Message #597

From: David Smith <djs314djs314@yahoo.com>
Subject: A new simple result
Date: Sun, 09 Nov 2008 05:06:34 -0000

Hello everyone,

I have been busy with other things lately, but I
took some time last night to solve a simple problem
which occurred to me. It did not involve any advanced
reasoning (just high-school geometry), but I think
it might be of interest to the group.

I wanted to know what size cubes (in cubies per edge)
of all dimensions were theoretically constructable,
i.e. what cubes were not definitely inconstructable
without the pieces falling out.

It turns out that for dimensions higher than 3, two
different types of cubes are theoretically constructable -
those in which the pieces would not fall out regardless
of how one turns the faces, and those in which the pieces
would only fall out when the corners are sufficiently
far from the center of that face. All other cubes
would be inconstructable, in that the pieces would
definitely fall out no matter how you rotate the faces.

For the first type of cube, a d-dimensional cube with
n cubies per edge is theoretically constructable
if the following inequality holds:

(d-1)(n-2)^2 < n^2

This means that in three dimensions, up to 6x6 cubes
could be constructed (we are not considering cubes
that are not cube-shaped, so the 7x7 and higher V-Cubes
would not count), in four dimensions, up to 4x4 cubes, in
five through nine dimensions, up to 3x3 cubes, and in
dimensions higher than nine, only 2x2 cubes. For the
second type of cube, the matter of whether one could be
theoretically constructed reduces to the previous formula
using one lower dimension, or:

(d-2)(n-2)^2 < n^2

The reason for this is that the type of rotation which
keeps the pieces closest to the center of that face is
a 90 degree coordinate-axis aligned rotation.

Concerning the second type of cube, the closer the two
sides of the first inequality, the farther the corner
pieces could be from the center of that face without
falling out. An interesting thing to note is that
equality holds in the first inequality for a 3^10 cube, so
the corner pieces would only fall out when the corners
are the farthest distance possible from the center of the
face (or sufficiently close to that distance, depending
on how well the actual cube mechanism was designed).
In a 3^11 cube, equality holds in the second inequality,
which means the pieces would definitely fall out, but
only when they are the farthest possible distance from
the center of the face using a 90 degree coordinate-axis
aligned rotation (i.e. at 45 degrees).

I hope this simple yet interesting result was of value
to the group. Tomorrow, I plan to continue working
on the analogous permutation formulas for five-dimensional
cubes that I discovered for four-dimensional ones.

All the best,
David