Message #648
From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Dimensionality Notation and Other Cubing Terminology
Date: Fri, 06 Feb 2009 21:37:06 -0600
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
- 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!
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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN">
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Dimensionality Notation<br>
and Other Terminology Considerations<br>
for Treating Higher Dimensional Analogues<br>
of Rubik’s Cube<br>
<font size=+1>
by David Vanderschel<br>
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<h2>Introduction</h2>
<p>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.
<p>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.
<p>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>
<p>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>
<p>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.
<p>The way I use the concept of dimensionality for an object does not
correspond to what I would call "<strong>dimensional extent</strong>"
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.]
<p>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.]
<p>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>
<p>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.
<p>Sub-cubes are always defined <em>relative</em> 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.
<p>A <strong>sub-1-cube</strong> 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.
<p>For 1 < k <= n, we inductively define a <strong>sub-k-cube</strong>
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.)
<p>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>
<p>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>
<p>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.
<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.)
<h3>Sub-k Positions</h3>
<p>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 <strong>sub-k position</strong> 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>
<p>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".
<p>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.
<p>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.)
<p>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.
<p>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>
<p>An <strong>order-m n-pile</strong> 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>
<p>An <strong>n-cubie</strong> 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
"<strong>facelets</strong>" (to distinguish them from faces of the
puzzle).
<h3>n-puzzles</h3>
<p>An abstract definition of a Rubik analogue n-puzzle can be given as
follows: An <strong>order-m n-puzzle</strong> consists of m^n
n-cubies arranged in an order-m n-pile.
<h3>n-stickers</h3>
<p>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 <strong>stickers</strong>. 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 <strong>axis</strong> of
the sticker. A sticker on an n-puzzle is called an
"<strong>n-sticker</strong>". 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>
<p>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 <strong>slice axis</strong> of the slice so determined.
We also say that the axis of a slice is a <strong>fixed axis</strong>
for the slice. If the constant value on the slice axis is zero, the
slice is said to be a <strong>center slice</strong>. Otherwise, the
slice is said to be an <strong>external slice</strong>.
<p>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).
<p>For a given slice, we define its "<strong>level</strong>" 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.)
<p>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>
<p>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 <strong>face-slice</strong> 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>
<p>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>
<p>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."
<p>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>
<p>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.
<p>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>
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