Message #805

From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Re: [MC4D] Dimensionality Notation and Other Cubing Terminology
Date: Wed, 09 Dec 2009 16:05:09 -0600

I heartily concur with all of the points in Don’s note quoted
below. As I wrote initially, "Both of these suggestions are very
tentative.
I am open to alternative suggestions." Clearly I was groping for
some external authority on this one. Roice and I discussed this
at some length privately. We were both looking at the N-cube
page at Wikipedia: http://en.wikipedia.org/wiki/N-cube. It has
inconsistencies that neither of us were happy with. I must admit
that I did not even know about the Polytope page to which Don has
pointed us. It solves the problem! It solves it better than
anything I suggested. Borrowing the polytope words for the
special case of n-cubes is not a problem. In fact, it helps to
emphasize that something new is going on here. The connotations
of "facet" and "ridge" are, for me, quite good. I like that it
provides a separate set of words depending on whether you are
coming down in dimension from the parent object or up from
vertices. With the consistency of "k-face" in the polytope
context, I can even give up on my "sub-k-cube" terminology, which
I had invented only so that I could be unambiguous about the
rest. In d dimensions (as in the Wikipedia article) a k-face is
what I was calling a sub-(d-k)-cube.

I will replace the paper with a modified version which is in line
with the polytope terminology from Wikipedia. However, there has
been so little interest shown so far that I don’t think the paper
is as important as I imagined when I wrote it. I think that
there is such a small number of people in the whole world who
actually care about these issues at a deep theoretical level that
it is counterproductive to try to get uniformity on how to
express all the possible relevant thoughts. If someone has
something new and interesting to report, we can expect that he
will explain his notation and terminology. If such notations and
terminology work well, others may be expected to jump on the
bandwagon; but there is no need to try to anticipate what needs
to be expressable. So I will also try to simplify the paper down
to what is more immediately relevant. However, this is also a
good time to jump in if anyone else has some suggestions on what
should go in there. I am reluctant to extend it beyond cubes at
this time. The old version will remain in the Group’s Files area
for a little while.

I once felt strongly that we should use "n" for the top level
dimension in which we are working, as "n-cube" is a common way of
talking about higher order cubes with generic dimension. But I
now see that "d" does have a good following also. Should we
switch to "d-cube"? Personally, I think it is less likely to
produce the correct semantic reaction on first viewing.

Regards,
David V.

—– Original Message —–
From: "Don Hatch" <hatch@plunk.org>
To: <4D_Cubing@yahoogroups.com>
Cc: "Don Hatch" <hatch@plunk.org>
Sent: Thursday, December 03, 2009 5:07 PM
Subject: Re: [MC4D] Dimensionality Notation and Other Cubing
Terminology

Hi David,
Just a couple of small points in all of this…
You propose the new term "hypoface" for a sub-2-cube…
It makes sense (as does the discussion of "hypo" and "super"),
but I feel like this new term is unnecessary
since facet,ridge,peak are (fairly) standard
and clear unambiguous terms for the sub-1,2,3-cube
respectively…
http://en.wikipedia.org/wiki/Polytope
(and see the links at the bottom)
And, I’d definitely stay away from using "face" to mean
sub-1-cube
(or anything else if possible)…
"face" is so traditionally overloaded that it will always be
confusing,
and we have the very clear term "facet" which can be used instead
to mean sub-1-cube.
Don

On Fri, Feb 06, 2009 at 09:37:06PM -0600, David Vanderschel
wrote:
>
> <p>For 4-cubes and 5-cubes, we lack words for some of their
> sub-cube
> types. I think it would be helpful to have such words,
> especially
> since they can also be used to name cubie types for order-3
> puzzles.
> For a sub-2-cube of an n-cube when n>3, I propose
> "hypoface". For a
> sub-(n-2)-cube of an n-cube when n>4, I propose "superedge".
> (Note
> that the "hypo" prefix suggests going downwards in
> dimensional extent,
> while "super" suggests going upwards. I would have preferred
> "hyper"
> to "super", but it would probably be helpful if no two names
> started
> with the same letter.) [Both of these suggestions are very
> tentative.
> I am open to alternative suggestions. (An alternative for
> this pair
> might be "subface" and "hyperedge"; but "subface" does not
> quite have
> the right ‘feel’ about it for me.)]
>
> <p>If you were to apply the above new names to lower
> dimension cubes,
> you would discover the following: For a square, a hypoface
> is a
> corner and a superedge is the whole square. For a 3-cube, a
> hypoface
> is an edge and a superedge is a face. For a 4-cube, a
> hypoface and a
> superedge are the same thing. (Multiple names for the same
> sub-level
> is a situation which already existed: E.g., on a square,
> face and
> edge would be the same.)