# 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