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Date: Tue, 4 Jan 1994 13:48:03 -0500
From: der Mouse
Message-Id: <199401041848.NAA12863@Collatz.McRCIM.McGill.EDU>
To: cube-lovers@ai.mit.edu
Subject: Re: Which is the Real Start?
> The net is so wonderful about answering questions, here are a few
> more:
I am hardly more than a dilettante of the Cube, but I can perhaps offer
a few suggestions, since this seems to me to be largely psychology and
3-D geometry rather than Cube group theory.
> 1. Take a standard 3x3x3 Rubik's cube, and remove the corner and
> center labels to make an Edges Cube. [...] Scramble the cube.
> Give it to a cubemeister to solve. How will the cubemeister know
> if the cube is solved? In other words, how will the cubemeister
> distinguish Start from Pons Asinorum?
> One answer is that the cubemeister cannot. [...] Another answer
> is that either one is Start -- that there are two Starts.
Obviously, it is correct to state that without some a priori knowledge
of the cube's coloring, the cubemeister cannot tell. As for whether
you call them both Start:
> However, if you like this answer, and if you identify the identity
> with Start, you are in the disquieting situation of having a group
> with two distinct identities (grin!).
Not at all. All you have to do is consider the group elements to be
equivalence classes under not only whole-cube rotation but also
reflection.
If you take a(n ordinary) cube and rotate the whole thing a
quarter-turn, the result is not essentially different from the
original - most programs and virtually all humans would consider them
"the same". Taking the stand that Start and P.A. are the same on the
Edge Cube means only that on the Edge Cube you consider a single group
element to consist of not only a position and all those reachable by
whole-cube rotations, but also those reachable by reflections as well.
The group-theoretic identity is then neither Start nor Pons Asinorum,
but rather the equivalence class containing both those (and 46 other
elements).
> 2. [...] Just what is the 2x2x2 cube? Or more correctly, how do you
> know when it is solved?
When you have achieved any of the 24 elements of the class that we lump
together as Start.
> With any size of cube, if you restrict yourself to quarter-turns,
> by definition you cannot rotate the cube in space as a single
> operation.
I'd argue the 1x1x1 breaks this statement :-)
What's more, it's not clear what "quarter-turn" includes: it usually
doesn't include slice turns on the 3-Cube, but on the 4-Cube and
higher, they must of necessity be included.
> Yet, a simple quarter-turn sequence such as RL' does rotate the
> 2x2x2 cube because it is faceless. Is Start of the 2x2x2 operated
> on by RL' solved?
Yes, I would say so. I would hope most people would.
> If so, you can argue that the 2x2x2 has 24 Starts. Most people
> would not. They would argue that there is only one Start, and
> that 2x2x2 cubes that differed only by a rotation are equivalent.
Right. I would say that RL' produces a cube that is precisely as
solved as that produced by RR' is - that on the 2x2x2, R and L are in
some sense the same thing. My position would be that there is only one
Start on the 2-Cube, and it is an equivalence class with 24 members.
> 3. Combining #1 and #2, I *think* that most people would argue that
> Start and Pons Asinorum on the Edge Cube are not equivalent, but
> that simple rotations of the 2x2x2 are equivalent. If I am
> correct about "most people", why?
I would say that Start and Pons Asinorum on the Edge Cube can be looked
at as mathematically equivalent (though they need not be, if you
choose) but are not intuitively equivalent. Physical objects generally
cannot be turned into reflected versions of themselves; they normally
*can* be turned into rotated versions of themselves. Thus, rotations
"feel" equivalent, but reflections don't.
> 4. [...] Since Start and Pons Asinorum differ only by a simple
> reflection, why had not my version of M-conjugation declared them
> to be equivalent?
I'm too lazy to answer this; I no longer have the messages describing
exactly what your M-conjugation operation is online. Presumably, you
implemented some intuitively-reasonable operation, and it produced
identical results for rotations but not reflections.
> 5. What is a reflection, really?
Ouch. Mathematically, this is easy enough: given a center of reflection
P in Cartesian 3-space, one computes the reflection of a point p as
P+(P-p). All the things one thinks of as reflections can be
represented as this operation compounded with rotation and/or
translation.
> Here is an exercise to illustrate the question. Take two
> identically colored and oriented 3x3x3 cubes. On one, perform F
> and on the other perform F'. Examine the two cubes, plus their
> images in a mirror. Why are there four distinct cubes rather than
> only two?
There are certainly four cubes - or at least four cube images. For
there to be only two distinct cubes, one would have to identify some of
them with one another. However, the only operations (on the cube as a
whole) that will allow identifying two of them are (1) reflection and
(2) recoloring. If your mathematical treatment considers reflections
or recolorings to be equivalent, then mathematically, there are only
two distinct cubes. Neither of these operations "feels" trivial,
though, so the four cubes all "feel" distinct.
> At this point, I can't help but note Martin Gardner's famous
> mirror question in Scientific American many years ago: why does a
> mirror reverse left and right but not up and down?
(rot13 for those who would rather think about this for a while.)
Nf abgrq va jungrire vg jnf V ernq gung dhrfgvba va, vg qbrfa'g - vg
erirefrf sebag-gb-onpx (jurer "sebag" naq "onpx" ner qrsvarq va grezf
bs gur fhesnpr qbvat gur ersyrpgvat). Jul guvf *nccrnef* gb nzbhag gb
erirefvat yrsg naq evtug vf n zber vagrerfgvat dhrfgvba, naq vg nzbhagf
gb nfxvat jung xvaqf bs fcngvny bcrengvbaf jr cresbez jvgubhg abgvpvat
(pbafvqrevat gurz abbcf). Va gur pnfr bs n ersyrpgvba bs n crefba, gur
bcrengvba jr'er cresbezvat jvgubhg abgvpvat vf gung bs znccvat
crefba-vzntr bagb frys-vzntr ol ebgngvba, fb nf gb (1) znc urnq bagb
urnq naq srrg bagb srrg naq (2) znc obql-sebag gb obql-sebag naq
obql-onpx gb obql-onpx. Ersyrpgvba, pbzcbhaqrq jvgu guvf ebgngvba,
*qbrf* erirefr yrsg-gb-evtug.
> 6. [...]
I'm not qualified to comment.
der Mouse
mouse@collatz.mcrcim.mcgill.edu