Message #454
From: Mark Oram <markoram109@yahoo.co.uk>
Subject: Re: [MC4D] Noel conquers the 4^5!
Date: Tue, 01 Apr 2008 14:47:17 +0000
Thanks guys for the comments/feedback: I found them all very interesting.
Certainly there are more things to keep track of as the dimensions go up, but I wonder if that necessarily implies that the moves HAVE to be more "complex", or just that there are longer sequences needed? Obviously one can solve, say, the 5-D cube by small steps; moving each class of cubie into place one by one with small sequences of moves. It just takes longer and needs more patience. Also, it may not be the quickest (or most elegant) means. If nothing else, I assume a 4-D (or 5-D etc) being could always resort to such an approach. Crucially, however, this does not exclude the possibility that a 4-D being could have a fundamentally superior insight, and might be able to see the most efficient sets of moves more intuitively than us every time.
I hadn’t considerd the practicalities of how the 3-D or 4-D cubes would be assembled physically in 4-D space, and again it may be that the engineering hurdles are equivalent regardless of the actual dimension of the space in question. It reminds me of the initial confusion form many people (myself included) when they had seen the 3-D cube for the first time, wondering how it could even be made without everything falling apart once it was turned! Still, clearly the 3-D cube is physically possible,a nd I have no doubt that equivalent ones are possible - at least in principle.
Of course, it is easy to imagine that 4-D and 5-D hypercomputer screens could represent any of these cubes with ease, but again the issue of how to usefully represent extra dimensions on a screen with a fixed maximum number of dimensions would occur there as well. (I have always felt that a large part of the genius of Charlie, Melinda, Remigiusz and Roice was finding a way to do this for the 5-D cube on a 2-D screen, so thanks again guys!)
One final point: are the 4-D/5-D hypercomputers still just Turing machines at heart, or is it possible that there can be some fundamentally more powerful means of computing in the higher dimensions? (Or am I now wandering too far from the main group topic???)
— On Tue, 1/4/08, Jenelle Levenstein <jenelle.levenstein@gmail.com> wrote:
From: Jenelle Levenstein <jenelle.levenstein@gmail.com>Subject: Re: [MC4D] Noel conquers the 4^5!To: 4D_Cubing@yahoogroups.comDate: Tuesday, 1 April, 2008, 4:08 AM
I have found by teaching other people to solve the 3x3x3 rubix cube that the hardest part of solving it is to figure out how the puzzle moves in 3 dimensions. I think this may be because they have trouble seeing the puzzle in 3 dimensions. They see 54 individual stickers instead of seeing the 26 peices. You can tell this because someone new to the cube will pick one color and try to get as many of that color on one side as possible, without regard to what pieces get knocked out. Now it may sound odd that we have trouble thinking 3 dimensionally even though we live in a 3D world, but there are a lot of 2 dimensional things in our world. This Computer screen is 2D The layout of our streets is 2D. Even the buildings we live although they are 3D simply consist of a bunch of 2D floors stacked on top of each other. It would be interesting to see whether a society run by monkeys would be better at solving these insane puzzles than we are since they
have more spacial minds.Wait I had a point. Whether a four dimensional creature would be able to intuitively understand a 3x3x3x3 would depend on how they thought and what there world looked like which is something we have no way of even imagining. My guess is that they would have to figure it out just like we did, but it would be easier because they could hold the thing in their hands and see it form all angles.
On 01 Apr 2008 02:01:35 +0000, David Vanderschel <DvdS@austin. rr.com> wrote:
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 isa sense in which it is not so. The sense in which itis clearly true is that there are more things to keeptrack of as the dimension goes up. Consider the following for the 3x3x3x3 puzzle: Becausethe possiblities for reorienting a hyperslice of the4D puzzle are so much richer, the orientation of anyhypercubie can be changed to any of its possiblestates - with the hypercubie remaining in the sameposition - simply by twisting any one of thehyperslices which contain it. (An (external)hyperslice is a 1x4x4x4 set of
hypercubiescorresponding to a hyperface, and it reorients like a3D cube.) In the 3D case, we lack the flexiblityrequired to achieve an analogous capability. Given aset of fairly simple moves that will isolate any givenhypercubie from one of the hyperslices in which itlies into another hyperslice parallel to the first andotherwise leaving the first unchanged, you wind with arather general and easily understood approach to doinganything.>… 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 2Dpuzzle. (I.e., stays together when constrained to liein a hyperplane one dimension down from that of theuniverse in which it exists.) Clearly any piece canbe translated without hindrance in the directionperpendicular to the 3D hyperplane containing the 3Dpuzzle.Regarding the perception of the problem by beings inother dimensional spaces, I have posed the reverseanalogous question - wondering what solving the 3Dpuzzle would be like for a 2D being. Indeed, my ownsimulation of the 3D puzzle will produce a displaythat corresponds to what a 2D being could perceivewhen the 3D puzzle is implemented in a manneranalogous to MC4D, so you can try your hand at 3x3x3solving from the perspective of a Flatlander.
Thoughthis unusual capability is not the main value of myprogram, I’d be interested in feedback from anybodywho tries it: http://david- v.home.texas. net/MC3D/Regards,David V.
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