Message #483

From: David Smith <djs314djs314@yahoo.com>
Subject: Re: [MC4D] Introduction to the 4D_Cubing Group
Date: Wed, 30 Apr 2008 02:20:02 -0000

Roice,

Thank you very much for your reply! I really appreciate the two
resources you kindly pointed out to me. I have actually already read
the paper "The Rubik Tesseract", which is what got me very interested
in the generalized problems of n^4 and n^k Rubik’s Cubes. I got
the idea to also try supercubes and super-supercubes from the
n^3 formulas page I referred to in my previous post. That page
inspired me to rediscover the same formulas that Chris Hardwick did.
An interesting thing about 4D super and super-supercubes I
realized is that any hypercubie with more than 1 hyperfacelet can be
twisted in the ways Keane and Kamack showed in their paper. However,
center hypercubies, with only one hyperfacelet, can be oriented in
24 different positions, and they undoubtedly have certain
restrictions related to the other hypercubies. I want to figure out
the regular cube first though!

As for the other resource you mentioned, the author of that book has
a preprint version available for download (at
http://web.usna.navy.mil/~wdj/books.html) which appears to be
very complete, despite the fact that it is not the actual book.
I have already studied it, and it was a good introduction to group
theory and how it relates to the Rubik’s Cube.

The difficult part of this task is not discovering the formula,
but proving it is correct and not just an upper bound. That is,
once I have ruled out the impossible permutations, I must show
that all of the remaining permutations are actually possible.
(Keane and Kamack actually admit they did not do this for the
3^5 calculation, but state that they are very confident it is
correct.) I suppose I could use the results from that paper, the
ones where they show using a computer program, that all of
the remaining permutations are possible. However, this would
be difficult to expand to higher dimensions. Therefore, I am
writing my own computer program, whose sole purpose is to convert
4-dimensional Rubik’s cubes into cycle notation, that is, labeling
every hyperfacelet with a unique number and listing, in cycles,
where each hyperfacelet goes when each hyperface is rotated in each
necessary direction. (I tried doing it by hand at first - not
recommended!) I am currently sorting out bugs in the program.
When it is working, I pan to take the output it provides and
put that into the Computer Algebra System GAP. I would then be
able to directly calculate the number of permutations of any
specific 4D cube, but more importantly, I will be able to show
how different types of hypercubies can interplay with the rest
of the cube. (example: On the 3x3x3x3, can I actually swap
two hyperfacelets of a 3-colored hypercubie without affecting
the rest of the cube?) I then plan to generalize those results
to any sized cube, perhaps by induction.

Once again, thank you Roice, for your quick and detailed reply!
I am glad I have someone else to discuss these things with.

Best Regards,

David

— In 4D_Cubing@yahoogroups.com, "Roice Nelson" <roice3@…> wrote:
>
> Hi David,
>
> Very nice to make your acquaintance, and welcome! I’m sure others
will have
> more to say, but I thought I’d point you to two resources on the
mathematics
> I’ve seen in the mean time. The first (because it is shorter) is
a paper
> Melinda pointed me to because it had the calculation for the
number of
> permutations of the 3^5. It is titled "The Rubik Tesseract" and
you can
> find it here:
>
> http://udel.edu/~tomkeane/RubikTesseract.pdf
>
> Secondly, there is a book by David Joyner called "Adventures in
Group
> Theory". It is a course on group theory focused around the 3D
Rubik’s
> cube. Though it doesn’t discuss the higher-d puzzles, it could be
a useful
> source for extending the relevant mathematical skills.
>
> I’ll look forward to hearing about your investigations. As
someone else who
> finds this very interesting but lacks related formal mathematical
training,
> I’ve always felt lacking in the group theory aspects of these
puzzles…
>
> All the best,
>
> Roice
>
>
> On Mon, Apr 28, 2008 at 9:29 PM, David Smith <djs314djs314@…>
wrote:
>
> > Hello, everyone! My name is David Smith, and I hope to
> > be a contributive member to this unique and highly
> > interesting group. My interests besides the cube are
> > mathematics, physics, chess, computer programming,
> > and retrograde analysis (a very remarkable type of
> > chess problem).
> >
> > Reading the posts made by various members has gotten
> > me very interested in the solving of higher-dimensional
> > cubes. It seems like a very challenging task, and
> > congratulations to everyone who has solved a four or
> > five-dimensional Rubik’s Cube! I will definitely try it
> > soon. What I am interested in now, however, is the
> > mathematics of the Rubik’s Cube. I wonder if any of
> > you also have an interest in this area?
> >
> > I am currently working on an interesting problem -
> > finding a formula for the number of reachable
> > configurations of the NxNxNxN Rubik’s Cube. I
> > believe I will have an answer to this question
> > soon, so I would be very glad to share it with
> > the group, or perhaps only the members who are
> > interested in mathematics, if any. The paper
> > written by Eric Balandraud on the MagicCube4D
> > website has been very helpful, but I am
> > currently stuck on a minor detail with his
> > calculation of the number of permutations of
> > the 5x5x5x5 cube, but I believe I will
> > discover my error soon.
> >
> > After this, I want to find the same formula
> > for 4-dimensional supercubes and super-supercubes.
> > (see http://www.speedcubing.com/chris/cubecombos.html
> > for a definition of these terms) After that, I
> > will (perhaps foolishly!) attempt to find formulae
> > for cubes, supercubes, and super-supercubes of
> > any size and any dimension.
> >
> > I apologize if this post has been too long, but
> > I wanted to give a detailed introduction of myself
> > and my current tasks, and I hope that at least some
> > of you would be interested in discussing these
> > problems and their solution. I am trying to do
> > this without any formal mathematics training, so
> > my solution, when I find it, may be long but
> > relatively simple to understand.
> >
> > I wish to thank everyone who has contributed the
> > the theory of Rubik’s Cube knowledge, helped
> > in the creation of Rubik’s Cube software, or
> > otherwise done amazing things with Rubik’s Cubes.
> >
> > Happy Hypercubing!
> >
> > Best Regards,
> >
> > David
> >
> >
> >
>