# Message #452

From: David Vanderschel <DvdS@Austin.RR.com>

Subject: Re: [MC4D] Noel conquers the 4^5!

Date: Tue, 01 Apr 2008 02:01:35 +0000

On Monday, March 31, "Jenelle Levenstein" <jenelle.levenstein@gmail.com> wrote:

>Your forgetting that the complexity of the moves

>required to solve the cube increases as you add

>dimensions,

Most folks seem to believe this, but I think there is

a sense in which it is not so. The sense in which it

is clearly true is that there are more things to keep

track of as the dimension goes up.

Consider the following for the 3x3x3x3 puzzle: Because

the possiblities for reorienting a hyperslice of the

4D puzzle are so much richer, the orientation of any

hypercubie can be changed to any of its possible

states - with the hypercubie remaining in the same

position - simply by twisting any one of the

hyperslices which contain it. (An (external)

hyperslice is a 1x4x4x4 set of hypercubies

corresponding to a hyperface, and it reorients like a

3D cube.) In the 3D case, we lack the flexiblity

required to achieve an analogous capability. Given a

set of fairly simple moves that will isolate any given

hypercubie from one of the hyperslices in which it

lies into another hyperslice parallel to the first and

otherwise leaving the first unchanged, you wind with a

rather general and easily understood approach to doing

anything.

>… By the way would a 3x3x3 cube be possible to make

>in a 4D would or would it just fall apart. It could

>be analogous to the slide puzzles we make.

It would be analogous to an interlocking type of 2D

puzzle. (I.e., stays together when constrained to lie

in a hyperplane one dimension down from that of the

universe in which it exists.) Clearly any piece can

be translated without hindrance in the direction

perpendicular to the 3D hyperplane containing the 3D

puzzle.

Regarding the perception of the problem by beings in

other dimensional spaces, I have posed the reverse

analogous question - wondering what solving the 3D

puzzle would be like for a 2D being. Indeed, my own

simulation of the 3D puzzle will produce a display

that corresponds to what a 2D being could perceive

when the 3D puzzle is implemented in a manner

analogous to MC4D, so you can try your hand at 3x3x3

solving from the perspective of a Flatlander. Though

this unusual capability is not the main value of my

program, I’d be interested in feedback from anybody

who tries it: http://david-v.home.texas.net/MC3D/

Regards,

David V.