Unscrambling the cube

Why the 2-generator subgroup is interesting:

because it resembles closely a two-body system. In \( \mathit R \) the set of relations for this subgroup, which is any two successive terms in the cyclic permutation of A = URFDLB.
There are two 4-dimensional numbers or elements of \( M_4 \), and the relations for an individual 'free' element \( g \in G\), i.e. any singleton element \( X \in A \), has a signature (mod 4).
Two co-rotating "frames" \( Xg_1, \,Yg_2 \) will generate a pressure which corresponds to a \( dW/dt \).

Consider phases of the lunar orbit, which are independent of the position of the earth in space, but depend only on the gravitational energy of the local earth-moon system.

Assign the beginning and ending phases called "new" and "full" to states \(s_i \), that represent even divisors of some lunar function (of being "full"), and the phases "first quarter" and "last quarter" to odd-i states. So you quarter the phases of the moon as \( s_1, s_2, s_3, s_4 \) and assign "the identity" to say, i=4.

Now define a "j-space" which maps each state to a coordinate system (say, on the surface of earth); this assumes a "first derivative" for each state exists in time and space.as much as there 'will be' a first appearance (in the map) of a lighted crescent, as the moon moves out of the earth's shadow - this event, for example, measures the occurrence of the end of the "new" phase).

So there are two 4-dimensional numbers, interacting in space and time and there's a record of these phases (quite a long record, really), in the space and time (and even dummies can see that 3 + 1 add up to 4).

 

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Reality check:

\( G \) is interesting (when N > 1), but \( G^{-1} \) is more interesting = "solving a scrambled Rubik's cube is more interesting than the inverse".

Corollary: the inverse of G, in R (the full group) is a contraction = finding a path through G that 'recompresses' the number of permutations to \( ~ N = 1 \). This models a gravitational constant, of sorts, for G|R. (thats: "general Rubikivity").

Ah yes. The ball of test particles initially at rest is the unit cube, in \( C_6 \)
Now assume each corner section's inner face is under stress, twisted up against the 'cubicle", a la Hofstader, it occupies. You can twist two corners (always) via the torquing maneuvre above, so you can generate twist in a single corner with a three-faces maneuvre Then each corner piece is about to leave its cubicle and fly out into space - an homogenous adiabatic fluid medium - the expanding cube or "big -bang' at t = 0.

There are components Px, Py and Pz applied to a corner 'part' of the puzzle. Now the at-rest ball of radius r is equal to the expanding ball of radius R (see J. Baez "The meaning of Einstein's Equation" http://math.ucr.edu/home/baez/einstein/node3.html). There is an abstract 'work-function' for discrete cycles in the structure (i.e. "causality") which means Px is a map of "torsion in the n-ball". Since, all operators "store torque" algebraically - there is a trail for single-step moves or turns m, you can "do a + or a -" on m, and you can "forget" the cube, come back later and you know which way the face was turned. This "turn information" vanishes when the moves get to 2 over the same face.
There is a need to find a commutator that maps X and Y to the same (homogenous) domain - that is assume the earth and moon, being locally bound by gravitational pressure, are in the same vector space, or X -> Y', so that Y -> X' (depending which frame you are "on").

 

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Doh!

just when you thought you had something figured out, along comes the 5th dimension 'exploding' all over the place

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But ('\slaps_forehead'), you recall that the physical construction of such a gadget is impossible in 3-dimensions ('\whew'). All you can do is slice up imaginary 'extra" dimensions that turn 2d areas (of color) into 3d volumes, then 4d hypervolumes. In a mathematical sense the extra dimensions are still in the category of spatial extent so in 3d + time you have 4-degree vertices in a graph, in 4d you have 5-degree vertices (and gravity is 10-dimensional) in 5d you have 6-degree vertices. Once you select a time-dimension there is one less spatial out-degree.

So ask: why do the 3d gadgets stop at N = 7? Probably a mechanical limitation and design problem.
Why do the n-dimensional gadgets stop at n = 5? Hmm.

 

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VIscosity:
resistance to flow.
Flow:
momentum equivalent of pressure.

The computer models are 'frictionless' but a real puzzle isn't. In the \( N^d \) when N = d = 3, the centers have four sides that meet four inner edges. Torquing down say, 4 sides around a center makes the center rotate. This represents the 'freedom' a fluid has to absorb pressure - the centers rotate away from the torque. If you could make the inner surfaces frictionless a ratcheting mechanism would also be needed to stop the inner pressure expanding the cube.

Of course with a real Rubik's cube, there is some friction and you can imagine there's a one-way ratcheting mechanism. A ratcheting cube puzzle would need a reset for the ratchet, in case of lockups.
So anyway, there are 4 skewed edge pieces (that show 2 facets) around each center facet, after the four faces around the center are twisted slightly. The centers rotate (on the 'top' and 'bottom' faces), the edge and corner pieces translate and rotate because they have two and three 'free' facets, respectively.

 

Writing a computer code to solve a Rubiks cube has nothing to do with modelling physical friction of an actual cube, it's about working out the combination of twists which return the cube to its initial state. It's a group theory problem, nothing to do with physical mechanics. You're simply rambling and posting nonsense, none of which gets you anywhere near the group theory needed to solve the problem. Precisely as I said many posts ago.

Personally the only way I've managed to do a Rubik's cube was to accidentally step on it and then put it back together. I now know how the inside of the cube works. Shame I lost one of the pieces....

 

For FUCKS SAKE.

I KNOW the Rubik's cube is a group of symmetrical objects (bleedingly obvious).
Writing a computer program to model the frictional and compressive aspects of one of these, is a perfectly legitimate pursuit. It has NOTHING to do with permuting the facets or group theory; it has everything to do with the physics of the cube, what torque does to it and so on. Then you can THINK about how these actions apply, singly and in pairs or triples - since, when N = 2 and d = 3, there are only three pairs of complementary \( M_4 \) components.

Have you studied the 2-generator group at all? The cross group? What does the tension in the torqued-down cube actually represent, in order to "solve" the skewing "problem" (not the permutation problem) what would you do, step on it, or use it for a rectal exam?

Let's see you show that \( M_5 \) isn't a subgroup, smart-arse.

 

Yet you haven't actually gone about writing a code to do that.

But then it comes down to the physical make up of a cube, ie how the pieces interlock, what material they are made of, what forces are being applied where. Nothing in your posts has considered the physical construction of the cube, you've just waffled about groups. Groups tell you which bit to turn and twist. Mechanical models tell you the torque required given the physical object of the cube. A cube 10 km in height and made of granite and lined with gravel is going to be harder to turn than one made of light plastic, 10cm across and well oiled.

You haven't even said things like "Let the coefficient of friction between the internal faces be k" or "Inside the cube only those unit cubes on the outer edge have internal faces". If you break up a cube (as I mentioned I did once) you find the internal components are all different kinds of shapes and have different surface areas other than just unit squares. You failed to mention this.

As usual, when I point out you're talking about something irrelevant the crank turns around and says "Oh so you know how to do this do you? What's the answer!". Where did I say I did a lot of group theory or knew how to solve it? Where did I say I know the mechanics of such devices to compute torques? Nowhere. I did have a group theory book whose first 3 chapters were on the Rubik's Cube but I've long since given it away. If you were not just waffling BS you'd have explained how all that stuff you're saying relates directly to the thing you claim to be addressing. Instead you try insults and demanding I solve a problem I never said I could.

Given the internal working of a Rubik's cube computing a realistic model of the forces involves in its turning are only going to be possible using a computer model of the cube, a CAD model. And if you're putting such huge time and effort into making a realistic CAD model of something do it for something more practical and interesting. Rubiks cubes are group theory examples, not mechanics examples.

See you when you get unbanned.

 

Just to satisfy my LEGO fetish..

http://www.youtube.com/watch?v=3QOvEG27Gt4

 

That is brilliant!

 

http://www.youtube.com/watch?v=i25cfdcum7U&NR=1&feature=fvwp

 

"I think we're going to have a problem here..."

I think this person is trying to say the Rubik's cubes are made out of group theories. I think the fact that the puzzles, like any puzzle, are physical and have mechanical properties like any puzzle, so they are computational examples, and also mechanical examples. I haven't been able to find any references, during my enforced leave, to any examples of cubes or anything else, built out of a group theory except another group theory.

Personally I find this statement mind-numbing and about as naive as you get from people's grasp of problem spaces, what a theory is vs what the theory is based on.

But as we say in the Gastrology Dept. "let's move on, shall we?"

Now most people who have managed (unlike some of us) to solve a scrambled cube, maybe more than once, know about the twist. The twist re-orients pieces but leaves them in the correct place.
A place, is just a "coordinate location" for a piece; there are different sets of pieces (fairly obvious) although each square looks like the side of a smaller cube, this is (obviously) an external.

Then finding a solution, using detective strategies, rather than stepping on a cube, we find that having at least two (doubling the problem) is more helpful than having only one (because, if you scramble the only example you have, you leave the identity behind, or "hide" it in a jumbled-up pattern). Then with two cubes, you can scramble one of them (or get the kids to), Now you can try to solve one, and you can slowly scramble the other and try to make them look "the same".

This means finding a node in a graph where the doubled cube has equivalent color patterns (an isomorphism/isometry). The location of this node is a number of steps from either the scrambled or unscrambled one of the pair, so they meet and you see one is "redundant". It is most definitely about developing algorithms (I have several).

The algorithms are more interesting than the cube because different words in the operator set can get to the same node, taking different paths. Most of these details are obvious, you can figure all this out and stay close to the solved state (i.e. experiment with a handful of moves, and back to observe results). The twist is a fundamental "parity function", which acts to restrict cycles.

There is:
generators {g,g1,g2,...,gn} are coordinators; relations {r,r1,r2,...,rn} are coordinates. F and Q are the full-face and quadratic 'root' (restricted to M1) phase spaces. F is in M4. MnMn is a subgroup of Mn (clearly?!).

Want to relate m,n to M: M -> Mn (?) Have T(m,n) map for K the corners with 3 facets and the twist. K is a divisor of G at N > 2. Restrict K in M (ignore 3-facets, use a 2-facet translation algebra coordinatized to the 1-facets which have rotational symmetry in M, from 0,..,4 2-facets align (pivot on a 1-facet). Restrict the centers (desticker them) and coordinatize by choosing a single facet color as the pivot. <G|R> is a map of coordinates and a set of transforms between.

So M is just a symbol that you can take to mean something like "an M-dimensional space", and the subscripts k to mean "k numbers in a M-dimensional space" or "a k-dimensional number in M".
The \( M_4 \) are obvious - there are slice groups in the original which vanish in the N=2 version (the pocket cube). These have 4 centers and 4 edges (1-facets, and 2-facets) around an axis of rotation. In this case a middle-layer slice is obviously in \( M_4 \).

p.s. the cube solving robot is an example of a mechanical solution to a mechanical problem - awesome eh?

 

Conjecture: (see how I'm being all logical)

A computer program or algorithm is a theory. The theory is rewritten in a "high-level" code, that the machine parses into "low-level" code. This is a test of the theory, by "locating" a computer (that is, finding, renting, buying, building, etc) and parsing it with rewrite rules. The theory is an outer form of an inner one, that tests it in this fashion: if the hierarchy can be commuted to the lower level, the theory "works".
Corollary: the halting problem is a hierarchy problem. (Thanks Alan...)

The locating "operation" that coordinatizes the program "operator" has a direct correspondence to another general kind of theory (thanks, Al...); running a program after locating a machine (M) to test it, is equivalent to multiplying a node in a graph by 1 (the machine identity \( I_M \)). A transition in M is a parallelization, equivalent to "go to M".

Ed: conjecture #2: the medium of the cube has two "flavors", which are the substructure and its tension and torsion (see above), and a passive component which is the coordinate (-ization) "table", in six faces, that the substructure torsions. The centers are in C1, and the center of the cube is in C0. I'm using simple indices unless I "raise" them or otherwise, the numbers after the C, simply mean "of colors", so the passive elements are inertialess, where the central "missing" 27th piece is "colorless".
The centers remain fixed, like poles; in fact you can treat the fully-stickered original, as a polarized form of M.
(should be a few buzzers ringing, in Al's place...).
That is to say, the colored stickers (stickering) are entirely 'inertialess".

Proof: remove one of the stickers from a center (1-facet). The polarization is unchanged, this remains true if the opposite face's center is de-stickered.

 

More BS: a material particle in relative motion, is equivalent to a material state in the machine-space of the Rubik group. When the slice function "goes nuts" and N gets large the permutation maps absorb large amounts of memory; the information volume increases according to a geometrical function in N, and 4 and 5 dimensions of information exist; ipso facto there are indices k up to 5, and clearly the faces group is in F5, because you can remove the passivity with the inverse of the stickering function, there are 5 colors to compare with the one at "infinity".
Corollary: in R, the stickering function is a field of 5 elements, which are "comparators", or contrast functions of inertialess color.
For the rest: "when you look at a color, you project it behind you to infinity, there are 5 colors to compare it with".

 

This illustrated how you're not anywhere near as mathematically informed as you'd like people to believe.

It's quite clear from what I'm saying that the Rubik's cube is something to teach people group theory, not mechanics. The complexity of the inner workings of a Rubiks cube is such that to model its physical structure would be complicated and then you might as well go with something more useful like the gear system of a car. The inner workins of the cube are irrelevant to the group theory of solving a cube hence why you can use it as a nice example.

If you are seriously trying to make this point then all you do is show your ignorance.

It isn't modelling friction and torque, it's using group theory to say "I have to twist this bit in order to move that to there". Your "Let's model the friction" thing would require you to know precisely the level of force applied by the robot. It just turns its bits 90 degrees, which is a much simpler thing and representable by a group generator.

The rest of your posts just ramble, not actually addressing the issue of getting an algorithm to solve it.

 

Deconstructing an apparently logical statement (in a set of statements)

You can do this with rules of implication, where you use relations, viz:

The inner workings of the cube are not irrelevant to using the cube as a computational device - like all devices it requires a mechanical solution, which is a configuration number. The configuration number of the cube is a formula that includes N and a number (up to 6) of colors applied to sections that N generates.

I have the friction and tension modeled thanks. And the shear stress. Shear is in fact, the freedom available in the structure (to, you know, rotate the faces since you can't rotate the inner structure - I experimented with "internal rotations" in 3d and found I seem to need at least one more spatial degree of freedom to turn any part of the structure inside out).
The tension is provided by inner spring mechanisms in each pivot. The torsion I make in the structure is recorded for me by the colors. These are the physical characteristics of the cube.

For someone who hasn't "bothered" with it, you seem to have an awful lot of discouraging things to say. What's the matter, don't you like puzzles and games? Have you ever played a strategy game or tried to win at gambling, say Blackjack or Poker? Any luck at Roulette the domain of operators who fall into "patterns of use" you can sometimes exploit, given you know a little about stats and mechanical systems that "lapse" into a pattern?

Note that if you "ignore" the mechanics of a roulette wheel and the person operating it you miss the subtle clues - you can tell if they are deliberately altering the action they apply at different times; some practise doing this and consciously try to avoid falling into a pattern. But they must get bored, and boredom is a pattern. If you can find a sufficiently bored croupier you can exploit the fact that a wheel is never perfectly balanced, and the croupier can unconsciously return the wheel to a previous state. The "method" is far from accurate, but you don't go to a casino to do exhaustive tests of your strategy.

I notice Alphanumeric, that you've posted exactly zero algorithms, don't you have any?
I mean, bitch away by all means, as if it matters for god's sake.

 

Actually, they are. You're only interested in the configuration of the external faces. If you want to use it as some kind of computational device then you're interested in the state of the external faces and how twisting the cube alters the faces. That's group theory stuff. The mechanical issue of 'how do the internal bits slide past one another, what's the torque required etc" is something you'd need to do in CAD.

The external faces are a physical representation of a group theoretic object. Using them to compute things has nothing to do with physically modelling the internal 'gears' sliding past one another.

So you have a CAD model which computes the friction between various faces as the cube is twisted? Or are you just saying "It takes X joules to twist from configuration 1 to configuration 2" ?

The colours record the configuration. They have no other physical meaning. Altering the colours doesn't change the energy in the system (which there isn't any) or anything else.

I don't have to know how to solve a Rubiks cube to see neither do you and that you're approach the problem in a completely hap-hazard and random manner. Your whole "Don't you like games?" thing is just you trying to have a go at me for some unjustified reason because you can't actually produce any relevant retort.

There's an entirely different problem to solving a Rubiks cube. If you're so up to speed on such things and a whiz at mechanical problems why not turn you attention to the Roulette wheel. It'd be more lucrative.

No, because I am not interested in doing such a thing. I think the modelling of the internal mechanics of a Rubik's cube is an entirely separate problem to constructing an algorithm to solve the cube itself. And I have other things I'm spending all my time doing.

The fact I'm not interested in solving the problem doesn't mean I can't point out that your talk of the internal mechanics of a cube has nothing to do with solving the problem of the colours of the external faces. You haven't even solved that yet, so you haven't provided an algorithm either and you claim you're actually trying.

The fact you complain I haven't solved the problem after I've pointed out your approach seems utterly pointless is a classic calling card of a hack. You don't need to solve a problem to spot errors in other people's methods. If you are as familiar with mathematics as you'd like people to believe you'd grasp this.

 

Why do I "need a CAD model?" Is it ok with you if I write something else?
Is it ok with you if I just assume the torque is mostly equivalent over face moves (i.e the medium is isotropic), so I just tell myself - "the cube is tracking torquing forces which I of course apply tangentially".

The thing is, I know that any given permutation has an unknown number of words that will permute the same, so "counting moves" (reducing the amount of work) only applies when you want to optimize.

I don't have to learn how to play a piano, to tell that you've never learned how. The fact you find this "uninteresting" enough to come along and have a moan suggests, to me, that you find playing music uninteresting too.

But I'm also afraid you are mistaken - how could you possibly know that I don't know how to solve a permutation puzzle? It did take me a little while and a bit of thinking, and it was probably before you went to primary school.

Let's see, you're probably in your early twenties (I can tell from your attitude), and that means you probably weren't actually born, when I first solved the cube, which these days takes me, in a relaxed mode, about less than an hour. I don't rush it because my aim is to examine the partial solutions, when I know I'm close to the initial or "factory reset" state.

I haven't even solved "WHAT" yet? I have an algorithmic solution, which I developed over 25 years ago ("shit, are you really that old?". "yep").

You haven't done or said much have you? Apart from general moaning and bitching about nothing in particular.

The fact you keep complaining about something, which I'm afraid still eludes me, suggests you are something of an idiot.

You're the sort of hack who gets off accusing other people of having defects you have yourself, you hack.
Your complaints seem utterly pointless and mostly a lot of gibbering about nothing in particular. What is the point of posting a load of stuff about mathematics? How many readers apart from big important YOU, would give a stuff?
I notice this forum is stocked with, shall we say, contributors with a varied set of ideas about "stuff". Why do you think I "need" to post anything for YOUR benefit, my man?

Since you've admitted you find the puzzles uninteresting, why should I bother even responding? (to someone who reminds me of a person who bought a grand piano so they could say "well, if I learned to play, I would probably be quite good at it, but the piano looks nice, don't you think?").

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If you're serious about modelling the mechanics of the cube then you need something which can compute forces and energies involved when you don't turn the cube perfectly. If you try to twist a cube two different ways at once it locks up. Can you model that? Or are you just saying "It takes X energy to move side 1 by 90 degrees"? In which case you can simply compute it from the sequence of twists done to solve the problem. Would you like me to explain?

Vkothii, you and I have crossed paths before. Even your current incarnation has replied to a thread of mine in the maths forum. Please don't try to pretend you don't know I've studied maths.

I've yet to see you do anything like that in this thread. Not to mention posts of your in other threads, such as the entropy/Lorentz transform one, further suggest you just talk nonsense in the hopes of suckering people.

Age does not automatically impart knowledge or wisdom.

So if you're up to speed on group theory and you have an algorithm, let's see it.

And the only reason I'd say "Shit, you're really that old?!" is because your behaviour of having at go at people who question you suggests you're younger than me. I'd expect someone in their 40s to be a little wiser but then, as I said, age does not automatically impart knowledge or wisdom.

Maths degree, masters and a PhD in theoretical physics, which resulted in published work. You?

I am happy to give indepth discussions of the area my PhD is on, over in the maths and physics forum, if you want to discuss it. Not just the wordy explanations but the algebraic details. I've got nothing to hide. I don't start threads on topics I don't know about trying to convince people otherwise. On the rare occasion I do post a thread its about something specific related to a problem I'm considering and I'm generally quite precise about what I'm after.

Quite, so why are you posting a load of stuff about mathematics?

Can you point to the post where I said that? Because I didn't. The problem is quite interesting, its just I don't have the time or the inclination to have a major crack at it right now. I have a passing interest in group theory but nothing huge or complicated, just the stuff needed to do theoretical physics.

Where have I claimed I'd be good at it if only I tried? Nowhere. Whenever a crank started inventing such narrative its always a sign they are out of anything other than vacuous insults.

 

Ok, for the people who know some math, and how to play with it:

The Moebius transformation is a subgroup; the quaternions are a natural isomorphism. M12 the Mathieau group and Golay codes. I have codes for several odd cycles, and I've noticed that cycle lengths of order 63 and 105 are common. I have a nice 4-cycle that has a 2-cycle subgroup which is exactly one-half of the (symmetric) word of the 4-cycle.

Investigating words you compose in 1,2 and 3 letters exposes some of the underlying group structures since the program rewrites your codes in an optimized form. Then, you have the initial redundant encoding and an optimization of it to compare. The rewriting is according to rules so the optimizations are again, part of the whole "Turing process".

I use a standard word a lot which is formed generally from X'YZX' and complements like YZ'X'Y. This looks like a figure or character of some kind - it appears because of the F2L method I use which leaves a column of unsolved elements along one edge, and a lower layer I can leave in the FTM while I apply QTM moves to 4 other faces, maintaining the "up" face's color as a coordinator.

The algorithm starts in FTM and moves to QTM on the F2L, after I solve the corners and 6 edges are placed; I can only rotate the "sides" by a QT, initially where the "driveway" is located. I can turn the side faces recursively as long as I do so in the same direction. I can restore this by inverting the moves, and the word I mentioned is an algorithm that I need to use during this (but I might truncate the code). Then I can always tell when I'm 4 moves from the solution.

Ok then?

 

I'm one. Let's go through what you said....

Subgroup of what? And the transformation isn't 'a subgroup', the transformations, ie \(z \to \frac{a z+b}{cz+d}\) for \(\left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL(2,\mathbb{C}\) can be related to a group, the SL(2,C) group of 2x2 complex matrices is unit determinant. Well, technically its PSL(2,C). So you fail to explain yourself and your choice of wording is incorrect.

'Isomorphisms' are maps between things. The quaternions are a set with particular properties. There's an isomorphism between sl(2,c), the Lie algebra of the group SL(2,C), and the quaternions but the quaternions are not more an isomorphism then '5' is a map. Again, you word it in such a way it suggests you read something about SL(2,C) and its relation to quaternions and due to a lack of understanding couldn't parrot it back properly.

Oh really, be specific. How do 'you' have it, when its nothing but a group theoretic fact such groups exist? What do you 'have' it with regards to?

To solve a Rubik's cube you're written a self editing, self optimizing code? This I'd like to see.

And you aren't exactly going to learn a lot about group theory if you consider 1 letter words, are you?