Thread: "Dimensionality Notation and Other Cubing Terminology"

From: David Vanderschel <DvdS@Austin.RR.com>
Date: 06 Feb 2009 21:37:06 -0600
Subject: Dimensionality Notation and Other Cubing Terminology

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:
http://games.groups.yahoo.com/group/4D_Cubing/files/

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!

______________________________________________________________________

 <center> Dimensionality Notation and Other Terminology Considerations for Treating Higher Dimensional Analogues of Rubik's Cube by David Vanderschel </center> <h2>Introduction</h2> With the advent of a simulator for a 5D analogue of Rubik's Cube, it has become apparent to me that there is much potential for confusion resulting from careless use of 3D terms like "edge" or "face" to talk about the analogous abstract constructs in higher dimensions. What I am offering in this note is a notational technique that allows one to use such words unambiguously in contexts in which all three of 3-space, 4-space, and 5-space possible meanings are relevant. I also introduce some additional terminology which is relevant in the context. Some of this presentation takes a tutorial sort of approach, as that is the most natural way to introduce the notations and the ways of talking about what is going on. For folks who have yet to read this note, the nature of the difficulty being addressed may not be fully apparent yet. However, by the time you reach the end, you will have seen complex examples in which it becomes clear that the distinctions being made are important. One thing I have noticed in reading the messages on the mailing list for 4D_Cubing Group is that there is rarely any attempt to discuss technical issues at what I would consider a deep level. I think one of the reasons for this is the absence of an agreed upon language for carrying on such discussions. In technical contexts, a specialized language is often referred to as "jargon" and many people take "jargon" in a pejorative sense. However, such a negative attitude towards jargon is not justified when the abstract situation being discussed is sufficiently complex that the discussion cannot be unambiguous without such jargon - or, worse, without complex circumlocutions. To the extent that some folks have attempted technical discussions, I have found them to be difficult to follow because each writer has his own means of expression which may be meaningful to the writer but for which the writers have incorrectly assumed that other readers would understand. My attempt to address the problem discussed in the preceding paragraph will not be effective unless folks try to follow it. I do not wish to put myself in the position of trying to impose a particular solution to the problem; but I have probably spent more time thinking about how to address it than have others. I want to be able to regard what follows as something that folks can agree on. Thus I will be very happy to consider revising it to incorporate better techniques for addressing the problems I am trying to solve. Thus, folks should not hesitate to suggest alternatives. However, I hope folks will restrict such suggestions to what can clearly be demonstrated to be improvements as opposed to cases of "my way" which happen to be different. If there are effective means of addressing some of these issues which already have a good following and of which I am unaware, I will be happy to embrace them once I have been shown "the way". <h2>Dimensions</h2> <h3>Dimension-Neutrality</h3> It is useful to be able to talk about some of the concepts in a way that does not assume any particular dimension. Thus, for example, we should be able to talk in a generic sense about "n-puzzles". Much of what I have to say here is said in a sufficiently general way that it applies to dimensions greater than 5. This includes a dimension-neutral definition of an n-puzzle and a definition of what is a twist for an n-puzzle. <h3>Dimensionality Notation</h3> One of the main things I advocate is liberal use of what I call dimensionality prefixes and suffixes. Dimensionality prefixes in contexts like "n-cube" and "n-space" are already familiar. We will also use suffixes, as with "edge-4", to be explicit about dimensionality for certain adjectives. The way I use the concept of dimensionality for an object does not correspond to what I would call "dimensional extent" for the object. What I mean by the latter is the smallest n such that the object can be mapped 1-to-1 into n-space in a way that preserves its topology. E.g., a triangle has a dimensional extent of 2 while a cube has a dimensional extent of 3. [There is probably an accepted term for this concept which I apparently don't know. So, if somebody would clue me in, I'll fix this.] Many of the objects for which we will use the dimensionality concept are derived from parent objects. Generally speaking, such an object's dimensionality number is inherited from its parent. E.g., a corner of a 4-cube would be called a 4-corner; or we could talk about an edge-3 position relative to a a 3-cube. [There is a potential for controversy here as some folks would prefer to call a face of a 4-cube a 3-face since it is a 3-cube. I think it is more consistent to specify the full dimension of the ancestor cube on which the 'part' lies.] Dimensionality numbers appear most naturally as prefixes on nouns and as suffixes on words used as adjectives. It is not necessary to continue to apply dimensionality numbers once sufficient context has been established to make sure that there is no ambiguity. (But it is often better to err on the safe side and show them if there could be any doubt on the reader's part.) Many examples will arise in the following. <h2>Decomposing the n-cube</h2> <h3>Sub-cubes</h3> 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. E.g., it is useful to be able to talk about what would be an edge-3 position with respect to an order-2 3-puzzle, which has no edge-type 3-cubies. Sub-cubes are always defined relative to a single ancestor n-cube which is centered on the origin. We will assume here that the coordinate values of its corners all have magnitude 1. The ancestor n-cube will also be referred to as the "sub-0-cube" of itself. A sub-1-cube of an n-cube is any face of its parent n-cube and it is itself an (n-1)-cube. However, the sub-cubes are not centered on the origin. They have distinct positions and orientations in n-space. For 1 < k <= n, we inductively define a sub-k-cube of an n-cube to be a sub-1-cube of a sub-(k-1)-cube of the ancestor n-cube. A sub-k-cube of an n-cube is an (n-k)-cube. Again, sub-k-cubes have position and orientation in n-space. (Exception for sub-n, i.e. corners, which do not have orientation.) The following table summarizes terminology in common use for various sub-levels: <pre> Names for sub-k-cubes of an n-cube ancestor dimension -> 0 1 2 3 4 5 sub-0 point line segment square cube tesseract, hypercube 5-cube sub-1 endpoint edge, side face face face sub-2 corner edge ? ? sub-3 corner edge ? sub-4 corner edge sub-5 corner </pre> In general, for sufficiently large n: <pre> A sub-n-cube of an n-cube is an n-corner. A sub-(n-1)-cube of an n-cube is an n-edge. A sub-1-cube of an n-cube is an n-face. </pre> Note that the words "face", "edge", and "corner" are being used in many different dimensionality contexts even though the corresponding objects have different nature depending on dimensionality. This is a potential source of ambiguity which can be avoided by appropriate use of dimensionality prefixes and suffixes. 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.)] 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.) <h3>Sub-k Positions</h3> The sub-cube concept will not actually come up all that much, but it does lead to a way of talking about certain positions with respect to an ancestor n-cube which definitely are useful. We define the midpoint of a sub-k-cube to be in sub-k position with respect to the ancestor. <pre> Names for sub-k positions of an n-cube ancestor dimension -> 0 1 2 3 4 5 sub-0 point center center center center center sub-1 end edge, side face face face sub-2 corner edge ? ? sub-3 corner edge ? sub-4 corner edge sub-5 corner </pre> To be complete, most of these names should require following with "midpoint"; though, in practice, this does not have to happen. So it is understood, for example, that a sub-2 position relative to a 3-cube is the midpoint of an edge of a 3-cube, which can also be referred to as an "edge-3 position". Assuming a size 2 n-cube centered at the origin and looking at the coordinates values of various sub-k positions, we see that k is actually a count of the number of non-zero coordinates for such a position. E.g., a sub-n position of an n-cube has n non-zero coordinates - i.e., all coordinates have magnitude 1, corresponding to the n-corners. The midpoint of a sub-0-cube is always the origin. For an order-3 n-puzzle, all the n-cubies' locations are sub-k positions and there is a direct correspondence between the number of stickers on an n-cubie and the sub-level of its position. I.e., n-cubies in sub-k positions have k colors. However, for other orders besides 3, it is still useful to be able to refer to sub-k positions relative to the puzzle in spite of the fact that they no longer correspond to n-cubie positions. Eg., it does not make sense to refer to a 2-color position relative to an order-2 5-puzzle, but you can refer to a sub-2 position relative to the 5-puzzle. (For orders other than 3, the cube on which the sub-k positions lie is no longer of size 2, but this generalization is straightforward.) Note that I am not advocating "sub-k" language as a replacement for the familiar position names. Indeed, I encourage continuing to use them - but with the affixing of dimensionality numbers whenever there is the slightest possibility of ambiguity (or when the writer just feels the reader could benefit from a reminder about the dimensionality of the context). Furthermore, I am even encouraging the invention of new words to fill in the "?" marks in the tables above. On the other hand, the "sub-k" language makes it possible to talk about sub-positions in a generic sense that does not assume a particular sub-level or even a particular dimensional extent for the ancestor cube. Furthermore, it gives us a parameter value to talk about the level of 'subness' for a given type of position. Though the concept of that parameter is simple - the number of non-zero coordinates among the coordinate values for the location - that is awkward to say without having defined "sub-k position" terminology. Finally, in the case of order-3 puzzles, the parameter is the same as the commonly used parameter corresponding to the number of stickers (or colors) on cubies which occupy the corresponding positions - which is again a fairly clumsy circumlocution without the "sub-k" terminology. <h2>Piles and Puzzles</h2> <h3>n-piles</h3> An order-m n-pile of some objects is defined without regard to the dimensional extent of the piled objects. The n-pile concept requires that the objects have positions in n-space. The number of objects in an order-m n-pile is m^n. Furthermore, we require that the displacement between objects along any coordinate axis of n-space be 1 and the that the pile be centered at the origin. Thus the coordinate value for an object on any of the n axes must come from the set { -m/2 - .5 + k | k = 1, 2, ..., m }. For odd orders, these are integers; and, for even orders, odd multiples of .5. In the familiar order-3 case, we are talking about a value of -1, 0, or +1 for any coordinate values of the objects' locations. <h3>n-cubies </h3> An n-cubie is an n-cube of size 1. It is a subset of n-space of dimensional extent n. Cubies are located by the positions of their centers. The faces of a cubie are called "facelets" (to distinguish them from faces of the puzzle). <h3>n-puzzles</h3> An abstract definition of a Rubik analogue n-puzzle can be given as follows: An order-m n-puzzle consists of m^n n-cubies arranged in an order-m n-pile. <h3>n-stickers</h3> For each coordinate axis for which the cubie's coordinate value is the maximum or minimum (+1 or -1 in the order-3 case), the cubie has an exposed facelet. Such exposed facelets are 'decorated' by the attachment of colored stickers. The dimensional extent of a sticker is the same as that of the facelet to which it is attached - n-1 for an n-cubie. The coordinate axis perpendicular to the facelet or sticker is referred to as the axis of the sticker. A sticker on an n-puzzle is called an "n-sticker". Note that an n-sticker does not have a dimensional extent of n - in fact, an n-sticker is an (n-1)-cube of size 1. <h3>n-slices</h3> Consider an order-3 n-pile (or n-puzzle). The set of n-cubies for which the coordinate value on one of the axes is a constant is called an "n-slice". We call the axis for which the coordinate value is constant the slice axis of the slice so determined. We also say that the axis of a slice is a fixed axis for the slice. If the constant value on the slice axis is zero, the slice is said to be a center slice. Otherwise, the slice is said to be an external slice. We need to be a little more careful with higher order puzzles. These have multiple internal slices. Furthermore, for odd orders, there is a center slice, the properties of which are sufficiently different from the other internal slices that this distinction is well worth making. I propose that we use the word "wing" to describe things that are neither centered or at either end. Thus we can have "wing edge" vs. "center edge" (neither a corner) or "wing slice" vs. "center slice" (neither an external slice). For a given slice, we define its "level" as its distance from the nearest parallel external slice. So an external slice is at level 0, the next slice in at level one, etc. (It would be rare, but you could specify a level greater than m/2 relative to a particular face.) By ignoring the slice-axis and considering only the n-1 remaining coordinate axes, we can regard an n-slice as an (n-1)-pile. Note that the objects in the pile are of greater dimensional extent than the dimension of the pile itself. This is not a problem, for, in the (n-1)-pile we are only treating the n-1 coordinate axes other than the slice axis. The (n-1)-piles comprising n-slices can be considered to be centered on the origin independent of whether the (now ignored) coordinate was zero or not. <h3>Face-slices</h3> Given an n-face and regarding it as an (n-1)-pile, we could consider taking an (n-1)-slice of the (n-1)-pile. We call such a second level down slice a face-slice It is the intersection with the face of the stickers on an n-slice which is perpendicular to the face. For higher orders, these are useful for talking about how various 1-color cubies are placed within a face. Relative to a given face we can talk about external face-slices, wing face-slices, and center face-slices. E.g., "This sticker is at the center of a wing face-slice." <h2>Twists</h2> A twist for an n-puzzle consists of applying a given reorientation of the (n-1)-cube to a set of n-slices with the same axis. In general, a twist may be applied to as many slices as desired and still be considered a single twist unless it is applied to all the slices with a given axis, in which case it is simply a reorientation of the whole puzzle which does not count as a twist at all. A center slice twist is equivalent to performing the opposite twist on the other slices and then applying the original twist to all slices - the latter being a reorientation of the whole n-puzzle. In practice, successive twists of the same slice set or its complement (with respect to the set of slices with the same slice axis) should be regarded as a way of specifying a single reorientation of the slice set and not as separate twists. <h2>Final Thoughts</h2> <h3>Illustrative Examples</h3> Now we can assert things like the following: "A 4-slice of the order-3 4-puzzle can be treated as 3-pile, in which context we can refer to the 3-corners of a 4-slice." "A 3-corner of a center 4-slice is in an edge-4 position relative to the 4-puzzle." It is very useful to be able to speak and write unambiguously about 3-space positions relative to a 4-slice which is regarded as a 3-pile. As can be seen from the example above, a 4-cubie's relative position in a 4-slice in 3D terms depends not only on the cubie's 4D type (or number of colors) but also on whether the slice is an external slice or a center slice. So words like "corner" and "edge" have the potential to be ambiguous without the occasional dimensionality prefix or suffix. A more familiar example can be seen with a center slice of a 3-puzzle: "A corner-2 of a center 3-slice is in an edge-3 position relative to the 3-puzzle." Here is something much more complex in the context of the 5-puzzle: "To specify a reorientation of a 5-slice, we can decompose the 5-slice into 4-slices by picking a second fixed axis in addition to the 5-slice axis. That second fixed axis serves as a 4-slice axis relative to the 4-pile which comprises the 5-slice. Then, in the spirit of MC4D, we can specify a pivot cubie (or pivot position) in one of the 3-piles which make up the 4-slices of the 4-pile. The corresponding reorientation of that 3-pile can then be applied to all three 3-piles which make up the 5-slice." This specifies in a fairly general way a reorientation of the 5-slice. It is meaningful to talk about such a pivot cubie as being in, say, an edge-3 position relative to its 3-pile. Depending on whether its 3-pile is a center 4-slice of the 5-slice and whether the 5-slice is a center 5-slice of the 5-puzzle, the "edge-3" 5-cubie could have any number of colors from 2 to 4. <h3>n-puzzles are not n-cubes.</h3> One thing I am trying to avoid here is referring to an n-puzzle as any sort of cube. I think that is a misconception. An n-puzzle actually consists of many small n-cubes (n-cubies) held in a certain spatial arrangement with specific constraints on how that arrangement may be modified. The sense in which a "face" of the whole puzzle exists is weak. At best, a face of an order-m n-puzzle can be regarded as a locus in n-space which contains m^(n-1) n-stickers. It has no 'physical' existence even from this abstract point of view; but the stickers do, and the set of stickers which lie in the face is dynamic. Note that what MC4D draws for a face of the 4-puzzle is actually a representation of the 3-pile derived from the 4-stickers which lie in the face. Each opposite pair of faces uses a separate fixed axis - namely the one which is perpendicular to the faces. (It is tempting to treat the stickers in the 3-pile for a face as representing the 27 4-cubies which lie in the external slice corresponding to the face and to which the stickers are stuck. Unfortunately in MC4D, center slices are difficult to 'see' in this sense.) <hr> <address></address>  Last modified: Fri Feb 06 20:09:37 CST 2009  </body> </html> ______________________________________________________________________ <hr/> From: Roice Nelson <roice3@gmail.com> Date: Sat, 7 Feb 2009 18:23:07 -0600 Subject: Re: [MC4D] Dimensionality Notation and Other Cubing Terminology --00163646bfcc16bd2004625d42fd Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit 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<http://udel.edu/~tomkeane/RubikTesseract.pdf>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 now... Best, Roice On Fri, Feb 6, 2009 at 9:37 PM, David Vanderschel <DvdS@austin.rr.com>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: > http://games.groups.yahoo.com/group/4D_Cubing/files/ > > 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! > --00163646bfcc16bd2004625d42fd Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable <div>Hi David,</div><div> </div><div>Thank you for providing a framework= to talk about notation. There is a lot suggested here, and I'm s= ure there are many thoughts among the group. I am going to keep my re= sponse short by limiting myself to feedback on the piece type names for now= . I'll stick to m^n notation below.</div> <div> </div><div>To poke and prod the suggested naming, I want to ask: &= nbsp;is "4-edge" unambiguous? The dimensionality prefix cla= rifies 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 edg= e-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 usi= ng the term "edge" at all (except in the ubiquitous 3D case). &nb= sp;"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 piece= s might well be (n-2)C pieces. (Another argument against "face" i= s to avoid it being overloaded, as it is already used in the context of puz= zle faces.)</div> <div> </div><div>Yes, the ambiguity is resolved if we all agree an "= ;n-edge" always has n-1 colors as in your tables. But the termin= ology 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'?".</div> <div> </div><div>This is why I have really liked the 1C, 2C, 3C, etc. pi= ece type designations that have evolved. I find these identifiers ext= remely useful and clear. Bonus that they are short to write out. &nbs= p;It is often the case that interesting properties we want to observe tend = to be associated with piece types irrespective of dimension, and these labe= ls work well for that. So for example, we can make observations like = David Smith did "apse: collapse; ">concerning the permutations of 2C and 3C pieces on an odd= cube". How would one say that with the terminology you&#= 39;ve laid out? Maybe with "face and hypoface pieces" (lack= ing dimension prefixes), but who wants to keep coming up with and rememberi= ng names for new piece types as we climb the dimension ladder? Admitt= edly, (n-1)C is a bit clunky compared to n-edge, but is unambiguous and the= scheme is infinitely extensible.</div> <div> </div><div>se: collapse; ">separate; ">Maybe a mix is in order? Your dimensional prefixes plus n= umber of colors, e.g. a "4-2C piece"? </div> <div> </div><div>se: collapse; ">separate; ">Also, for those that haven't seen it, the <a href=3D"http:/= /udel.edu/~tomkeane/RubikTesseract.pdf">Rubik Tesseract paper</a> calls 2Cs= "dyads", 3Cs "tryads", and 4Cs "tetrads". &n= bsp;Thought that was a relevant bit of history here.<div> </div><div>Well, I've already written more than intended so I'l= l stop at this point for now...</div><div> </div><div>Best,</div><div>Ro= ice</div><div> </div> <div class=3D"gmail_quote">On Fri, Feb 6, 2009 = at 9:37 PM, David Vanderschel <<a href=3D"mailto:DvdS@= austin.rr.com">DvdS@austin.rr.com</a>> wrote: <blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p= x #ccc solid;padding-left:1ex;"> <div style=3D"background-color:#ffffff"> <div style=3D"width:655px"> <div style=3D"width:470px;margin:0;padding:0 25px 0 0;float:left"> <div> 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: <a href=3D"http://games.groups.yahoo.com/group/4D_Cubing/files/" target=3D"= _blank">http://games.groups.yahoo.com/group/4D_Cubing/files/</a> 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<br= > 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" becam= e "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<br= > 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! </div></div></div></div></blockquote></div><br= ></div> --00163646bfcc16bd2004625d42fd-- <hr/> From: David Vanderschel <DvdS@Austin.RR.com> Date: 08 Feb 2009 00:13:56 -0600 Subject: Re: [MC4D] Dimensionality Notation and Other Cubing Terminology On Saturday, February 07, "Roice Nelson" <roice3@gmail.com> wrote: >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? No. On a 4-puzzle an edge piece has 3 colors. My proposal does not leave this ambiguous. See the first table. It strikes me that this use of a dimensionality prefix is not even all that useful because it is likely that, in most contexts where we want to talk about an edge cubie, we have already established the dimension of the puzzle. When I have applied dimensionality to "edge", it was being used as an adjective in a context more like your use of "edge-3" below and in which other dimension objects existed to create the potential for confusion. >(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). Yeah, but you got complicated on what I regard as a fairly straightforward issue. >I think one could argue for both, hence I avoid using >the term "edge" at all (except in the ubiquitous 3D >case). You could argue for both. However, I don't like the ambiguity, so I have chosen one. We can argue about whether or not my choice was a good one. If folks can agree about the choice, then the ambiguity disappears. If you are arguing that the normal interpretation of the words is at odds with the definition I gave, that is an objection worth considering. However, I don't actually see that. When I elaborate on the sub-k positions, it seems to me that I am using the term "edge" in a consistent way across multiple dimensions. >"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.) Again, use of dimensionality prefixes on cubie type does not strike me as being a big help. It seems that your issue has nothing to do with the dimensionality prefixes or suffixes but with using words like "face" and "edge" to describe cubie types. However, you are picking this up in a context where I had not even started talking about cubies yet. I was not talking about puzzles. I was talking about an n-cube. I was talking about naming positions relative to an n-cube. I was not talking about naming cubie types. As a matter of fact, looking over the document, I don't see anywhere where I was talking about cubie types. >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'?". So we explain it. This is precisely the purpose of having such a document to help out newcomers. If you somehow fail to think this use of the words is logical, then it is necessary to just take it as an arbitrary definition. However, I don't think it fails to be logical, and I do think a newcomer would have no difficulty adopting the view offered. It seems to me that your concern for the newcomer is an argument for the existence of such a document. If the document only says what the newcomer could have surmised without it, then we did not need it for that particular purpose. >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. Consider this paragraph from the document: For an order-3 n-puzzle, all the n-cubies' locations are sub-k positions and there is a direct correspondence between the number of stickers on an n-cubie and the sub-level of its position. I.e., n-cubies in sub-k positions have k colors. However, for other orders besides 3, it is still useful to be able to refer to sub-k positions relative to the puzzle in spite of the fact that they no longer correspond to n-cubie positions. Eg., it does not make sense to refer to a 2-color position relative to an order-2 5-puzzle, but you can refer to a sub-2 position relative to the 5-puzzle. (For orders other than 3, the cube on which the sub-k positions lie is no longer of size 2, but this generalization is straightforward.) I pointed out the correspondence that exists in the order-3 case. However, I was trying to emphasize that the sub-k positions do not have to be regarded as cubie positions. I did not mention "face" or "edge". I did mention color counts. I think you have gotten hung up on cubie type names when what I was talking about was mostly positions (points) relative to a single n-cube. I like the "dC" way of speaking myself. ("d" is for digit.) There is a problem with it, however: I define the "type" for a cubie as being determined by the set of locations in the puzzle to which it can be transported. In this sense, the number of colors is not sufficient to determine type. There is more to it than that in higher order puzzles. There is an issue here. If we use "type" to indicate only the sticker count, then what should we call the more precise type that depends on which set of locations the cubie can occupy. "Subtype"? I would prefer to say that two cubies are of the same "type" only if they can both occupy the same position. I would suggest then that we take the attitude that "dC" is identifying a _set_ of types, all of which have the same number of stickers. I could also use the word "class" (of types) in this context. E.g., one could refer to the "2C class of cubie types" or "cubies with types in the 2C class". There are certainly contexts where the number of colors is the only issue and subtype can be ignored. Then "dC" is simply descriptive. >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? As I mentioned, I have no objection to the "dC" way of identifying sets of cubies. In this case, it is a good way. Since I agree that "dC" is an OK way of speaking, I propose to add a paragraph to describe this convention. >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. Again you want to make this be about cubie type names when, in fact, I was naming subcubes of an n-cube. Yes, in the order-3 case it may be natural to borrow those names for cubie types; but I would not necessarily expect it. I don't think it is all that important to have a name for a sub-2 cube or a sub-3 cube of a 5-cube. But corner, edge, and face are all very useful. There have been times when the hypoface concept would have simplified what I was saying; but it is rare enough that explaining around is probably OK. (Note that 2C pieces for the order-3 4-puzzle are in hypoface positions.) >Maybe a mix is in order? Your dimensional prefixes >plus number of colors, e.g. a "4-2C piece"? Absolutely. I would never have discouraged it in the first place. But, again, this is not a very good example. Well ... maybe it is OK. E.g., you might be comparing properties of 2C pieces in the 4D and 5D contexts. >Also, for those that haven't seen it, the Rubik >Tesseract >>paper<http://udel.edu/~tomkeane/RubikTesseract.pdf>calls >2Cs "dyads", 3Cs "tryads", and 4Cs "tetrads". >Thought that was a relevant bit of history here. Yes. It might have been a good thing had the MC4D folks picked up on the work of Kamack and Keane when they started. The "__ads" terminology is not bad and it is consistent with "dC". Going on to pentads and hexads is not such a big leap. That article is very profound. I wish I had read it many years ago. I have spent a lot of time rediscovering much of what they had worked out in detail 27 years ago! Regards, David V. <hr/> From: David Vanderschel <DvdS@Austin.RR.com> Date: 08 Feb 2009 18:50:49 -0600 Subject: Re: [MC4D] Dimensionality Notation and Other Cubing Terminology 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: >I think we are generalizing by saying n-puzzle, when >it would be just as easy to say n-cube, which is the >ultimate shape of these puzzles (MC4D & MC5D). The problem with that is that we do need to be able to talk about a simple n-cube also when we are talking 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. <hr/> From: Don Hatch <hatch@plunk.org> Date: Thu, 3 Dec 2009 18:07:07 -0500 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: > > 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.)] > > 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.) <hr/> From: "David Vanderschel" <DvdS@Austin.RR.com> Date: Wed, 9 Dec 2009 16:05:09 -0600 Subject: Re: [MC4D] Dimensionality Notation and Other Cubing Terminology 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: > > 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.)] > > 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.) <hr/> <div style="text-align: center;"> <a href="http://www.superliminal.com/cube/cube.htm">Return to MagicCube4D main page</a> <a href="http://www.superliminal.com/">Return to the Superliminal home page</a> </div> </body> </html>