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.