Unscrambling the cube

The 4 letter word above, means I have a giant meson on the corner opposite the aligned elements making up the figure (along 3 edges and meeting at the corner piece). Now I can translate this figure around the cube in any of 3 ways, so it gets decomposed and recomposed in a different position.

It's a kind of filter, because I can, by translating it with rewritten words (the rewrites transform coordinates) reposition or scramble other elements, because of the 4-letter structure of the algorithm I can stop at any point of a translation (set of moves) and flip an edge or corner, with an odd or even cycle, then reconstruct the X'YZX' figure. Sometimes I leave part of it unreconstructed so I can use the place "in the figure" to move another element. I also think of it as a kind of 3d stacking operator.

After the initial construction, either from Z = 1 or if anywhere else, by getting to an F2L partial solution. It appears to be linked to the use of the FTM on one face, during a solve, and the single "stack" in between two faces that is left unsolved during the F2L.

That's what the torquing down maneuver is about - characterizing a part of the cube (but, dang, it didn't turn the cube inside out).

I see Alph has decided to waste more of his precious time, probably gibbering away about what an idiot I am., You have to wonder - what TF does this guy do for recreation? Look for kids to beat up?

 

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Ah, so because someone points out your mistakes you're being persecuted and I just like 'beating people up'?

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If you say something wrong pointing it out isn't 'persecution'.

I really don't get why people come to discussion forums and get upset when someone says something other than "Wow, that's amazing" to them. Particularly when they know they are just making crap up.

 

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To "transform" the cube into a Moebius polynomial, you need a configuration number.
The first step is to coordinatize the vertices, using symbols. Use the integers and infinity. You need to arrange the numbering so that some transitions "go through" the cube from one corner to an opposite corner, here is a reference:

http://www.permutationpuzzles.org/rubik/crossgp.html

This is like joining vertices in a discontinous way (you can't really "get" from one corner to the other through the cube, you have to go around the cube). And the cross group yields the polynomial above, so you can "write" different numbers in the new labeling "basis".

The \( M_{12} \) is a construction that you get pre-packaged; of course you use it - even if you don't characterize it like mathematicians and computer scientists do. The trick is understanding how to exploit the "Moebius flip".

 

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To some people the Rubik's puzzles are nonsensical.
This is because some people are concerned about the "lack" of proper mathematical formalisms, I mean, the thing doesn't have any numbers on it...

Anyone who uses one is doing math though. You in fact need to develop a personal science of the cube to understand it, in your own way. Then if you can read, you can find out that other people have done this too; now it's a matter of understanding their "science of the cube". Some of these other people also know a bit of math. So you can understand just how nonsensical that is as well.

Because color is abstract you can say it's (almost) anything. It could be a "charge" of some kind (color potential) or a scale (a cube with a different "area" for each face), or just an index - the default.

The cube, minus the coloring, is already a coordinate system, and so the colors are the coordinates (really!); you eliminate whole groups of coordinates, just by moving the faces so that you get a "match" somewhere, you iterate the "matching" formula.

If you have two coins there are 4 possible outcomes of a toss: {ht, th, hh, tt}. Map these to the 4-cycle of a singleton operator X. Now if X is randomized it is equivalent to the phase space of two coins. This turns out ('meh') to be quite handy.

Ed: equivalence is fairly straightforward: the colors are equivalent to coordinates; a single X from the operator set is equivalent to a phase space (two coins are one kind of equivalent). Now, we look at someone's notes and add some reminders (to oneself) about equivalence and inertia to try to develop a sense of this in the phase space:


"...being at rest on the surface of the Earth is equivalent to being inside a spaceship (far from any sources of gravity) that is being accelerated by its engines."
* Inside a spaceship, a small ball of test particles will shrink as the particles attract each other, unless external forces act on the particles.

"From this principle, Einstein deduced that free-fall is actually inertial motion."
* Acceleration mg/(m) is inertial motion.

"By contrast, in Newtonian mechanics, gravity is assumed to be a force."
* Forces are a formal variable in Newtonian mechanics.

"This force draws objects having mass towards the center of any massive body. At the Earth's surface, the force of gravity is counteracted by the mechanical (physical) resistance of the Earth's surface."
* The surface of the earth has a mechanical resistance (R), which is the tension and stress due to gravitational compression.

"In Newtonian physics, a person at rest on the surface of a (non-rotating) massive object is in an inertial frame of reference."
* On a rotating sphere with mass, a particle at rest on the surface is NOT in an inertial frame of reference - the coordinate system is non-inertial and cannot be freely chosen.

-The Equivalence Principle, wikipedia.org

 

Holonomy and curvature

Holonomy is from the Greek holos meaning "whole" and the root nom (this is according to Wiktionary), and the latter is related to the notion of "naming or numbering".
Also to geodesy and surveying. Romans built straight roads for absolute bloody miles, using a fairly simply constructed device, the gnomon.
This employed, or deployed plumb weights and a revolving upper arm to first align a central pole upright, then sight along two plumb lines to a distant observer.

You can read all about holonomy here, and this is a nice pretty picture:

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I tried to do a sort of explanation:
Transport around a sphere as in the above diagram, is the classical example of anholonomy introduced by the curvature of the sphere, when the transported object has an initial direction, which in the case of being directed towards a pole of the sphere from the equator (any point lying on the diameter), then back to a different point changes the direction.

If the object is a vector with magnitude as well as direction, the transport to a pole and back transforms the vector (an operation on the vector space), the change in direction depends on the path over the sphere alone.

Thus there are three sections in such a path to the pole, from the pole, and back to the start point (x). Step 1 (T1) transports the vector over a geodesic curve = 1/4 the [oops, circumference projected to the] diameter so that it is parallel to a line on the equatorial plane, step 2 returns it to the equator along an arbitrary direction, along another 1/4 [circumference projected to a] diameter (the vector is now an outer derivative of the intersection of two planes through the pole that each bisect the sphere like the equatorial ), the remaining step that returns it to x_initial depends only on the angle at the pole between T1 and T2, so that T2 -> T0 is an arc subtended by the polar angle between T1,T2 a spherical triangle.

The ascent T1 (pitch) is parallel to the geodesic curve, the descent T2 (roll) is orthogonal to the angle subtended, and the final anholonomy is a result of the curvature of each connected step Tn plus "transport" = pitch, roll, yaw.

In other words a vector pointing in an arbitrary direction at the pole (with an infinite circle of directions) parallel to the equatorial plane of the sphere, which is transported to the equator along any geodesic other than parallel to the diameter it is pointing along at T0 at the pole, will 'preserve' the angular difference, when it returns to the pole.

 

I see you're still having trouble saying anything about the actual topic or explaining how you copy and pastes from Wikipedia have any bearing on the topic at hand.

I happen to know a fair bit about holonomy and while it can be vaguely related to Rubik's cubes the problem is that a Rubik's cube is not defined by a vector on a sphere, it is defined by the configuration of the entire cube. You can use the notions of holonomy to shift around a particular square but you won't end up with the original cube and just that square rotated. Further more, due to its symmetry you can only have a discrete holonomy group and rotations by 90 degrees on a square don't change it so while the vector in your Wiki picture changes (\(S^{n}\) has holonomy group SO(n)) the Rubik's cube equivalent would not.

 

So, you're saying SO(n) wants the cube, but it can't handle the cube?

I am trying to keep a straight face just now, btw.

 

Clearly basic holonomy is beyond you.

The holonomy is \(S^{2}\) is SO(2). It is a continuous symmetry group, as \(S^{2}\) is a smooth space. The cube does not have the symmetries of a sphere. You can rotate a sphere, about some axis, by 10 degrees and the system is unchanged. You can't do that with a cube. The symmetry group of an n-cube is \(S_{4}\). This is a little more complicated for a Rubik's cube since it has transformations which leave the cube's corners unchanged but which change the layout of the cube's exterior.

Your comment "So, you're saying SO(n) wants the cube, but it can't handle the cube?" illustrates all I've been saying about you. you blindly spew stuff you either don't understand or just make up.

Yes, I imagine you making a complete ignorant arse of yourself, despite repeatedly being proven to be a hack, is very funny to you. Your sense of humour is obviously well developed....

 

Does a spherical triangle have any symmetries of a sphere?
Clearly, humor is beyond you. You wouldn't know a joke if it fell out of a jug of beer and all over you.

 

Why am I bothering with this pompous twit?

Of course part of the surface of a sphere has spherical bloody symmetry - it's got a curved surface that fits on a sphere, right?

Maybe there's more to the fact that an equilateral triangle fits over a corner of a cube than I first thought.

 

No, it is a closed loop within a sphere. Particular spherical triangles have particular symmetries but your question is just the same as "Do triangles have symmetries" but in spherical geometry. Some do, some don't. Those that do don't all have the same symmetry.

Because you make claims and parrot things you don't understand and I point them out because, unlike you, I happen to actually know something about the topic. I hand in my PhD thesis tomorrow and I have entire sections devoted to holonomy in it.

The triangle in your picture doesn't have spherical symmetry. It's not a sphere of any dimension. It's a closed path in a sphere.

Holonomy is about the transformations a vector or spinor undergo through parallel transport along a closed loop within a space. The space is unchanged, the vector or spinor is moved through it and transforms due to the properties of the space. The twists and turns of a Rubik's cube actually change the space. The alterations to a Rubik's cube and holonomy look superficially similar but they are not quite the same. Not least because you have multiple choices of which bit to twist in a cube so there are multiple sections you can do the kind of action seen in your picture to.

 

'sigh'

for the cubemeisters: this thread is not about restoring a messed up Rubik's cube to the initial setting ("state"). There are enough clues already about doing this particular thing.

This one, is about why the cube is deeply-sliced, and what having the frame means, compared to what not having a frame means (trawl backwards over the thread, to see the example). Thus, by construction the frame vanishes. So what?

The cubes in the 3d group are "numbers". The first mechanical solutions are a sequence of constructions from N = 1 (not sliced, the backwards or trivial step T(n-1)) to N = 7.
What kind of numbers? N is the number-type for counting sections along an edge of the sliced cube, so that when N = 2 there are a square number of sections per face, 2(N-1) slices per face, etc. N can be seen as a configuration number, that constructs another number, the number of registers - if you can keep this low, you get "good" algorithms, patterns the operator (guess who) can recognize.

Much as Turing recognized what writing and reading (on, you guessed it, paper) is like a program that computes a number.

 

Nice changing of the subject when you realised you're in over your head. If I'm such a 'pompous twit' then surely you can put me in my place?

 

Well of course, I couldn't possibly comment.

Sorry about changing your subject all over again.

Who thinks they know something about digital and analog computers and computation, would that be... ME?
Why do we get units of squared and cube seconds, respectively in acceleration formulas or power transfer ones?

You can only construct unital cubes of dt when you formulate (i.e. write) rules for power transfer (or rewrite other units), intensity etc which is why real computation requires the storage of a volumetric form of energy - i.e. digital or analogue machines.

Or, we have to conclude that computation is Galilean, by default.

 

Wow, just wow.

I guess I don't even need to point out your BS, you do it yourself by going utterly over the top.

 

Who else thinks the Rubik's cubes and their n-dimensional analogs are computational devices or gadgets?

In what sense is any device Galilean? Well in the sense it isn't Lorentzian, I'd say.

So what class of device is it?
"
I. Thermodynamic Limit. Let a set of physical systems {Sa} range over some index set such that all systems are governed by the same mechanical theory, or "power law" (power/intensity laws define computational boundaries). Let Wa be an observable associated to Sa.
Suppose |Sa| is the complexity of the system, or the number of bits required to describe it, so depends on the number of components and interactions between them. We unitarize the components in a standard basis for observables W, i.e. seconds and metres.
Now suppose in {Sa} a family of systems for which two conditions hold:
(i) Calculating a minimum energy of Sa: minWa is NP-hard, any known method will be exponential in |Sa| (as the #components increases without bound).
(ii) the mechanical theory predicts that, dynamically Sa reaches its minimal energy state in dt, which is bounded by a polynomial in |Sa| as it approaches infinity.

II. Initial Segment. Asymptotic behaviour and thermodynamic limits are ignored in favour of a practical consideration: Are there instances of NP-complete problems that can be solved by an appropriate analog system?"

But let's not get to hand wavy here. The cubes have entropy of information and Shannon's Laws therefore apply. The number of bits is an easy problem to calculate an answer for, but not the number of paths to say, the Cayley diameter of Sa and back (or any node in the system), which is the "machine limit" for a group of stacked & colored cubes.

The group of cubes is grouped twice, once by the frame and once by colored stickers - hence the symmetry of a cube's geometry is restored when it gets regrouped, so the mechanics are an abstraction of the group - so an abstraction is a computation too. The relations are all about breaking and restoring a cube's symmetry ("freely") - this is an obvious connection to digital computation, where we use charge symmetry, say.

 

Greetings to both of you:

I just like to point out that the math knowledge in this thread is seriously worrisome , and I hope this will call some public attention to the serious national crisis in mathematical education. If you both can admit your errors, you will have contributed constructively towards the solution of a deplorable situation. We have been following this thread carefully, and all I can say is that there are many irate mathematicians in here.

LC, Ph.D., Los Alamos National Laboratory.
Los Alamos, New Mexico.

 

LC: please note that, I sometimes intentionally descend into errors and mistakes, this is so as to correct them.
I recall that even professors and lecturers make errors and mistakes too - this is to catch out students whose job it is, by default, to point them out.

For instance the hotly debated Mobius subgroup. This is on the vertex set but as I tried to naively point out, you have to have a path through the cube which actually implies you have to slice it from vertex to vertex. The default group isn't sliced this way, so that the vertex map is in an "imaginary" slice group (not that this prevents you labeling the vertices as if it's real), which also traverses some faces from "corner to corner" , and if the cube is a unit, that means a 2,3 triangle group is somewhere in the imaginary slice group, because of the geometry.

The map is another slice function, or a regrouping function - a morphism no less.
So the transformation rules here are again just a way to rewrite some "information content", i.e. total entropy.

 

Here is another conjecture, in the trail.

I've proposed that the puzzles speak a language, that you have to learn (to interview the alien). That nouns and verbs, conjugations etc exist. This is what I conjecture is a natural linguistic morphism, and also a natural (neuronal) simplification - we can all iterate cubologisms.
The complexity of |G| (or any |Sa| "machines" generated, up to index group as above) has an included group of complexity-reducing morphisms, or operations that exchange complex configurations for simpler ones. (The "exchange principle" appears to be a fundamental kind of operation, in cube-space)

This morphism is reflected in the general development of theories T of "solution" that converge on F2L plus alterations. Thus the initial segment problem converges to a simplification -> initial F2L, plus a remaining, less complex segment.
At the end of initial segment for theories corresponding to F2L segmentation, we have a remainder theorem having "switched the fabric". So this morphism is also like a Turing reduction of S in the basis a (algebraic number), with "free" index variables {ijklmn...}.

:bugeye:

 

I have one question regarding a term I've never heard before. What is a Möbius polynomial? You only mention it once, where you say you need to 'transform the cube into a mobius polynomial'.