial-twist
problem. We can see this in the simulation of the 2x2x2 by a 3x3x3.
If you try to simply ignore the center slabs, you can end up with the
corners being aligned with each other but with a center slab twisted
by 45 degrees. This makes it impossible to turn the corners except in
the plane parallel to the oblique slab. If we shrink the center slab
enough that it becomes unnoticeable, we will still be unable to twist
the cube except in one direction except by breaking the center slab.

The first solution to the partial-twist-problem is to select one of
the eight near-central cubies, a cubie that abuts the center slabs on
three sides. We then glue the adjacent parts of the center slabs to
that cubie. Then when we turn along the center slice(s), the glued
part of the thin center slab will follow the selected cubie, and will
push the rest of the thin center slab along. This is a modification
of the solution that is taken inside Rubik's Revenge, as I described
to this group in my Invisible Revenge article of 9 August 1982.

I like this solution except for one thing. It destroys the symmetry
of the cube, by selecting one specialized octant that the center slab
must follow. There is one more solution, though, that keeps the cube
symmetric, which is *even* *sillier* than the thin center slab itself.
Let us now visualize the center slab. It has the corners removed, so
it is in the shape of a disc. The disc is cut in a grid pattern by
the cuts from perpendicular planes. Now suppose we cut each slab in a
second grid pattern, with the grid at a 45 degree offset from the
original. With such a center slab, the cube can be twisted if each
slab grid is in the correct position, or if some are at a 45 degree
offset from the correct position.

And how shall we prevent turns of less than a 45 degrees? Gears!
Embed tiny gears in each fragment of the center slab, that contact
tiny toothed tracks in the adjacent slabs on both sides. This will
force the center slab to turn at exactly half the angular rate of one
half of the cube with respect to the other. Thus when the off-center
slabs of the cube are aligned, the center slab will be at one of the
positions that allows twisting.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil