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3x3 Rubik's Cube Patterns and Particle Physics

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Greetings!

I have been investigating different kinds of Rubik's Cube patterns and their symmetry families for the last couple of years and I have discovered a great deal of information that I have seen nowhere else. It was long ago noted that the ways you can twist corners on a Rubik's cube closely paralleled the ways you can construct particles with quarks. So far, nobody has pursued this as far as I can tell. I have found many more ccorrespondences than this and indeed I am acquiring a deep understanding of the often subtle and complicated ways seemingly simple Rubik's patterns relate to each other. My discoveries are best understood by comparing the different families of particles that theoretical ophysicists have come up with or hypothesized may exist. Basically, I identify a symmetrical pattern on the Rubik's cube as being equivalent to a subatomic particle. What kind of particle depends on the symmetry of how the colors are exchanged and the kind of orbit the cube must be in for that pattern. If this seems simple, consider the 6-way checkerboard pattern that everybody knows, you just need 3 half slice moves to make it and it has reflection symmetry on all 3 axes through the centers. I have met few people who know any others than this one. It might surprise you to know that there are 28 other possible checkerboard patterns (counting color permutations) on a given Rubik's cube falling into 5 different symmetry families. The reason's behind this are what I am investigating,

It is best if you use one of the speed type cubes. I use the Rubik's Speed Cube, in fact about 6 different ones so I can study different symmetry versions of the same basic pattern side by side, You should be able to disassemble the cube and reassemble it into a different orbit. It is also nice if you can pry off the center facelets and exchange opposite ones to creeate mirrored patterns. The speed cubes seem to be the ones you can do this with. The Rubik's New Cube is not designed to be disassembled and is not really useful for this purpouse.
I keep finding more and more things about Rubik's cube patterns and I would appreciate it if somebody would be interested in assisting in the work of discovery. I am especially interested in anyone willing to look for the rarer patterns in the Magnetic Monopole andG.U.T. families.
Anyway,

Let me know if you are interested! Math Bear ^,..,^
 
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#2
It is also nice if you can pry off the center facelets and exchange opposite ones to creeate mirrored patterns.
just so you know, if you exchange two opposite centers you have changed the color scheme of the cube. No sequence of legal moves could create that situation, so that may be something you want to consider.
 
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Particle Physics and the Rubik's Cube

just so you know, if you exchange two opposite centers you have changed the color scheme of the cube. No sequence of legal moves could create that situation, so that may be something you want to consider.
But that is exactly the idea! I have no interest at all in solving the cube, I am exploring pattern symmetries. Particle physics is governed by strict rules that parallel those of the Rubik's cube. Those patterns accessible from the basic cube correspond to the Standard Model of particle physics. Theoretical physicists believe that at high energies one or more of the rules that usually hold can be violated and you can create new particles not in the Standard Model. deliberately disassembling the cube and making some normally forbidden change lets you access patterns you normally can't achieve. Thehighly symmetrical patterns in the "one-pair-exchanged" or (as I call it) Fermionic Orbit all have counterparts in the regular (or as i call it, "Bosonic" Orbit) but different symmetry. Most commonly the Fermionic patterns have flip symmetry (top bottom are exchanged and adjacent side faces) whereas the Regular patterns have rotational symmetry (the colors rotate + -120 degrees about an axis through two corners). This corresponds to the SuperSymmetry partners of particle physics. A single pair exchnage corresponds to spin 1/2 in particle physics. Normally, you can only chnage a particle's spin by an integer amount. This corresponds to the rule in Rubik's cube patterns that you can only change a cube's state by an even number of pair exchanges. In particle physics, there is a belief that it may be possible to change a partcles's spin by 1/2, creating a dramatically different new particle that is the "partner" of the old. Particles with integer spin values are called "Bosons" and those with fractional spin (always 1/2, 3/2,5/2 etc. in SuperSymmetry theory) are called "Fermions". The two kinds of particles have very different properties in physics. In the case of the Rubik's cube, all the Regular patterns are Bosons because the number of exchanged pairs is always even and in the case of the one-pair-exchnaged cube patterns, they are all Fermions because they have an odd number of pair exchanges corresponding to an odd number of spin 1/2 spin value units.
It is fun to explore the consequences of this. The 6-way "Eye" pattern in the Regular cube orbit has rotational symmetry and is a Boson. The corresponding Supersymmetry "partner" in the "one-pair-exchnaged" orbit has flip symmetry and is a fermion. If you create the Worm pattern on the regular Rubik's cube, it clearly has rotational symmetry and of course, like all regular patterns is a Boson. Try the challenge of coming up with the corresponding pattern on a cube in the fermionic (one pair exchnaged) orbit, It will have flip symmetry but otherwise look the same. You are doubtless familiar with the "cube within a Cube pattern with strongly rotational symmetry? The sorresponding SuperSymmetry pattern on the Fermionic cube has flip symmetry, but if you look at the colors of the 2X2 cube imbedded in the 3X3 cube, they are all the opposite colors of the larger cube it is embedded in. The visual difference is very striking! On the other hand, with the "snake" pattern (like the worm but the pattern runs straight on one pair of opposite sides), both the Boson and Fermion pair have flip symmetry and look almost identicle, The diifference is very subtle. You can have fun handing the cube in the fermion pattern to an experienced cubist and asking him to "solve" it. Of course however he tries to solve it, he will always come up with a single pair that cannot be removed. It should deeply puzzle him!
Try finding some other familair patterns on the Fermionic cube!

Note: Exchnaging two centers on a cube invokes a parity transformation and creates what I call a "mirror" cube. This corresponds to what physicists are calling "Mirror" matter (parity rversed matter). Mirror matter does not interact with ordinary matter except through gravity, the only force neutral to parity effects. There is increasing evidence that this may be the identity of "Dark Matter", at least in part. Parity has profound effects both in particle physics and Rubik's cube patterns. a discussion on that must wait for later.
 
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#4
First of all, I am a pure mathematician, so I have almost no idea of physics. But I try to understand, what you are talking about:

It might surprise you to know that there are 28 other possible checkerboard patterns (counting color permutations) on a given Rubik's cube falling into 5 different symmetry families.
I counted 14, that can be reached by making arbitrarily many moves and optionally exchanging two edge pieces. So I am not quite sure, whether we are talking about the same thing.
To make this clearer, this is what I counted: Divide the cube in (centers and corners) and (edges). For the cube to be in a checkerboard pattern, these two subsets must stay in the same configuration, but be rotated against each other. So at first there are 24 possible different rotations but some of them leave a face one-colored, so they must be omitted. In total there are 14 rotations left.
However as your number is exactly twice as big as mine I have the feeling we are talking about neraly the same thing.

Thehighly symmetrical patterns in the "one-pair-exchanged" or (as I call it) Fermionic Orbit all have counterparts in the regular (or as i call it, "Bosonic" Orbit) but different symmetry. Most commonly the Fermionic patterns have flip symmetry (top bottom are exchanged and adjacent side faces) whereas the Regular patterns have rotational symmetry (the colors rotate + -120 degrees about an axis through two corners).
Usually speedcubers are refering to the six sides of the cube as UDLRFB for up down left right front back. So by flip symmetry you mean the pattern is invariant under U <-> D, F <-> L, B <-> R? And also we call two patterns the same if there is a bijection of colors (together with a rotation?), that make them identical?


I also have trouble understanding the cube-in-a-cube pattern example. What you are describing as the corresponding Fermionic one-pair-exchange state seems to be reachable only by echxanging two centers.

If I understand you correctly, you search for some theorem, that goes approxiamtely like: For any normal state of the cube, there is a one-pair-exchange state, so that every face of the normal cube looks by bijecting colors as the corresponding face on the one-pair-exchange cube. This "theorem" clearly doesn't hold for the solved cube. Also I didn't specify, whether you can change your color bijection on different faces of the cube or if it has to be the same on every face. So still some improvements must be made. But all together it seems to be an interesting problem to consult.
 
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#5
I'll about to not reading all the details of what you're talking about, but it sounds like what you're really doing (under the hood, so to speak) is relating particle physics to the symmetry group of the cube. That is, simple group theory may do a much better job of explaining what you're looking into. Rubik's Cube patterns, for obvious reasons, are also affected by the cubical symmetry group, but I don't think the Rubik's Cube as a puzzle, or its specific constraints on corner rotation parity and so on, will add anything above and beyond what group theory gives you.
 
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#8
I disagree with counting permutations that can only be achieved by physically taking out pieces. If you do that, you're not exploring Rubik's Cube patterns, but simply putting together 26 blocks, each with stickers on them.
 
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#9
I disagree with counting permutations that can only be achieved by physically taking out pieces. If you do that, you're not exploring Rubik's Cube patterns, but simply putting together 26 blocks, each with stickers on them.
He's not interested in exploring cube patterns. He's interested in drawing analogies between orbits on the Rubik's cube and those in particle physics. In group theory, an orbit is an a group for which state in the group can reach any state after a sequence of legal actions. If no sequence of legal actions can take you from one state to another, then the two states are in different orbits. For example, twisting a corner makes the cube unsolvable with just face turns, so the solved state and a state with one twisted corner are in different orbits. Because of that, taking out and removing pieces is exactly what he wants.

On to OP's question, it feels like you're taking two fields to which group theory may be applied, and trying to use that as an anchor to combine the two. Personally, this is one of my pet hates. In my own field (differential equations in fluid mechanics), some people tend to take a mathematical model specifically developed for one field and then directly apply it to another, and get the wrong answers. What they really should be doing is taking the very well developed mathematics, and applying that to the other field to get a better model for that field.

In other words, group theory is great for analysing the Rubik's cube, and may be great for particle physics (I don't know enough to say). It would probably be a mistake to force the theory of the Rubik's cube to fit particle physics though, when the raw mathematics of group theory would probably probably result in a model better suited to particle physics.
 
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I think you are getting way off base by trying to get theoretical far too prematurely. Working mathematical theory blndly on something that isn't abstract to begin with can be deeply trecherous.
And you do need to know something about basic contemperorary particle physics to appreciate what I am doing. Please check out a WikiPedia article or something. The correspondences I am finding are not exact in the mathematical sense but many things in particle physics have no precise mathematical formulation largely because they havent been discovered yet and nobody is sure which of many competing theories might be correct. If you want to discuss abstract mathematics, please communicate with me privately so you dont end up scaring away the other cubists on here. especially the hobbyists. Remember truth is ultimately not theoretical, theory is a way of describing truth.

Now, the first checkerboard results from applying a parity transformation to all 6 faces, reflecting the edge cubies of each face onto its opposite color face. This pattern is unique.
Now most people also know how to generate the 6-way cross with rotational symmetry. If you then rotate just the centers back again, you end up with a 6-way checkerboard with rotational symmetry. Since there are 4 corner axes and each can be rotated two different ways, clockwise or counter clockwise, you get 8 possible color permutations. To each of these 8 possibilities you can apply the first transformation to get 8 patterns with what I call "swirl" symmetry. This symmetry moves through all 6 faces in the same way the Worm pattern does. It is obviously NOT rotational. This gives you already 17 possible patterns with a regular cube. (Bosonic Orbit) Now go to the ferminoic orbit cube. Whatever you exchange inititially, transform it to exhanging two opposite edges of a center layer. Then flip all the edges in that center layer. This will leave all edges in the center layer with facelets of the same color facing each other on all 4 edge cubies. The centers will have a different color. Turn the middle layer until the edge cubies match the face they are in and the centers are rotated 90 degrees + or - in the horizontal plane, This is a 4-way eye pattern that is basic to all patterns on the Fermionic cube, There is a simple transformation that will transform the 4-way eye pattern into a 4-way checkerboard pattern of the same character. Once you have this 4-way eye pattern turned on an axis through 2 centers 90 degrees clockwise or anticlockwise. you can generate the 6-way flip symmetry by exchanging the edges of the two top and bottom faces and two opposite horizontal faces. The flip symmetry is what you get when you rotate the cube 180 degrees about an axis through two opposite edge cubies in a middle layer. Since there are 6 possible edge axes, there are 6 possible color permutations for the 6-way checkerboard pattern with flip simmetry. Now if you exchange the edge cubies on opposite horizontal faces, you get what I call "twirl" symmetry. The top and bottom faces are exchanged and the side faces rotated 90 degrees horizontally clockwise or anticlockwise. Since there are 3 axes through the centers with two rotational states, that gives you 6 possible patterns, 6 + 6 = 12 Fermionic 6-way checkerboard patterns. bAdded to the 17 from the regular (Bosonic orbit) you get 29 total. If you are not convinced of this then please try constructing all 5 families of patterns on your cube (Reflection, Rotational, Swirl, Flip and Twirl symmetry families) and count the permutation possibilities of each family and confirm that there really are 29 possible 6=way checkerboard patterns on a given Rubik's cube, counting both kinds of orbits (more orbits will be prsented later).
You really need more experience with your eyes and fingers of all this first. ^,..,^

Oh ok. Well have fun with math and stuff.
Sheesh! It's not about math really, that is a small part of it, It is mainly about understanding and exploring symmetry and the patterns that result, stuff you can see with your eyes.

Please don't be intimidated!

I'll about to not reading all the details of what you're talking about, but it sounds like what you're really doing (under the hood, so to speak) is relating particle physics to the symmetry group of the cube. That is, simple group theory may do a much better job of explaining what you're looking into. Rubik's Cube patterns, for obvious reasons, are also affected by the cubical symmetry group, but I don't think the Rubik's Cube as a puzzle, or its specific constraints on corner rotation parity and so on, will add anything above and beyond what group theory gives you.
.

That is not at all what I am doing! I love the many patterns you can create on the Rubik's cube and playing with them. But I am trying to find a way to classify them and understand how they relate to the underlying symmetries of the cube. I have found a striking correlation with the patterns of particle physics and it really helps in understanding all these different patterns, Group Theory cannot handle this, it works on too abstract a level. You need to work with CONCRETE examples of the groups. If you want to get technical the Group SU(2) underlies both the Rubik's cube and most particle physics. You have to exclude the Strong Force because it is described by SU(3) which is distinctly more complex. SU(2) is easy, it is basically the algebra of the rotations of a solid object in 3D space.
I am not interested in snowing anybody under with math! the best way to understand what I am doing is to use your eyes and fingers to duplicate what I am doing, Human understanding works from the concrete to the abstract and that is the only way you can appreciate what I am doing. You can leave out the particle physics stuff but it makes it SOOOO much kewler! Fiddle around with my ideas with your hands on a cube and you'll soon "get" it!

im guessing that pretty much all of the people on this website are nerds including myself but i dont think anyone is really that into it. maybe im wrong and if you find someone, good luck to you both
I am not trying to be "Mathematical" The real appeal of the cube is sensual and I am giving you tools for deep exploration of its patterns and their symmetry. I am especially interested in getting you "nerds" (and am I not a nerd?) interested in this stuff. It is so cool once you start getting it! I think a nerd with a good experience of the cube will get farther understanding my ideas than some professor who doesnt know anything practical about tthe cube.
Keep in mind I am dealing with radically new and original ideas about the Rubik's cube. when something is really new and original, most people have trouble understanding it or actively misunderstand it. Hold back your judgement until you have had a chance to play around with it. That is why I am on this board and not one dealing with something theoretical.....

I disagree with counting permutations that can only be achieved by physically taking out pieces. If you do that, you're not exploring Rubik's Cube patterns, but simply putting together 26 blocks, each with stickers on them.
Huh? That doesnt quite make sense to me.
Please elucidate.

------

Why do you say that?

I am totally interested in exploring cube oatterns and it is basically my purpose here. But I am really fascinated by the way that these families of patterns relate to families of particles in physics and exploring these relations add a lot of insights to both kinds of studies. You can leave out the partice physics stuff if you wish, but it is so much more fun and intriguing if you include them.. Keep in mind, that I have presented only a small amount of my researches so please don't jump to conclusions prematurely. And please don't go on about Abstract algebra! It provides helpful insights here and there but it is not the foundation of my ideas. Abstract algebra tends to really scare people!

Cheers,

Math Bear

^,..,^
 
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I do not think there is much new to discover there. The possible 33 symmetries of cube patterns are completely classified here http://kociemba.org/symmetric2.htm.
If you also are interested in antisymmetric patterns, the 131 possible pattern types are listed here http://kociemba.org/antisymmetric.html
Yeesh! Can anybody but a professional mathematician make head or tail of that? I have no idea whether your claim is valid or not. I have no interest in debating with mathematicians (at least not on here). I am interested in presenting my ideas to normal mortals and I want to keep theoretical claptrap and obfuscation to a minimum. The preferred scientific focus is on particle physics and anyone on here can get all they need to know by reading a few Wikipedia articles on the subject. I'll make some suggestions later. Technical knowledge is not needed! Please do not keep bringing up abstract algebra on here. I may mention group theory a bit, but it will be attuned to the typical cubist not to a PhD thesis. the only way you will be able to genuinely understand my ideas is to get a rubik's cube in your hands and play with it. You cannot get there through abstract theoretical mathematics. Do not assume that a formal theoretical description of a subject constitutes genuine comprehensive insight into it.

Sincerely,

Math Bear


P.S. I have been told that replies to more than one person should be combined into one maessage through editing.
Is this true?
 
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#13
If you're going to be this rude, condescending, and ridiculous, you can't expect to get any real answers. You can't seem to decide whether we're mere "cubists" who can't possibly understand undergraduate-level concepts or out-of-touch mathematics professors talking about ideas that belong in a Ph.D. thesis, while also pushing anyone from our community who knows what they're talking about away, because their ideas are too difficult (not for you, for ourselves). You post nonsensical, vague walls of text full of casework that would take ages to untangle even if we knew exactly what you were talking about. You try to bring up particle physics while having a clear disdain for theory. And even worse, you have a disdain for abstract algebra, which has been understood for decades to describe the Rubik's Cube (a permutation group) and its patterns extremely well without even getting into difficult ideas. When you refuse to look into this kind of stuff it is clear that your ideas are not "radically new and original", but just the result of a novice reinventing the wheel - at least as far as the cube portion is concerned.

If you really want to understand more or have a discussion, come back with a little willingness to learn and a lot less insulting the forum's intelligence. And yes, posting two times in a row (let alone six) is generally frowned upon.
 

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#14
Is there any particle physics analogy to the 12 orbits of the cube pieces with center positions fixed? For example, with all center colors and positions fixed the sum of clockwise corner twists can be congruent to 0, 1, or 2 (mod 3). The sum of edge flips can be congruent to 0, or 1 (mod 2), and the permutation parity of the corners can either match or not match the permutation parity of the edges.

This gives 3*2*2=12 orbits for the assembly of a Rubik's cube's pieces, assuming center colors and positions are fixed.

P.S. I have been told that replies to more than one person should be combined into one maessage through editing.
Is this true?
On this forum people will usually quote multiple people, and address them, in one larger message.
 
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#15
If you're going to be this rude, condescending, and ridiculous, you can't expect to get any real answers. You can't seem to decide whether we're mere "cubists" who can't possibly understand undergraduate-level concepts or out-of-touch mathematics professors talking about ideas that belong in a Ph.D. thesis, while also pushing anyone from our community who knows what they're talking about away, because their ideas are too difficult (not for you, for ourselves). You post nonsensical, vague walls of text full of casework that would take ages to untangle even if we knew exactly what you were talking about. You try to bring up particle physics while having a clear disdain for theory. And even worse, you have a disdain for abstract algebra, which has been understood for decades to describe the Rubik's Cube (a permutation group) and its patterns extremely well without even getting into difficult ideas. When you refuse to look into this kind of stuff it is clear that your ideas are not "radically new and original", but just the result of a novice reinventing the wheel - at least as far as the cube portion is concerned.

If you really want to understand more or have a discussion, come back with a little willingness to learn and a lot less insulting the forum's intelligence. And yes, posting two times in a row (let alone six) is generally frowned upon.
I back this up.
 
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#16
The preferred scientific focus is on particle physics and anyone on here can get all they need to know by reading a few Wikipedia articles on the subject.
Seriously? Understand particle physics from a few Wikipedia articles? Hahaha...

It might surprise you to know that there are 28 other possible checkerboard patterns (counting color permutations) on a given Rubik's cube falling into 5 different symmetry families. The reason's behind this are what I am investigating.
If you want to understand the reasons, you can't do better than read Herbert Kociemba's work. The stuff about symmetry that he linked to is really not that complicated or difficult to understand.

In fact, all it is a list of symmetry types, and if you click the "yes" in the "More Information" column, you get a page listing how to reach all the patterns with that type of symmetry, with a nice little java applet that shows you what the pattern looks like so you don't even need a cube with you.
 
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#17
I have often posited some relation between the fundamental nature of dimensionality and the cube. However, you will find little help on this site. These are not intellectuals on this site. None have created anything. They read recipes and then try to repeat them as fast as they can. They have trolled me and others, and it is part of their culture to stomp on something they don't get or understand. Speak to a real mathematician. He or she will understand.
 
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However, you will find little help on this site. These are not intellectuals on this site... Speak to a real mathematician. He or she will understand.
I'm sorry, but have you even read the thread? Real mathematicians have indeed understood and responded with useful information, yet Math Bear has asked to specifically not get theoretical in this thread...
 
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To all the offended math purists, I will hold back and let you cool off. I did not want you guys to dominate the discussion and scare everybody else off.
I am aware of lists of symmetries but they overwhelmingly deal with the regular (Bosonic) orbit of the cube. I largely deal with the other orbits so it is not that useful to my purposes. The realtionships between particle physics and the Rubik's cube are not mathematically exact! Technically speaking, don;t expect the relationships to be isomorphic or homomorphic. They are instead analogous but quite striking for all of that. This lack of mathematical exactness makes group theory less useful than it might be otherwise. I will develop my own informal language for describing the symmetries, but it will be strongly geometric rather than algebraic. Mathematics has two foundations, geometry and algebra. They reflect the two most highly developed aptitudes of the human mind, vision and language. To reallly understand a primary mathematical system, you need to use both modes of understanding. Western mathematics developed its enormous power because it found a way to link the geometric and the algebraic modes through the use of the Cartesian coordinate system. " Mathematics without geometry is blind, mathematics without algebra is dumb"(not my words, anybody know whose?). I suppose if dolphins were ever to develop a system of mathematics they would probably base it on harmonics, to match the unique power of their hearing. Anyway, modern mathematics has become largely incomprehensible to the general public in part because there is a huge emphasis on the algebraic (linguistic) modes of description nad an often extreme avoidance of the geometric (visual) modes of description. The Rubik's cube is fundamentally a geometric object and you need to emphasize understanding it geometrically first. Thus, while I am establishing the basics of my ideas I intend to avoid Group Theory and Abstract Algebra and maybe save them for later, after the whole thing has been properly presented. That will take a while....)

Now, if you dissasemble the cube and reassemble it randomly, you will only have a 1 in 12 chance of ending up with a cube that you can solve all the way to the "start" configuration, This is because you are not completely free to rearrange the cubies but must obey certain restrictions. If you exchange pairs, you can only exchange an even number at a time. If you flip edges, you can only flip an even number at a time. If you twist corners, the total amount of twist can only add up to an integer. You violate these restrictions and mix and match the results you get the 12 orbits. If none of these restrictions is violated, you end up with the Prime orbit which is the type of cube you have in the Start configuration. I call this the Bosonic orbit because an even number of pair exchnages is analogous to a particle with integer spin, and thus can be regarded as a "boson". Each pair exchnage corresponds to 1/2 spin. The next orbit has one pair exchanged, and the total number of pair exchanges of any pattern in this orbit must be an odd number. Any pattern in this group corresponds to a Fermion and has half-integer total spin. The introduction to the Wiki article on Supersymmetry is easy enough for anybody to understand the gist and some later parts are technical enough to appeal to the math geeks. It would help if you were to play with patterns and in particular with correlating patterns in the Prime orbit with those in the Fermionic orbit. Try using two cubes for this, one in the Prime orbit and one in the Fermionic orbit. the next orbit is one flipped edge or an odd number of flipped edges.After playing around with this for a long time, I decided it is analogous to magnetic poles. All particles have both a "north" and "south" pole or pairs of such, No particle has only a "north" pole or only a "south" pole. Such a particle would be called a "magnetic monopole" and some theories (G.U.T.) suggest they may exist, though none have been found. I only have one pattern for this, a version of the 6way-C (or 6-way U) pattern that appears like any of the others but if you examine the color symmetry, it has a bizzare, twisted lop-sidded symmetry that you won't find on any standard list of Rubik's cube symmetries. it is possible to create a new orbit by combining the previous two into an orbit of cube patterns with an odd number of pair exchnages and an odd number of flipped edges. I only know one such pattern and it may be regarded as a "Fermionic Monopole" to draw an anology with particles. Please read the intro to the Wikipedia article on "magnetic monopole" for a better picture.
The next orbit involves the corner cubies. (note: this correspondence goes all the way back to the 1st Scientific American article on the Rubik's cube) If you count a clockwise twist of a corner as +1/3 and a counterclockwise twist as -1/3, then the total twist must add up to an integer, never a fraction. This is a close analogy to the restrictions on how quarks may be combined to form a particle. 1/3 or 2/3 electric charges are associated with each quark, but quarks must combine in a way that the total electric charge of the particle is an integer. The mathematics used to describe the strong force that governs quark interactions is different from that used for Sypersymmetry. I only know two patterns in this orbit but they are very striking. They are both bosons and involve rotational symmetry through the corners.) Some have theorized the existence of particles that would have an extra quark and thus fractional electric charge. Patterns based on a cube in the single twisted corner orbit (actually 2 orbits, depending on whether the twist is clockwise or counterclockwise) correspond to these particles. I call this orbit the "quark" orbit. I haven't found it necessary to distinguish between the clockwise and counterclockwise versions. The other orbits are the possible combinations of the orbitals involving flip, pair exchange and corner twist.
So far, I am working with the Prime, Fermionic, Monopole orbits (both Boson and Fermion), and the two trivially different Quark orbits. This is only half of the 12 possible orbits and I would be very interested if anybody finds patterns corresponding to any of the othe others. There is also a whole realm of 12 orbits belonging to the Mirror cube and I have some patterns I have discovered in the Mirror Boson and Mirror Fermion orbits. I will save the discussion on these until I get into the weighty subject of parity.

Now, In describing cube moves, I use the F,B,R,L,T,D system and a cubies position is defined in terms of the corners, not edges. It makes it a lot easier to understand the patterns if you assume the corners are fixed but the centers can move. To describe moves, I just use the letter corresponding to the face you want to turn and add + (clockwise), - (counterclockwise) and 2 (180 degree turn). The face letter should be capitalized. It can be preceded by a small leter "s" (slice), "a" (antislice) and "c" (turn the whole cube). Thus, turn the bottom face counterclockwise is "D-", a clockwise antislice turn parallel to the right side is "aR+" , turning the top face 180 degrees is "T2" and a 180 degree slice move through thevertical layer parallel to the Front side is "sF2". I am not sure who to credit this system to but it feels familiar. Anyway, it is easy to use.

The basic symmetries of a cube govern the basic patterns, These are: the rotational symmetries of rotating the cube through a corner to corner axis, either 120 degrees clockwise or counterclockwise. Since there are 4 such axes, there are 8 posibble corner rotations; If you turn the cube about an axis through a pair of centers, you get 3 possible rotations )1/4 turn, clockwise, 1/2 turn and 1/4 turn counterclockwise. Since you have 3 center axes, you get 9 possible rotations. I prefer to keep the three 1/2 turn patterns separate from the six 1/4 turn patterns as the 1/2 turn patterns can also be describled as reflections (4 way reflection) and are distinctly different. Finally, if you flip over a cube 180 degrees through an axis through 2 diagonally opposite edges in a center layer, you get flip symmetry. It results in top and bottom being exchnaged and 2 pairs of adjacent faces on the sides.

A simple basic pattern is one that involves only one kind of cubie, is the same on all 6 sides or on 4 horizontal adjacent sides (6-way and 4-way) and no more than 2 colors should appear on any side. A pattern meeting all these criteria but involving 2 different cubies on each side(both the same color) is a compound basic pattern. Such patterns can be decomposed into two simple patterns. Think of it as two subparticles composing a more complicated kind of particle.
if a pattern is not identical on all relevant sides or involves twisting or moving corners or has more than 2 colors, then it should be called a "complex" pattern.

The well-known 6-way Checkerboard pattern does not fall into any if the above categories as it is the result of a forbidden parity transformation of the cube. Each side is the mirror reflection of its opposite side. This is forbidden because a Rubik's cube is not identical to its mirror image(i.e. it has parity). However, edge cubies are identical with their mirror images and if a move involves nothing but edge cubies it can violate parity restrictions. I call such patterns "forbidden" for short but they are not forbidden for the edge cubies, but calling them that reminds us to be careful with such moves. This corresponds to the fact that particle interactions involving only the weak force can violate parity restrictions too. Issues involving parity are often subtle and deep and I will save them for later.

All simple basic patterns invoving the Prime (or Bosonic) orbit all have "partners" in the Fermionic (odd pair exchange) orbit. Bosonic patterns have either corner rotation (6-way) or 1/2 center turn (4-way) symmetry. Their Fermionic partners have either Flip (6-way) or 1/4 corner turn (4-way) symmetry. This is a deep correspondence to particle physics. SuperSymmetry theory assigns a "superpartner" to every basic particle of the Standard Theory. If the standard particle is a Fermion, its superpartner is a Boson and if the standard particle is a Boson, its superpartner is a Fermion. The simple basic patterns of the cuNow, if you dissasemble the cube and reassemble it randomly, you will only have a 1 in 12 chance of ending up with a cube that you can solve all the way to the "start" configuration, This is because you are not completely free to rearrange the cubies but must obey ccertain restrictions. If you excahnge pairs, you can only exchnage an even number at a time. If you flip edges, you can only flip an even number at a time. If you twist corners, the total amount of twist can only add up to an integer. You can mix these restrictions and the result is the 12 orbits. If none of these is restrictions is violated, you end up with the Prime orbit which is the type of cube you have in the Start configuration. I call this the Bosonic orbit because an even number of pair exchnages is analogous to a particle with integer spin, and thus can regarded as a "boson". The next orbit has one pair exchanged, and the total number of pair exchanges of any pattern in this orbit must be an odd number. Any pattern in this group corresponds to a Fermion and has half-integer total spin. The introduction to the Wiki article on Supersymmetry is easy enough for anybody to understand the gist and some later parts are technical enough to appeal to the math geeks. If you play with patterns and in particular with correlating patterns in the Prime orbit with those in the Fermionic family, It would help to use two cubes for this, one in the Prime orbit and one in the Fermionic orbit. the nest orbit is one flipped edge or an odd number of flipped edges.After playing around with this for a long time, I decided it is analogous to magnetic poles. All particles have both a "north" and "south" pole or pairs of such, No particle has only a "north" pole or only a "south" pole. Such a particle would be called a "magnetic monopole" and some theories (G.U.T.) suggest they may exist, though none have been found. I only have one pattern for this, a version of the 6way-C (or 6-way U) pattern that appears like any of the others but if you examine the color symmetry, it has a bizzare, twisted lop-sidded symmetry that you won't find on any standard list of Rubik's cube symmetries. it is possible to create a new orbit by combining the previous two into an orbit of cube patterns with an odd number of pair exchnaged and an odd number of flipped edges. I only know one such pattern and it may be regarded as a "Fermionic Monopole" to draw an anology with particles. Please read the intro to the Wikipedia article on magnetic monopole for a better picture.
The next orbit involves the corner cubies. if you count a clockwise twist of a corner as +1/3 and a counterclockwise twist as -1/3, then the total twist must add up to an integer, never a fraction. This is a close analogy to the restrictions on how quarks may be combined to form a particle. 1/3 or 2/3 electric charges are associated with each quark, but quarks must combine in a way that the total electric charge of the particle is an integer. The mathematics used to describe the strong force that governs quark interactions is different from that used for spin. I only know two patterns in this orbit but they are very striking. They are both bosons and involve rotational symmetry through the corners.) Some hvae theorized the existence of particles that would have an extra quark and thus fractional electric charge. Patterns based on a cube in the single twisted corner orbit (actually 2 orbits, depending on whether the twist is clockwise or counterclockwise) correspond to these particles. I call this orbit the "quark" orbital. I haven't found it necessary to distinguish between the clockwise and counterclockwise versions. The other orbits are the possible combinations of the orbitals involving flip, pair exchange and corner twist.
So far, I ma working with the Prime, Fermionic, Monopole orbits (both Boson and Fermion), and the two trivially different Quark orbits. This is only half of the 12 possible orbits and I would be very interested if anybody finds patterns corresponding to any of the othe others. There is also a whole realm of 12 orbits belonging to the Mirror cube and I have some patterns I have discovered in the Mirror Boson and Mirror Fermion orbits. I will save the discussion on these until I get into the weighty subject of parity.be form pairs in exactly this way. So do many complex patterns. The compund basic patterns are different. Usually, if one kind of cubie forms a Bosonic pattern then so does the other, Like wise, if one is Fermionic so is the other. Now two Bosons combine to form a Boson because even plus even equals even (number of pair exchnages). Likewise two Fermions combine to form a Boson as well because odd plus odd also equals even. Anyway, the combined particle in either case is a Fermion and both the basic partner and the superpartner belong to the Prime orbit. You can distinguish them by examining their component sub patterns. , Both bosons for a Standard pattern and both fermions for a Supersymmetric pattern. Again, this parallels particle physics where two fermions combine to become a boson.

All simple basic patterns invoving the Prime (or Bosonic) orbit all have "partners" in the Fermionic (odd pair exchange) orbit. Bosonic patterns have either corner rotation (6-way) or 1/2 center turn (4-way) symmetry. Their Fermionic partners have either Flip (6-way) or 1/4 corner turn (4-way) symmetry. This is a deep correspondence to particle physics. SuperSymmetry theory assigns a "superpartner" to every basic particle of the Standard Theory. If the standard particle is a Fermion, its superfermion is a Boson and if the oarticle is a Boson, it's superpartner is a Fermion. The simple basic patterns of the cube form pairs in exactly this way. So do many complex patterns. The compund patterns are different. Usually, if one kind of cubie forms a Bosonic pattern then so does the other, Like wise, if one is Fermionic so is the other. Now two Bosons combine to form a Boson because even plus even equals even (number of pair exchnages). likewise two Fermions combine to form a Boson as well bexause odd plus odd equals even. In Particle theory, two fermions likewise combine to form a boson. Thus, the supersymmetry pairs of the compound basic patterns all are bosons in the Prime orbit, You can tell which is Standard and which is Suppersymmetric by looking at the subpatterns.
Two common 6-way simple basic patterns in the Prime orbit (both bosons of course) are the rotational 6-way Eyes pattern and the Rotational 6-way Checkerboard pattern (not the Reflection Checkerboard!). Both patterns are in the corner rotation symetry family. their Fermionic counterparts are theFlip 6-way Eyes pattern theand yhe Flip 6-way Checkerboard, both possible only in the Fermionc orbit. Note you can create a compound basic 6-way Crosses pattern with rotational symmetry by combining the two bosons and of course its a boson. If you combine the two fermionic patterns you get the compound basic Flip 6-way Crosses pattern but this is also a boson though made up of two fermions. There are two 4-way crosses in the Prime orbit. One has Center Half Turn symmetry and the other Center 1/4 turn symmetry and of course both are bosons. The 1/2Turn pattern is the Standard cube particle because it is made up of two bosons, both of what should be familiar but the 1/4Turn pattern is composed of two fermions that you are probably not familiar with.

For "home work" I recommend putting your cube into the one-pair-exchange or fermionic orbit and try creating all of the fermionic patterns I have described above. It will really help clarify for you what I am talking about. you might try finding some of the fermionic partners of the prime orbit complex patterns. if you really feel ambitious. They are not always easy to find. Try finding the 3rd type of the Prime orbit 6-way Checkerboard and the 2nd type of the Fermionic 6-way Checkerboard pattern.
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Cheers,

Math Bear ^,..,^
 
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cmhardw

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I think it's neat that there are many parallels between the way the symmetries and pieces interact on a 3x3x3 cube to particle physics. I have heard this idea mentioned before, but your explanation here is the most detailed one I've seen explaining the parallels. Thanks also for describing the possible ways to interpret the 12 orbits of assembling a cube as they compare to particle physics.

Does physics have a parallel to the 4x4x4 cube centers? The four center pieces of any color are indistinct from each other, and thus for them the concept of permutation parity makes no sense. Is there a particle or physical construct for which the parity is meaningless or non-existent?

Lastly, what about the supercube 3x3x3 where the rotations of the center pieces are noticeable? Let's start with the true supercube where all 6 centers have four distinguishable rotations each. There are also pseudo-supercubes for which only some centers have distinguishable rotations, or where some centers may have only two distinguishable orientations instead of four.

My thought is that at some point the analogy between the Rubik's family and particle physics breaks down, but perhaps it can extend a little further than the regular 3x3x3 cube?
 
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