Message #651
From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Re: [MC4D] Dimensionality Notation and Other Cubing Terminology
Date: Sun, 08 Feb 2009 18:50:49 -0600
On thinking more about it, I am not happy with the
response I made to Roice’s comments on the draft. The
problem is that Roice was coming from a such a
profound misconception about what he was commenting on
that the same point of confusion kept coming up
repeatedly. Thus my inline comment approach was
counterproductive. I want to follow up in hopes of
preventing others from adopting the same flawed point
of view. There are interesting new thoughts here too.
Roice’s concerns had to do with naming cubie types and
otherwise identifying them. But I had not set out to
address those issues at all! Period. Seriously!
I was not too concerned about this area of nD cubing
because we already seem to have conventions that are
working. However, I am not saying that those issues
do not need to be addressed, and I will try to add
some material in this area.
I scanned the document for occurrences of "name" and
there was only one which was in the context of cubie
types: "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."
The naming part was an afterthought and restricted to
the order-3 cases. I said they "can" be used to name
cubie types. This is reasonable since we have plenty
of precedents for it already. I was not actively
recommending such use, but I recognized its
inevitability.
The section Roice was commenting on was the one I
titled "Decomposing the n-cube". The first sentence:
"The concept I am going to introduce here provides a
way of talking about certain types of positions
relative to an n-cube in a way that is not connected
with n-puzzle-specific concepts." I actually
explicitly denied that I was talking about n-puzzles.
I was talking about an abstract n-cube.
It is a fact that we lack good techniques for talking
about the geometry of higher dimensional cubes. For
n>3, things get pretty messy when you start getting
down into the detailed structure of an n-cube. Roice
is so good at thinking about these issues that it may
not occur to him that the lack of a good language for
expressing such thoughts to others is a problem. I
surmise that, not regarding the lack as a problem, he
apparently failed to see that what I was really
providing was sort of an nD geometry lesson with
emphasis on the structure of the abstract n-cube.
I must imagine that Roice skimmed very lightly over
the introductory material, saw those tables with words
I intended for geometrical reference, and, in spite of
the titles on those tables, inferred erroneously that
it was about cubie types.
One thing that is especially ironic about this
confusion is that it is an area of confusion that I
was explicitly trying to avoid. Recall this
interchange from the day before:
On Wednesday, February 04, "rev_16_4" <rev_16_4@yahoo.com> wrote:<br>
>I think we are generalizing by saying n-puzzle, when<br>
>it would be just as easy to say n-cube, which is the<br>
>ultimate shape of these puzzles (MC4D & MC5D).
The problem with that is that we do need to be able to<br>
talk about a simple n-cube also when we are talking<br>
about an n-puzzle (not necessarily the same n's).
So even though I was making the point that talking
about an n-cube and n-puzzle are different and can
come up in the same context, Roice managed to take my
tutorial words about an n-cube as applying directly to
an n-puzzle.
OK. So much for the confusion. Now on to Roice’s
issue.
What Roice advocates about using the "dC" terminology
(e.g., "3C", "1C") is entirely consistent with my
motivations of dimension-neutrality. Furthermore, I
would take it even farther and admit it in a more
mathematical notation sense like "kC" where k is the
symbol for an integer-valued variable. E.g., "kC
cubies lie on sub-k-cubes of the n-cube corresponding
to the n-puzzle."
(A thing I need to add is the following: "In the
context of an n-puzzle, references to an n-cube are
assumed by default to refer to the n-cube whose
corners correspond to the centers of nC (or corner)
pieces of the puzzle.")
The "dC" terminology already plays well with what I
have written. E.g., I would like to be able to say
something like, "For the order-3 4-puzzle, the 2C
cubies are located at the centers of the hypofaces."
The word "hypoface" is _not_ being used to _name_ a
cubie type - it is being used to specify _where_ such
cubies lie relative to the n-cube. If I am denied
"hypoface" (or some other word for sub-2-cubes), my
backup is that "the 2C cubies are located at sub-2
positions with respect to the 4-cube." If I am denied
the sub-k concept, then explaining where these cubies
lie relative to the puzzle becomes very cumbersome.
Generalizing the order of the puzzle, we could say
things like, "m-2 (n-1)C cubies lie evenly spaced
along each sub-(n-1)-cube (between the nC cubies which
lie at either end)." Or, "For 4^4, there are 4 2C
cubies around the middle of each hypoface."
Regards,
David V.