Message #649

From: Roice Nelson <>
Subject: Re: [MC4D] Dimensionality Notation and Other Cubing Terminology
Date: Sat, 07 Feb 2009 18:23:07 -0600

Hi David,

Thank you for providing a framework to talk about notation. There is a lot
suggested here, and I’m sure there are many thoughts among the group. I am
going to keep my response short by limiting myself to feedback on the piece
type names for now. I’ll stick to m^n notation below.

To poke and prod the suggested naming, I want to ask: is "4-edge"
unambiguous? The dimensionality prefix clarifies that we are talking about
an edge on a 4-puzzle. Unfortunately, to me "edge" still has a great deal
of ambiguity surrounding it. Is it a 2C piece? (analogous to edges on the
3^3 because they share the same number of colors) Or is it a 3C piece?
(analogous to edges on the 3^3 because they have n-1 colors or because "they
are in an edge-3 position relative to the 3-pile of stickers comprising a
4-face" - tried to use the suggested terminology of the writeup there, not
100% sure I got it right). I think one could argue for both, hence I avoid
using the term "edge" at all (except in the ubiquitous 3D case). "Corners"
are less ambiguous so I do indulge using that label, but I dislike using
"face" pieces for similar reasons. Why should faces always be 2C pieces and
edges (n-1)C pieces? Face pieces might well be (n-2)C pieces. (Another
argument against "face" is to avoid it being overloaded, as it is already
used in the context of puzzle faces.)

Yes, the ambiguity is resolved if we all agree an "n-edge" always has n-1
colors as in your tables. But the terminology is not self-descriptive
enough to stand on its own, and so we’ll inevitably have to answer the
question of meaning for new members who will ask the question "why ‘edge’?".

This is why I have really liked the 1C, 2C, 3C, etc. piece type designations
that have evolved. I find these identifiers extremely useful and clear.
Bonus that they are short to write out. It is often the case that
interesting properties we want to observe tend to be associated with piece
types irrespective of dimension, and these labels work well for that. So
for example, we can make observations like David Smith did "concerning the
permutations of 2C and 3C pieces on an odd cube". How would one say that
with the terminology you’ve laid out? Maybe with "face and hypoface pieces"
(lacking dimension prefixes), but who wants to keep coming up with and
remembering names for new piece types as we climb the dimension ladder?
Admittedly, (n-1)C is a bit clunky compared to n-edge, but is unambiguous
and the scheme is infinitely extensible.

Maybe a mix is in order? Your dimensional prefixes plus number of colors,
e.g. a "4-2C piece"?

Also, for those that haven’t seen it, the Rubik Tesseract
paper<>calls 2Cs "dyads",
3Cs "tryads", and 4Cs "tetrads". Thought that was a
relevant bit of history here.
Well, I’ve already written more than intended so I’ll stop at this point for


On Fri, Feb 6, 2009 at 9:37 PM, David Vanderschel <>wrote:

> I have written a draft for a document which attempts
> to establish some terminology which I hope we can
> agree on for talking about cubing in various
> dimensions. I have written it as an HTML document
> using only rudimentary HTML features.
> I have uploaded the file to the files area for the
> Group, which is here:
> Yahoo prevents me from giving you a link to the HTML
> file itself. You can find it there as
> Dimensionality.html.
> I actually started these notes two years ago. I
> hesitated to offer them then for fear that my action
> would be perceived as being too ‘pushy’. However,
> remarks that Levi and Roice have made recently about
> needing agreement on terminology have led me to
> conclude that maybe the time now is right to try this.
> One of the reasons that I believe the place for the
> document is the Files area is that I expect the
> document to evolve. Right now, it includes quite a
> bit of discussion about my motives and justification
> for the effort. In the long run, such discussion can
> be removed. Furthermore, other aspects of notation
> and terminology may come to be included. I volunteer
> to maintain the document; but I am hoping that many of
> you will take an interest and make contributions.
> (Perhaps we need multiple documents in a folder, as
> there are other areas which could also use some
> ‘standardization’.)
> I am going to append the raw HTML of the current state
> of the draft to this email. The use of HTML markup in
> it is so rudimentary that the document is fairly
> readable in raw form. However, I do recommend reading
> it in your browser. The real reason for including the
> plain text here is to facilitate quoting for someone
> who wants to comment directly on the text of the
> draft.
> The only significant change I made in the draft from
> two years ago was to add more consideration for orders
> greater than 3, which have only recently begun to
> interest me.
> Historical note: When I wrote my 4D program several
> years ago, I was already aware of the ambiguities
> inherent in use of terms like "edge" and "face" in a
> context that includes objects of dimension higher than
> 3. My solution then was to introduce a whole new set
> of words for the 4D cubie types. I did it by forcing
> in an "h" to connote ‘hyperness’. "Cube" became
> "Hube", the name of the program. I wound up with
> Faysh, Ehdge, Cohrner, and Phage type hubies. (In 4D,
> the 2C type is intermediate in nature between 3D edges
> and faces, so I concatenated the beginning of "face"
> with the end of "edge" using "ph" for the "f" sound to
> get the "h" in there. That’s where my "Phage" type
> comes from.) I was happy with this because it allowed
> me to use "edge" and "corner" in their 3-space sense
> which (as I point out in the document) is not always
> consistent with their meaning in 4-space. However,
> when Roice introduced his 5D program, I realized that
> my approach was going to be very cumbersome when the
> number of dimensions was so variable. I then came up
> with the idea of liberal use of what I call
> "dimensionality prefixes" and "dimensionality
> suffixes". I have had enough experience with them now
> that I know this provides a fairly effective means to
> remain unambiguous in otherwise awkward situations. I
> think I would have had a hard time selling my funny
> names anyway. ;-)
> I am hoping that many of you will check out the draft
> and that a lively discussion will ensue here.
> Regards,
> David V.
> PS for Levi - I can assure you that the last
> subsection, "n-puzzles are not n-cubes.", was already
> in there 2 years ago!