# 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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<title>

<center>

<font size=+2>

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>

</center>

<font size=+0>

<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>

<address></address>

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