# 3x3 Rubik's Cube Patterns and Particle Physics



## Math Bear (Feb 8, 2015)

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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## lerenard (Feb 8, 2015)

Math Bear said:


> 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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## Math Bear (Feb 8, 2015)

*Particle Physics and the Rubik's Cube*



lerenard said:


> 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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## whauk (Feb 8, 2015)

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:



Math Bear said:


> 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.



Math Bear said:


> 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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## qqwref (Feb 8, 2015)

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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## lerenard (Feb 8, 2015)

Math Bear said:


> But that is exactly the idea!



Oh ok. Well have fun with math and stuff.


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## OliverSW (Feb 9, 2015)

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


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## brian724080 (Feb 9, 2015)

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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## AlphaSheep (Feb 9, 2015)

brian724080 said:


> 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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## Math Bear (Feb 9, 2015)

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. ^,..,^



lerenard said:


> 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!



qqwref said:


> 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!



OliverSW said:


> 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.....



brian724080 said:


> 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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## Herbert Kociemba (Feb 9, 2015)

Math Bear said:


> Why do you say that?
> 
> I am totally interested in exploring cube oatterns and it is basically my purpose here.
> ^,..,^



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


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## Math Bear (Feb 10, 2015)

Herbert Kociemba said:


> 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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## qqwref (Feb 10, 2015)

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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## cmhardw (Feb 10, 2015)

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.



Math Bear said:


> 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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## JemFish (Feb 10, 2015)

qqwref said:


> 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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## AlphaSheep (Feb 10, 2015)

Math Bear said:


> 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... 



Math Bear said:


> 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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## Alphalpha (Feb 10, 2015)

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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## AlphaSheep (Feb 10, 2015)

Alphalpha said:


> 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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## Math Bear (Feb 11, 2015)

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. 
0
Cheers,

Math Bear ^,..,^


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## cmhardw (Feb 11, 2015)

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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## Math Bear (Feb 11, 2015)

cmhardw said:


> 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?
> 
> ...



I have had no luck in trying to match other types of cubes to these correspondences
with physics. There seems to be something unique about the standard 3X3 cube that makes it
fit unusually well. I have tried exchanging adjacent centers but all that reasults is the
symmetry of the cube is broken. I have never thought about making the rotational position of
the center visible, but it doesnt really seem to show up in patterns. Maybe try out the proposed
alternate coloring schemes for the cube?
I am still exploring aspects of the analogy so I am not sure where it will
start breaking down. I fits remarkably well so far.
i wonder if it could be possible to design cubes to model different'theories
of pphysics?

Thank you for your ideas!

Cheers,

Math Bear


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## AlphaSheep (Feb 11, 2015)

I'm sorry I came across quite aggressively yesterday. But I'll just ignore the particle physics bit (because I'm not comfortable talking about it if we ignore the underlying mathematics), because even on it's own the field of cube patterns is fascinating.

You should learn to use visualcube the [cube] tags to show pictures of your patterns. It uses the UDLRFB notation described on the wiki.

You can then embed pictures of cubes in your posts using BB code like this:


```
[c[B][/B]ube]alg=D2UF2D2F'D2LD2FU2R2B2U'B2R'U2[/cu[B][/B]be]
```

which will show a picture of a cube like this:


It's much easier than trying to describe the cubes in words. Also, instead of using move combinations, you can colour faces directly, so you can even do patterns that aren't possible on a normal rubik's cube, although that doesn't seem to work with the [cube] plugin:







Also, here are some more interesting patterns for you to look at.



That last one is definitely my favourite. It's a third order checkerboard. 

Also, all of these are on Herbert Kociemba's site, and his site lists EVERY possible type of symmetry, even those "bizzare, twisted lop-sidded symmetry that you won't find on any standard list of Rubik's cube symmetries", as you put it. Also, although the examples he gives are only for the single legal orbit of the Rubik's cube, the symmetries are still valid for other orbits. I

Also, you should really give Cube Explorer a try. You can basically colour in faces to make a pattern and then with a click of the button, find ways to get that pattern on your cube.


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## Math Bear (Feb 12, 2015)

AlphaSheep said:


> I'm sorry I came across quite aggressively yesterday. But I'll just ignore the particle physics bit (because I'm not comfortable talking about it if we ignore the underlying mathematics), because even on it's own the field of cube patterns is fascinating.
> 
> You should learn to use visualcube the [cube] tags to show pictures of your patterns. It uses the UDLRFB notation described on the wiki.
> 
> ...



************************************************************************************************************************************************************************

Math Bear

Are you sure about this? I want you to try making the pattern yourself and see if it matches any of his proposed symmetries. It may not be a valid symmetry of a Normal cube.

Try making the monopole pattern yourself and examine the resulting patterrn symmetry carefully. See if you can figure it out. You'll soon see what I mean when I say it is strange. Then check if it is a symmetry Herbert Kociemba includes in his list. I would be most interested if there are any other known patterns that have a like symmetry. I would dearly love to find more "monopole) patterns (odd # flipped edges orbit). This is something of an obsession of mine. 

I have 4 new cubes on their way to me. This will allow me to present all the variations of some patterns in a neat row. I would be delighted to use the software you mentioned to show off my patterns. For example, there are 5 distinct types of the 6-way Eye pattern, all with different symmetries, 5 types of 6-way Checkerboard patterns and 5 of 6-way Crosses patterns. The last of course is the result of combining the first two. Can you find them all? ^,..,^ 

Now to do the "monopole" version of the 6-way C's pattern. start with a regular cube in the standard orbit. The apply the following maneuver: aR +, sF 2, aR - The cube should have the top and bottom centers exchanged and 2 opposite stripes on the middle layer. If you move the middle layer by a 1/4 turn, you will get a flip pattern, different if you go sT + than sT - Rotate the top layer T + . Now exchange the edge cubie Top Right with the edge cubie Bottom Right.Note that the exchange is skew because you rotated the top first. To complete the exchange, exchange two edge cubies in the Middle layer between Top and Bottom). Then turn the top layer back T - . You now have the C pattern on top and bottom. Now turn the middle layer so that the centers on the middle layer match the side facelets of the cubies you exchnaged between top and bottom. Now arrange as necessary (you may have to flip 2 middle layer edges) so that you get the C pattern on the faces that contain the side facelets of the two cubies exchnaged between Top and Bottom. Basically, you shold now have 4 C's. The exception is the two left back faces. They should have the Eyes pattern with the two centers exchnaged. Notice that if you flip the edge cubie in the middle layer separating these two back centers, you will then also have the C pattern on these two left back faces. The best way to do this is rotate one of the back faces 45 degrees then pry out the cubie of interest and flip it then push it back in and turn the face 45 degrees in the opposite direction. You should now have the C pattern on all 6 faces but with an unusual pattern of symmetry with respect to how the pattern is oriented. The color exchanges follow the flip syymmetry pattern. 
I think the oddness results from the skew exchnage between top and bottom. It seems to twist the rest of the pattern.
to explore the regular 6=way C pattern with full flip symmetry,. Start the pattern over again on a regular Prime orbit cube but this time after applying the aR +, sF2, aR - manuever, exchange two top and bottom cubies directly over each other WITHOUT turning the Top face. the other pair of exchanged edge cubies should be two opposite edge cubies in the middle layer, if you can manage the necessary conjugation pattern. Anyway, accomplish this even if you need an extra move. Now flip all 4 edge cubies in the middle layer, Rotate the middle layer until you get the 4-way Eyes pattern around the equatorial faces of the cube. You may need to exchange two parallel pairs of middle layer edge cubies to get this to come out right. Once you have the "4-way Eyes pattern, look for two opposite middle layer edge cubies, that if you flip them, will create 4 C patterns on the equatorial faces. The resulting 6-way C pattern should have a clear cut flip symmetry. 
It really helps if yuou can have the twisted pattern on one cube and the normal pattern on the other cube but matching the color exchanges in voved in the flip symmetry of each cube.


Anyway,

Have Fun!

Math Bear


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## Herbert Kociemba (Feb 13, 2015)

Math Bear said:


> Try making the monopole pattern yourself and examine the resulting patterrn symmetry carefully. See if you can figure it out. You'll soon see what I mean when I say it is strange. Then check if it is a symmetry Herbert Kociemba includes in his list. I would be most interested if there are any other known patterns that have a like symmetry. I would dearly love to find more "monopole) patterns (odd # flipped edges orbit). This is something of an obsession of mine.
> 
> I have 4 new cubes on their way to me. This will allow me to present all the variations of some patterns in a neat row. I would be delighted to use the software you mentioned to show off my patterns. For example, there are 5 distinct types of the 6-way Eye pattern, all with different symmetries, 5 types of 6-way Checkerboard patterns and 5 of 6-way Crosses patterns. The last of course is the result of combining the first two. Can you find them all? ^,..,^
> Math Bear



1. The symmetries a cube can have are completely independent of the orbit it is in. So it is impossible to find new kinds of symmetries different from the listed 33 types. But since in the "prime orbit" there are no realisations with exactly the symmetry O and Td, there might be realizations in some of the other orbits. I checked this with the result, that there are no cubes with exactly symmetry Td in the other orbits. There are 4 cubes with exactly symmetry O and the all lie in the orbit with a wrong corner twist. See the attached file below, you can load it into Cube Explorer to visualize them.

2. I also did a quick hack for the pattern search part and deleted the restriction that the solutions have to be in the prime orbit. I confirm that there are 5 checkerboard patterns now. I do not exactly know what you mean with "6-way eye pattern" but if you mean the 6-dot pattern, there are only two different kinds, and there are also only two types of "6-way crosses patterns". See the attached file below.


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## Math Bear (Feb 14, 2015)

Herbert Kociemba said:


> 1. The symmetries a cube can have are completely independent of the orbit it is in. So it is impossible to find new kinds of symmetries different from the listed 33 types. But since in the "prime orbit" there are no realisations with exactly the symmetry O and Td, there might be realizations in some of the other orbits. I checked this with the result, that there are no cubes with exactly symmetry Td in the other orbits. There are 4 cubes with exactly symmetry O and the all lie in the orbit with a wrong corner twist. See the attached file below, you can load it into Cube Explorer to visualize them.
> 
> 2. I also did a quick hack for the pattern search part and deleted the restriction that the solutions have to be in the prime orbit. I confirm that there are 5 checkerboard patterns now. I do not exactly know what you mean with "6-way eye pattern" but if you mean the 6-dot pattern, there are only two different kinds, and there are also only two types of "6-way crosses patterns". See the attached file below.



Hee hee! Gotcha! ^,..,^

I suspected you would be able to find the five 6-way Checkerboard patterns. But I was pretty sure you would not be able to find the 3 other corresponding patterns for the 6-way Spot and 6-way Ciross patterns. In the original scientific American article, the "Spots" pattern was called "Windows" and in later literature I heard it called "Eyes". I didnt knw the contemporary term for it., But it involves moving center cubies only. 

Parity refers to the tendency of geometric objects to come in mirror image pairs such that the right hand image is different from the left hand image ((I am speaking to the crowd BTW, not just you)) in the same way right hands are different from left hands. Parity is an intrinsic part of our Universe, and appears in such diverse fields as particle physics and biochemistry. subjects in which parity plays a huge role. It also shows up in coordinate systems. The traditional X,Y,Z 3D Carteesian Coordinate system most people use has rightward represented by the +X Axis, uoward represented by the +Y Axis and outward represent the +Z Axis. if you let your thumb point in thw X direction, your 1st finger point in the Y direction and the middle fimger point in the Z direction, the result matches the left hand and as a result, we call it a "left Hand" coordinate system. We have to be aware of parity even on very basic levels. 

In particle physics, parity restricts tthe kinds of interactions particles can undergo. Parity manifestations shouldn't show up out of nowhere, parity needs to be conserved. If the interactions involve the Strong, Electromagnetic or Gravitational forces, parity is conserved, but not in interactions invoving the Weak force. These clearly show parity alteration. Since the Weak force governs radioactive decay, and nuclear fission it is quite important. Check out the Wikipedia articles on "The Weak Interaction" snd "Chirality(physics)" for an introduction. 

Rubik's cube patterns involving the corner or center cubies respect parity but patterns involving the edge cubies can violate it. This may reflect the fact that the edge cubies are the only ones that look identical to their mirror image and thus have no parity. This is why you could find all 5 Checkerboard patterns on a "Normal" cube, three of the symmetries require parity transformations and since the Checkerboard pattern only involves edge cubies, it can be easily accomplished without altering the cube. For the Spot and Cross patterns, you must alter the cube. In 3D space, a "parity transformation" (converting from the left-handed to the right-handed pattern or vice versa) can be accomplished by flipping one axis, exchanging the + and - sides of the axis. You can also do this by flipping all 3 axes. Both are widely used for parity transformations. To parity transform the Rubik's cube you must exchange 2 centers on opposite faces of the cube. This is not in any of the 12 orbits and in fact has 12 orbits of its own, I call this a Mirror cube and the original, non-arity transformed cube the Normal cube. This transformation can be easily accomplished on a Speed Cube as you can pry off the centers individually but not with the original or Rubik's New Cube. I treat as the basic Start state of the Mirror cube the 6-way Spots pattern in which each center is exchnaged with its opposite color. This generates the first of the 3 Spots and Cross patterns you did not find. 


Now a note on the Start pattern of both types of cubes: The virgin Start configuration of the Standard cube has no exchanged cubies. It thus has zero spin. But it definitely has parity as you can see if you hold it up to a mirror. Now the basic definition of the Higgs Boson in physics, the infamous "god" particle (physicists rEALLY hate that expression!) is of a spin 0 Boson that has parity. This fits the characteristics of the Start pattern of the rubik's cube so we could call this the Rubik's cube Higg's particle. The 6-way spot pattern in which each center is exchanged for its opposite is for me the Mirror equivalent of the Normal cube Start pattern. It is a zero-spin (no pair exchnage) Boson of opposite parity. I call it the "mirror Higgs particle" and it corresponds to the Start configuration on the Normal cube. Yes, I know you could also just exchange a single pair of centers but the result is not as symmetrical and I do not consider it a proper particle, just a transformation.

Physicts have speculated on the existence of particles with reversed parity. This is called Mirror Matter (sometimes called "Shadow" or "Alice" matter). Now an electron generally is described as having "left-handed" parity. The positron has right-handed parity but it doesn't count as it has reversed charge and everything else. What they are looking for would be a negatively charged electron with a right-handed bias and an antiparticle with left-handed bias. this is by no means a trivial goal. Parity affects all the forces except gravity. Mirror particles interact with other Mirror particles through Mirror forces. even the photons are Mirror photons. Mirror matter would not interact with Normal matter at all except through gravity. This is in contrast to antimatter, which interacts forcefully indeed! Mirror matter basically doesnt interact. This would make it difficult to detect in particle experiments; There are some theories which suggest that at extreme energies, exotic phenomena would allow for at least slight interaction between Mirror and Normal matter, producing detectable effects. If mirror matter is a major component of our Universe then it should show up gravitationally and in fact is becoming an increasingly good candidate for Dark Matter. Please check out the Wikipedia article on "Mirror Matter" for a fascinating discussion. 

Anyway, allowing parity transformed cubes expands significantly the range of Rubik's cube patterns and permits us to complete families of patterns. 

There are 5 fundamental symmetries for a typical 6-way pattern, 2 Normal and 3 Mirror. All of these are possible on edge cubie only patterns without altering the cube as these are immune to parity restrictions. Anyway, the two Normal symmetries are "rotational", that is to rotate the cubes through a corner axis by 60 or 120 degrees These patterns all seem to be Bosons, whether in the Standard orbit or not), and "Flip", whereby you "flip" the cube by 180 degrees through an axis through 2 opposite edges in a middle layer. Simple basic flip patterns are Fermions but the compound patterns are often Bosons. The firstMirror symmetry has everything on each side exchanged with its opposite side, as with the Mirror Higg's particle. This will be called the "reflection" symmetry. The next has a symmetry pattern that crawls along all 6 faces of the cube in the same way the "worm" pattern does, It is the parity transformed version of the "rotational" pattern. I call it the 'swirl" symmetry.

To deal with the last symmetry, I need to discuss the 4-way symmetries. These all apply to 4 equitorial faces while ignoring the opposed top and bottom faces. The easiest is the symmetry in which each equitorial face exchanges cubies with its opposite side. This does not really count as a parity transformation as 2 such transformations put you back in your original parity. It is equivalent to a rotation about the vertical center axis of 180 degrees, so I call it the "1/2turn" symmetry. Simple basic patterns of this sort are Bosons. The next symmetry is to turn the cube 90 degrees about a center axis. This can be done clockwise or counterclockwise. Simole basic patterns of this sort are Fermions, but the compound patterns can be Bosons as two fermions become a boson. This is just called the "1/4 turn" symmetry. The last pattern has the 4 sides divided into two groups of adjacent pairs such that the member of each pair exchanges cubies (sometimes just facelets if the edge between flips) with theother pair. This creates a "side by side" effect pattern. This symmetry is actually the parity transform of the "flip" symmetry, moving the cubies on the top and bottom faces back to where they were. If it involves edge cubies only (like a 4-way Checkerboard) then the result is a Normal cube) pattern. I call this the "adjacent" symmetry. Simple basic patterns involving 1/4 turn symmety or adjacent symmetry are Fermions. obviously, the compound 1/4 turn symmetry 4-way Cross is a boson, because it is made up of two fermions. It is the Supersymmetry partner of the 1/2 turn cross which is made of two bosons. 
The best way to accomplish the parity transformation when only edge cubies are involved is to do the moves for the 6-way reflection Checkerboard (the one everybody knows) "sR2, sF2, sT2". This will give you a perfect parity transformation involving only the edge cubies. 

To do this with center cubies, you have to physically transform your cube to a Mirror cube. This is easiest if you pry off two opposite centers and exchange them. Then you can generate the 6-way symmetry I call the "twirl" symmetry. It is like the 4-way 1/4 turn symmetry but with top and bottom exchanged, It is the obvious parity transformed version of the "1/4 turn 4-way symmetry. It is also generally results in fermions. 

If you apply these to the 6-way spot patterns, you will get all five symmetry counterparts of the Checkerboard patterns and if you combine them together, you will get all 5 families of the corresponding Cross patterns. Counting color permutations as separate patterns, you will find that you get 29 different kinds of each pattern. There are 8 permutations of the "rotational" symmetry, 8 Permutations of the parity transformed "swirl" symmetry, 1 only of the parity transforming "reflection" symmetry, 6 of the "flip" symmetry and 6 of the parity transformed "twirl" symmetry, adding up to 29 in all. It would make a powerful visual impression to get a bunch of cubists to generate all 87 possible cube oatterns and display them properly organized. 

If anybody is really bored, they might try generating all the 29 patterns of one particular type. This might sound tedious, but there are simple transformations that will allow you to easily and quickly transform one color permutation into another. A speed cubist could do it quickly!

Cheers,

Math Bear ^,..,^


P.S. Sorry for such a lengthy submission, but I warned you "parity" was kind of a 'weighty" subject.
I won't need to submit such long messages again. Both long articles pretty much cover the heart of my discoveries.
Also, keep an eye out for articles on "mirror mattter" in the press. Since they have so conspicuously failed
to find evidence of Suppersymmetry , it is quickly shaping up as a hot new topic, especially to explain "dark matter".

P.S. I don't want them to give up on Supersymmetry either. For the first two years I was in college, I was an Organic
Chemistry major. Physicists seem to just assume that if you generate enough energy, you automatically get the kind of particle you want.
Chemists know that energy is often not enough to get what you want. You often need a catalyst of some kind. perhaps the physicists could try providing something similar in their experiments. I would try generating supersymmetry partners in the presence of an intense beam of neutrinos to try to create particles with abnormal values of spin.


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## Herbert Kociemba (Feb 14, 2015)

Math Bear said:


> For the Spot and Cross patterns, you must alter the cube. In 3D space, a "parity transformation" (converting from the left-handed to the right-handed pattern or vice versa) can be accomplished by flipping one axis, exchanging the + and - sides of the axis. You can also do this by flipping all 3 axes. Both are widely used for parity transformations. To parity transform the Rubik's cube you must exchange 2 centers on opposite faces of the cube. This is not in any of the 12 orbits and in fact has 12 orbits of its own, I call this a Mirror cube and



It seems that we talk of different things. If you start to exchange center stickers without exchanging the corresponding stickers on edges and corners, obviously in most of the cases you get an object which is no Rubik's cube any more. Even by disassembling and reassembling this object you will not be able to reach the "solved" state. That you can achieve more patterns in this way is obvious. I do not know which stickers you allow yourself to peel off and exchange, but in principle you can achieve 54!/(9!^6) different patters with your method. So you should be more specific and strictly define the operations you allow to build your patterns.


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## Math Bear (Feb 14, 2015)

*Huh???????*



Herbert Kociemba said:


> It seems that we talk of different things. If you start to exchange center stickers without exchanging the corresponding stickers on edges and corners, obviously in most of the cases you get an object which is no Rubik's cube any more. Even by disassembling and reassembling this object you will not be able to reach the "solved" state. That you can achieve more patterns in this way is obvious. I do not know which stickers you allow yourself to peel off and exchange, but in principle you can achieve 54!/(9!^6) different patters with your method. So you should be more specific and strictly define the operations you allow to build your patterns.




Why do I get the impression you have no idea what I am doing? 

I am not really exchanging the center stickers, I am exchanging the center CUBIES! I gave no interest in exchanging stickers, I do not regard that as valid or meaningful. As for what I am doing, you really need to go back and read my two long articles carefully. My concepts revolve around the 12 orbits and the parity reversed cube, which is really not the same thing as a mirror umage reversed cube. please read carefully before you comment.


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## ender9994 (Feb 14, 2015)

Math Bear said:


> Why do I get the impression you have no idea what I am doing?
> 
> I am not really exchanging the center stickers, I am exchanging the center CUBIES! I gave no interest in exchanging stickers, I do not regard that as valid or meaningful. As for what I am doing, you really need to go back and read my two long articles carefully. My concepts revolve around the 12 orbits and the parity reversed cube, which is really not the same thing as a mirror umage reversed cube. please read carefully before you comment.



Aren't you both talking about the exact same thing. Swapping 2 centers can be done by removing the stickers and swapping them (as Herbert Kociemba Stated), or by simply swapping the 2 center core pieces (Which is what I believe you are stating in the above quote). Both methods result in illegal cube positions.


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## Herbert Kociemba (Feb 15, 2015)

Math Bear said:


> My concepts revolve around the 12 orbits and the parity reversed cube, which is really not the same thing as a mirror umage reversed cube.



So I ask you again: Which kind of operations you allow to create your patterns? Exchanging arbitrary corners, twisting corners, exchanging arbitrary edges, flipping edges, exchanging opposite centers. Something else?


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## AlphaSheep (Feb 15, 2015)

Math Bear said:


> Why do I get the impression you have no idea what I am doing?
> 
> I am not really exchanging the center stickers, I am exchanging the center CUBIES!



If you remove the stickers from centre cubies, then they are, for all intents and purposes, indistinguishable. Therefore there is no practical difference between swapping centre cubies and swapping centre stickers. No need to get rude about it.

On many speedcubes, each cubie can be disassembled and then the parts can be reassembled in different combinations. This would allow patterns like the one below:





Would you allow operations like this? If not, why not?

I ask, because swapping centre cubies requires either disassembling the cubie (e.g. by popping a centre cap), or disassembling the core. This effectively results in changing the colour scheme of the cube, and the other pieces (edges and corners) will no longer match the colour scheme of the cube. If you're going to allow pieces that don't match the colour scheme of the cube, why not go all the way?

Also, allowing swapping of centre cubies drastically increases the number of possible orbits. I think there are 240 possible orbits, unless I've miscounted...


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## Math Bear (Feb 15, 2015)

Herbert Kociemba said:


> So I ask you again: Which kind of operations you allow to create your patterns? Exchanging arbitrary corners, twisting corners, exchanging arbitrary edges, flipping edges, exchanging opposite centers. Something else?



I started out my "project" (if you want to call it that, for me it was more like idle curiousity) several years ago as an attempt to analyze and classify the patterns on the Rubik's cube. I had read in the early literature on the cube about the possible connection between the Rubik's cube and particle physics, pointing out the similarity beyween the way corners could twist and the way that quarks combined to form a hadron. I also remember one early source speculating whether other orbits of the cube might have interesting patterns. I was aware of this but I didnt feel motivated to do anything about it. Anyway, I was very curious as to why, for example there were three different 6-way Checkerboard patterns but only two 6-way cross patterns and only one 6-way spots patern, The Cross patterns, for example, were composed of a Spot plus a Checkerboard pattern. I could see that the rotational Cross was composed of a rotational spot and rotational checkerboard pattern, both patterns I could easily generate. But the Flip cross was different, I was unable to create the corresponding Flip Spot pattern and to my surprise, could not get at the Flip checkerboard pattern either. The Flip cross was easy to generate so why not its components? Another problem was the 6 way Zig Zag ((or 6-way Diagonal) pattern. The 4-way pattern was easy to make. Turn the cube in your hands and do an antislice with each turn and after 6 such amaneuvers you had a paerfect 4-way Zig Zag. But wh6 faces y did it seem so impossible to have the pattern on all faces? (the pattern is that 2 opposite corners and the center in between are one color and the rest of the cubies another.) I gradually became aware that the symmetries of the Rubik's cube were incomplete. My first attempt was to expand the possible pattern symmetries of the Rubik's cube by exploring the (odd-pair-exchnage) orbit. This allowed me to extend the checkerboard patterns to all five possible 6-way symmetries. I was also aso able to get the flip Spot pattern. But I could find no more Cross patterns. There were none in the new orbit. I also became aware that there was a striking correlation between the even-odd pair exchange patterns and the way "spin" works in particle physics. The restriction on valid moves of the Rubik's cube that you could only exchange an even number of pairs corresponded to the rule in particle physice that you could only change a particle's spin by an integer amount. In fact, the basic unit of spin in particle physics was spin1/2 so this was equivalent to changing the spin by two "units" of spin1/2. Particles with integer spin were called "Bosons" and particles with net spin1/2 (or non-integer spin) were called Fermions. I started thinking of patterns in the Prime orbit as "bosons" and those in the (odd-pair-exchange) orbit aas "fermions". Allowing this significantly expanded the symmetries of the cubes and allowed the explicit expression of patterns that were only implicitly present on the Prime orbit. Thus, although the Flip cross was in the Prime orbit, it was made of two fermionic patterns that I could expresss explicitely in the Fermion orbit. The symmetrical patterns naturally come in Boson Fermion pairs (or double Boson double Fermion pairs for the compound particles) just like the predicted Boson-Fermion Supersymmetry pairs predicted by physicists for particles.

This seemed to me an important extension of Rubik's cube patterns. I became curious about the other orbits. The twisted corner orbit had already long been speculated about as being related to "quarks" in particle physics. By twisting one corner, I was soon able to generate the long sought for 6-way Zig Zag pattern. This convinced me that the addition of the "one twisted corner" or as I took to calling it, the "Quark" orbit was a valid extension of the Rubik's cube. patterns. I only found a few but I believe there are others to be found. The one-flipped-edge orbit frustrated me for a long time. It didnt really seem to contain any valid patterns, nor did it seem to correspond to anything obvious in particle physics. I finally observed a correlation betwen edge flips and magnetic poles. Magnetic poles can only be created in North-South pairs, just as edges can only be flipped in pairs. Physicists had long speculated on the existence of particles containing only one pole, North or South, and which would have net magnetic charge, North or South just as particles can have net electric charge, + or - .But such particles had neither did I have any very compelling patterns involving an odd number of flipped edges. I thus thought for a long time that the "odd-flipped-edge" orbit did not have any valid extensions for Rubik's cube patterns. Then, quite by accident I found a variation of the 6-way C (or U) pattern(I know of 4 kinds of these patterns) that naturally fell into this orbit and convinced me that there must be others so I decided this orbit was valid after all and called it the "Monopole" orbit. 

Now , the Quark and Monopole patterns were rare but I had a lot of Boson-Fermion pairs among my basic patterns. Some elements were tricky to find. I had no trouble finding the flip version of the "worm" pattern in the Fermionic orbit, but the corresponding "Snake" pattern was much harder to find. I knew of the flip Bosonic version, it was a well known pattern, but the corresponding Fermionic version was elusive. I originally tried to find it among the rotational symmetry patterns but there is no such thing as a "rotational" snake! I finally found it among the flip symmetry fermionic patterns. In this case, both the fermion and boson members of the pattern pair have flip symmetry which is natural to the snake pattern. In fact, the two patterns look almost the same, the difference is rather subtle. I loved the "2x2 cube inside the 3X3 cube" pattern. It was discussed in the original Scientific American article where it was called the "Giant Meson". The fermionic version was much trickier to generate and as you might expect had flip symmetry. It was strikingly different as the 3 colors of the embedded 2X2 cube were all different from the 3X3 cube it appeared to be embedded in.

I now had a comprehensive expanded theory of patterns of the Rubik's cube that occured in distinctive pairs that I called supersymmetry partners borrowing the striking anology with particle physics in which particles occured as Boson-Fermion pairs and were called supersymmetry partners. 

But I noticed there was something strange about edge patterns. the edge patterns could form in patterns that violated parity, patterns forbidden to the other cubies. The most famous edge cubie only pattern was the traditional 6-way Checkerboard pattern, equivalent to a parity transformation on all the edges, exchanging each with its opposite edge. I could also impose this transformation on the rotational version of the Checkerboard and the flip (fermionic) version of the Checkerboard to create two new 6-wy symmetries, the Swirl and Twirl symmetries. this made for 5 basic symmetries on the 6-way patterns and a little additional thought revealed that there were no additional ones possible. The basic list of possible symmetries was now complete. 

But there was a problem. The complete list of symmertries applied only to the edge cubies. For example, for 4-way patterns I had the complete list of possible 4-way symmeetries for the Checkerboard, 1/4 turn (fermion), 1/2 turn (boson) and adjacent symmetry 4-way Checkerboard pattern (also fermion). But I had no adjacent symmetry 4-way Cross or Spot patterns. 


My list of symmetries in cube patterns felt strangely incomplete! Originally, I did my research on an ancient model of the cube. It never occured to me to try messing with the centers. i was stuck for a long time on this. Then I found out about the new Rubik's Speed Cube and I realized you could switch centers on it. I ordered 2 so I could have one in the Bosonic Prime orbit and one in the Fermionic orbit. I primarily got them as they could be adjusted to run smoothly right from the beginning instead of dealing with a prolonged and fristrating breaking in period. I thought about the significance of the center cubies as they could be switched on my new cubes. Now, no legitimate manuever of the cube can alter the realtionship of the centers to each other. This property is taken advantage of in the construction of the cube. The center pieces are attached to a spindle which holds the cube together. This is just as well. If you look at the number of ways to arrange the centers, you come up with 30 different possibilities, occuring in 15 pairs that are mirror reflections of each other. these rearrangements are basically vain as far as Rubik's cube patterns are concerned, irreversibly breaking the symmetry of the cube. So I left it alone for the time being. Then I encountered the discussion on "mirror Matter" in physics, how evidence was slowly accumulating that it is a valid concept and an increasingly suitable candidate for "Dark matter". Mirror Matter requires that you perform a "parity transformation" on a regular particle, changing a "left-handed" electron into a "right-handed" mirror electron. The strange properties of parity in particle physics have puzzled and intrigued physicists for generations,and it is an active topic of research at the deepest levels. 
Now, the simplest way of accomplishing a parity transformation in a coordinate system is to flip one axis, letting the positive and negative directions interchange. The equivalent on the Rubik's cube is to exchange an opposite pair of centers. Now physicists and mathenaticians often prefer to fip all 3 axes as this accomplishes the same parity transformation but without creating a "priveledged" axis different from the others. Doing all 3 better preserves the original symmetry. This reflects the fact that the new coordinate axes can be rotated so that any pair can be regarded as the "flipped" one. Flipping all 3 makes this obvious. 

Now the Rubik's cube allows you to perform parity transformations on edge cube patterns, but only on the edge cube patterns. I would not want to try messing with the corners. In my system it is the corners which fundamentally define the faces and thus the kind of cube you have. Exchanging the corners for their mirror image equivalents creates an entirely new cube, not a valid pattern on the original cube. Likewise, allowing promiscuous exchange of the center cubies fundamentally breaks the symmetry of the original cube, creating a fundamentally new kind of puzzle. But what about exchanging centers only with their opposites? No matter how you go about doing this, you only end up with two possibilities, a Normal cube like the original or a Mirror cube with the centers in a parity transformed configuration. With the new speed cube, I could exchange a center cubie with its opposite by orying off the color tiles and exchanging them. This accomplishes EXACTLY the same result as unscrewing the two centers and exchanging them, it is just a lot easier to accomplish. This is not the same thing as peeling off stickers and exchanging them. In the great majorityy of cases, this will not accomplish the same result as exchanging two cubies. But if it makes you feeel better about it, could ahead and actually exchange the two center pieces. You can only do this with centers because they only have one facelet. 

By allowing the parity transformation on the centers as well as edges, I was able to again majorly expand the range of Rubik's cube patterns. I was also able to finally complete the basic patterns of the Rubik's cube. I now had 5 different kinds of Spot and Cross patterns as well as Checkerboard patterns, I now had all three kinds of 4-way symmetries. I now had the complete set of patterns, none missing. if I were to present a randomly chosen pattern from the full set of 87 possibilities, you would not necessarily be able to tell if it was a Normal or a Mirror pattern just be a quick glance. The new patterns fit seemlessly in with the old ones. The Mirror patterns also come in Boson and Fermion versions, just like the old ones. 

The fact that allowing the Mirror cube as a valid transform of the original cube by allowing the exchange of opposite centers allowed me to finally come up with a full and complete geometrical/transformational theory of all of the basic Rubik's cube pattern. There is quite simply no other way to accomplish this. I do not regard peeling off and switching stickers as in any way valid, this simple violates the integrity of the fundamental symmetry of the cube. All the changes I justified by the important results that were therby obtained. The same motivation was used to extend the range of particle theory. There is a powerful imperitive in the human mind to complete perceived symmetries. the existence of negatively charged electrons, by symmetrical extension, suggests the existenceof positively charged electrons and this led to the discovery of antimatter. The discovery that the electron has an inherent left-hnded parity bias has led to the idea that right-handed electrons exist and this idea may lead to the discovery of Mirror Matter. My discoveries are the result from a strong desire to complete the symmetries of the fundamental patterns of the Rubik's cube. I have done this using a fundamentally geometric descriptive language. I know you like to use group Theory, but this is fundamentally an algebraic language. Although 
it expresses a great many important facts and realtionships about the cube, It cannot fully express the fundamentally geometric nature of a fundamentally 3D geometric object like the Rubik's cube, at least not in a way that satisfies a normal human mind. 
I really want to bring the whole discussion on Rubik's cube patterns away ffrom a predominately algebraic approach to a fundamentally geometric approach. i welcome the insights that Group Theory can provide but these should NOT supplant the fundamental geometric insights that are of the very essence of the cube itself. Formal algebra does not suffice.

I am also struck by the striking similarities that exist between the patterns of the Rubik's cube and the different families of proposed
particles in particle physics, Please note that this is ANALAGOUS! It is in no sense exact as in the mathematical sense of "isomorphic" or even "homomorphic". For example, you could say that all known particles correspond to the Prime orbit of the cube, All the other orbits and the Mirror orbits are equivalent to the various speculative particles that have yet to be found. Yet most actual particles are fermions whereas all the patterns in the prime orbit are bosons. The analogy is striking anyway. Why should it surprise anybody that the world of real life particles would be more complex than the world of Rubik's cube patterns? Maybe somebody will be able to design some sort of exotic computerized cube that can fully embody a particle.

I renounce all further changes and expansions to the Rubik;s cube patterns. The symmetry is now complete. There is no need to further mess with it!

Cheers,

MathBear 

P.S. Most of you may not have heard of "mirror matter", but I suspect you will, especially with the current difficulties involving "supersymmetry". If it is the best candidate for "dark matter", then there just may be an alternate universe of mirror matter that interacts with our universe only gravitationally. A parity transformation can be accomplished by moving into a higher dimension, in our case the 4th Dimension and rotating in the new dimension. This is beginning to sound like something out of Star Trek! I better stop......

^,..,^


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## Herbert Kociemba (Feb 16, 2015)

Exchanging all three pairs of opposite centers is also indeed usefull since it does not change the symmetry of the pattern. Exchanging an odd number of opposite center pairs is btw. equivalent to leaving the centers unchanged and instead substituting all corrners by their mirror images.
Though Cube Explorer is not designed to find patterns in that "mirror space", it can be used to find all these patterns too. If you search for example for for the 6 cross patterns 

ABA
BBB
ABA

in the mirror space, you define the two patterns

ABA
BAB
ABA

and

ABA
BCB
ABA

and both allow them on all six faces simultanuously. There are not too may patterns -27 in this case - and you sort out all those manually where the pairwise exchange of all opposite centers does not give the 6 cross pattern. There are exactly three patterns left, so together with the two patterns in the standard space there are indeed five different types.

DB DL DF DR UB UL UF UR BL BR FL FR UFR URB UBL ULF DRF DFL DLB DBR //Oh{I}
DB DL DF DR UB UL UF UR BL BR FL FR FRU FUL FLD FDR BUR BRD BDL BLU //S6{D3d}
DB DL DF DR UB UL UF UR BL BR FL FR ULF UFR URB UBL DFL DLB DBR DRF //C4h{D4h}

If you put the three lines into a text file, you can load this directly into my program. I also appended the symmetry information for these three patterns.

In a similar way it is for example possible to show that there are 4 snake patterns. The first two are in the standard space, the other two in the mirror space.

FR FU UB UL DF DR BL BD RU FL BR LD FRU FUL FLD FDR BUR BRD BDL BLU //S6{D3d}
DL DB UB UL DF DR UR UF LB FL BR RF DLB DBR DRF DFL UBL ULF UFR URB //C2h(b){I}
UF UR DF DR UB UL DB DL FR BR FL BL UFR URB UBL ULF DRF DFL DLB DBR //D3d{I}
FR FU DF DR UB UL BL BD RU BR FL LD FRU FUL FLD FDR BUR BRD BDL BLU //S6{D3d}

Of course you have to use a special version of Cube Explorer, where the pattern search ignores the usual twist, flip and parity restrictions. Do not try to solve the results of the pattern generator in the main window, since the program will crash if the cube is not in the "prime orbit". 
You can download the special version here, if you want.


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## Math Bear (Feb 19, 2015)

Herbert Kociemba said:


> Exchanging all three pairs of opposite centers is also indeed usefull since it does not change the symmetry of the pattern. Exchanging an odd number of opposite center pairs is btw. equivalent to leaving the centers unchanged and instead substituting all corrners by their mirror images.
> Though Cube Explorer is not designed to find patterns in that "mirror space", it can be used to find all these patterns too. If you search for example for for the 6 cross patterns
> 
> ABA
> ...



Actually, I made my discoveries geometrically fiddling with the cube in my fingers. I now have a total of 6 speed cubes I ordered from the Official Rubik's Cube website. I like to try different variations of the same pattern then set them up in a row. I will need to explore to find the mirror versions of the snake patterns. I researched the additional paterns on the cube in part so I could have fun with the cube, I will resist the temptation to use a mathematical program to find the patterns but do it through exploration with my fingers on an actual cube. I usually start with one of the 6 Spot patterns then think it through from there. I know 4 different variations on the 6C's pattern. Two are in the Prime orbit, one is Mirror Fermion and one is in the Monopole (odd# of flipped edges) orbit. I am sure there are others..... and I shall have fun hunting them down. but thanks for researching this for me! It gives me an idea of goals to set.

I have been playing with the Quark orbit patterns, where you start by twisting a single corner. I only have two good patterns so far but they are apectacular and unlike the other patterns. This is because they follow a tetrahedral symmetry. The 8 corners on a cube are organized into two tetrahedrons pointing in opposite directions. Putting the cube into the Quark orbit emphasizes this symmetry. To make the 6 Diagonal (or 6 Zig Zag) pattern, start with the one twisted (or single quark) corner of the cube. Rotate the 3 adjacent centers to match the faces of the "Quark" corner. notice that the Quark corner connects to 3 closer faces by edge cubies and to 3 more distant corners by the center faces.The Quark corner and the 3 distant corners make up one tetrahedron. Rotate the 3 more distant corners to match the 3 rotated centers. This is the 6 Diagonal pattern. The 4 corners of one tetrahedron are connected by 6 diagonals making the pattern whereas the corners of the other tetrahedron, pointing in the opposite direction are surrounded by edges of the same color. 

You can make a really great pattern using these secondary 4 corners that are not part of the diagonal pattern. Try rotating each of these corners so the colors dont match any others on the new faces they are rotated to then rotate the 3 adjacent centers to match them. The result can be called the 6-Way Three Color Diagonal pattern. On each face you have the diagonal of one color with the 3 adjacent facelets of another color on one side of the diagonal and 3 facelets of another color on the other side of the diagonal. The result is three different colors on each side in a dynamic diagonal pattern. Very striking! I wonder what other tetrahedral symmetry patterns there might be in the Quark orbit? It will be fun to explore.

Anyway, Good Luck in your researches!

Cheers!

Math Bear ^,..,^


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## Herbert Kociemba (Feb 20, 2015)

There are lots of 6C patterns, I count 22 basically different patterns in the standard space, I did not look for patterns in the mirror space.

There are six 6-Way Three Color Diagonal patterns in the "prime orbit", so generators can be given

B' D U' L' B L R F' L2 U' F2 L F2 R2 U2 L F2 U (18f*) //S4{D2d(face)}
B F L R U' B' F' D L R U' L R D' B2 F2 U (17f*) //C2(a){C2v(a1)}
F L F2 D B' F R' F U' F D2 U F2 D' U F' R' (17f*) //S4{D2d(face)}
D L R B' F' D' L R U R2 D' U' F D U R2 (16f*) //C2(a){I}
D' R' B F' D L2 U R2 D F2 L' R' D2 F U' (15f*) //S4
D B' U2 L R B2 U' R2 D' L2 U' B F' R U (15f*) //S4

There are seven more patterns of this kind in the other orbits

LU RB RU LF RD RF LD LB DF UF UB DB UFR RBU UBL LFU RFD DFL LBD DBR //T{Td}
LU RB RU LF RD RF BL BD DF UF UB LD UFR RBU UBL LFU RFD DFL BDL DBR //C3{C3v}
FR FU BR BD BL FD FL BU RU LU LD RD FRU URB BRD ULF DRF BLU DLB FLD //C3{C3v}
LD RB RU LB LU LF FR FD UF DB UB RD UFR RBU UBL LBD LFU DFL FDR DBR //C1{Cs(b)}
FR FU LD LF RD LB LU RB RU DB UB DF FRU URB LDF ULF DRF RDB DLB LUB //C3{C3v}
RU LF LU RB LD LB RD RF DB UB UF DF UFR LFU UBL RBU LBD DFL RFD DBR //D2(face){D2d(face)}
RU LF LU RB LD LB FR FD DB UB UF RD UFR LFU UBL RBU LBD DFL FDR DBR //C1{Cs(b)}

and definitely none in the mirror space.

I think it will be difficult to find them all without the help of some program, but good luck and fun to find them manually!


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## Math Bear (Feb 21, 2015)

*More Talk on Patterns*



Herbert Kociemba said:


> There are lots of 6C patterns, I count 22 basically different patterns in the standard space, I did not look for patterns in the mirror space.
> 
> There are six 6-Way Three Color Diagonal patterns in the "prime orbit", so generators can be given
> 
> ...




I can vouch for the existence of a lot of three-color diagonal patterns in the Prime orbit. But they all have the same basic symmetry with trivial variations. Basically, the tetrahedral symmetry that I found in the Quark pattern version. None of the Prime Orbit patterns though is reducible to the 6-way diagonal pattern. That seems to require the Quark orbit in order to construct it. Once you have it (In 16 color permutations) then you can construct TriColor diagonal patterns, so there are a lot of these. 

I had no trouble finding the two additional snake patterns. They were both in the Mirror Fermion orbit. one had reflection symmetry and one had swirl symmetry. The two on the Normal cube have rotational (Boson) or flip (Fermion) symmetry. I guess you cannot have one in the twirl symmetry.

I recall another interesting symmetry I found once in the Prime orbit. I was exploring 6-way TriColor Stripe patterns and there seem to be quite a few of these. But I found one that I really liked. If you exchamge two adjacent corners and flip the edge in between, then it is like you flipped a whole column of cubies. Call this a "column flip". As long as the number of "column flips is even, then you are still in the Prime orbit. But an odd number puts you in theMonopole Fermion orbit. I am still looking for a decent pattern in that orbit. There is an obvious one but it is a little weak as a pattern, Anyway, you start by flippig two pairs of parallel columns that are skew to each other. Say, The Front to Back left pair and Right pair on the Top face and the Left to Right Front pair and Back pair on the Down face. Then you have to exchange a pair of edges (it will be obvious which these are) and you will have the tricolor stripe pattern. The symmetry is a "cupped' one". be is divided into two sets of three faces. one triplet might be The Top-Right-Down faces and the other the Front-Left-Back faces. The stripe pattern on all three faces of the triplet have the same three colors but permutated in going from one face to the other. The other triplet of faces will also have the same three colors, also permutated but the opposite colors of the first triplet. It makes for a very nice and satisfactory three color stripe pattern. It is the most symmetrical one I know. 

Can you find any kind of 6-way Diagonal patterns that are NOT in the Quark orbit (one-twisted corner)? I cannot!

Cheers,

Math Bear


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## Herbert Kociemba (Feb 21, 2015)

1. There only is one kind of 6-way Diagonal pattern as it is easily shown. Indeed it is in an orbit with corner twist <>0.

2. Concerning the snake pattern you guess that I "cannot have one in the twirl symmetry". You seem to have misunderstood me.
I claim that the four snake patterns I gave above with the symmetries S6,C2h(b),D3d and S6 (I omit the antisymmetry part of the classification here) give a complete classification and that any other snake pattern can be transformed to one of these by conjugation with one of the 48 cube symmetries.

It is a pity that you are resistant to all the knowledge about symmetries and some basic group theoretical concepts which would give you a clear understanding when to consider for example two patterns as basically the same. It also would prevent you from reinventing the wheel and thinking that you found something completely new with your "twirl symmetry". I suppose that you mean the symmetry type S6 or S4, but since in all your posts you did not give a single concrete example for a cube with "twirl symmetry" I just have to guess.

3. You write that all three-color diagonal patterns in the Prime orbit have the same basic symmetry. No, there are two completely different types, types C2 and S4 as I showed above. But again, if you are not willing to deal with some basic concepts we will continue talking at cross purposes.


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## Math Bear (Feb 22, 2015)

Herbert Kociemba said:


> 1. There only is one kind of 6-way Diagonal pattern as it is easily shown. Indeed it is in an orbit with corner twist <>0.
> 
> 2. Concerning the snake pattern you guess that I "cannot have one in the twirl symmetry". You seem to have misunderstood me.
> I claim that the four snake patterns I gave above with the symmetries S6,C2h(b),D3d and S6 (I omit the antisymmetry part of the classification here) give a complete classification and that any other snake pattern can be transformed to one of these by conjugation with one of the 48 cube symmetries.
> ...



For the record, what I meant by "fundamentally the same basic symmetry, I meant that you have 4 tetrahedrally arranged corners connected by diagonals. The other 4 corners, also diagonally arranged are each surrounded by their neighboring edge cubies, arranged so the color faces of the corner and edges match up. This is what I meant by the "same basic symmetry". I recognise that you can exchange the corners and their attendent edge cubies in various creative ways around the 4 points of the tetrahedron but I see no way you can depart from this basic pattern in creating any 6-way TriColor Diagonal pattern. Have you actually found a pattern that departs from this double tetrahedral arrangement? If so, pleease describe it in plain language we all can understand. I understand, for example, C2, C3, C4 as refering to a group of rotations about a center with 2,3, or 4 fold symmetry. The flip symmetry is C2, the rotation symmetry is C3 and the center turn symmetry is C4. I understand "S4" as referring to the group of the 24 possible permutations of four objects in a row. I cannot visualise what you are talking about. Please stop using jargon. 

I told you right from the very beginning that I was developing a GEOMETRIC language to describe the Rubik's cube patterns. You insist on using a jargon based on abstract algebra that is meaningless to the great majority of viewers on here. Group theory by itself does not constitute complete genuine geometric insight. You need a visual language for that. You so consistently use algebraic explanations that at times I feel like I am discussing the patterns with a blind man. Maybe you have forgotten how to see the cube?

Try not to be hostile. It is uncalled for in a discussion of this sort.
I was just agreeing with you in the comment on their being no snake pattern exhibiting the "twirl" symmetry, which as I recall I explained very carefully previously. It is simply the Mirror transform of the 1/4 turn 4-way pattern. The 4 horizontal sides show the 1/4 turn through the vertical center axis symmetry combined with a parity exchange of the top and bottom faces. these patterns are typically mirror Fermion orbit patterns except of course for the 6-way Checkerboard pattern which is in fact in the Prime orbit. I don't care what kind of argument you came up with out of group theory, that is quite simply not enough for me. I want to understand it GEOMETRICALLY!
Not that I doubted you for a moment but I tried taking the twirl symmetry 6 Spot pattern and developing a Snake pattern from it. I succeeded in doing so but it had reflection rather than twirl symmetry. It proved impossible to impose twirl symmetry on a Snake pattern and I now have geometric insight into this. The other additional Snake pattern has swirl symmetry which is a Mirror transformed version of the corner rotation symmetry.

To get a twirl version of the 6-way cross, first make the (Prime orbit) 1/4 turn version of the 4-way cross, then exchange the center and edges of the top and bottom faces. It is the parity trannsformation of the 1/4 turn symmetry that expands it into a 6-way symmetry. the twirl 6-way cross is a Mirror Boson, using my method of equating patterns and particles. The swirl snake pattern, on the other hand, is a Mirror Fermion as you have to have a 6 item cyclic exchange. Since the number of items is even, the equivalent number of pair exchanges is odd and therefore fermionic. Swirl patterns wind around all 6 faces of the cube and the swirl snake shows this vividly. 


I had a basic course in Abstract Algebra and Group Theory when I was working on my M.A. in Mathematics but I deliberately choose not to describe the Rubik's cube in this algebraic language. It has been done for many years and by many others. I want to develop a GEOMETRIC language for this, and one accessible to the great majority of the viewers on here.
Please respect my purpose.

Sincerely, 

Math Bear

P.S. The 4 types of Snake patterns are: 1. Rotational This is the familiar one from the Prime orbit. 2. Flip This one is from the fermion orbit. 3, Reflection This is from the Mirror Fermion orbit. Swirl This is also from the Mirror Fermion orbit. This is how I describe things in my geometric language.


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## obelisk477 (Feb 22, 2015)

Math Bear said:


> For the record, what I meant by "fundamentally the same basic symmetry, I meant that you have 4 tetrahedrally arranged corners connected by diagonals. The other 4 corners, also diagonally arranged are each surrounded by their neighboring edge cubies, arranged so the color faces of the corner and edges match up. This is what I meant by the "same basic symmetry". I recognise that you can exchange the corners and their attendent edge cubies in various creative ways around the 4 points of the tetrahedron but I see no way you can depart from this basic pattern in creating any 6-way TriColor Diagonal pattern. Have you actually found a pattern that departs from this double tetrahedral arrangement? If so, pleease describe it in plain language we all can understand. I understand, for example, C2, C3, C4 as refering to a group of rotations about a center with 2,3, or 4 fold symmetry. The flip symmetry is C2, the rotation symmetry is C3 and the center turn symmetry is C4. I understand "S4" as referring to the group of the 24 possible permutations of four objects in a row. I cannot visualise what you are talking about. Please stop using jargon.
> 
> I told you right from the very beginning that I was developing a GEOMETRIC language to describe the Rubik's cube patterns. You insist on using a jargon based on abstract algebra that is meaningless to the great majority of viewers on here. Group theory by itself does not constitute complete genuine geometric insight. You need a visual language for that. You so consistently use algebraic explanations that at times I feel like I am discussing the patterns with a blind man. Maybe you have forgotten how to see the cube?
> 
> ...


Idk about any of this mess, but you should know that your language is as confusing to me as (if not more confusing than) what HK is saying. So stop pretending like you're communicating with the masses


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## Herbert Kociemba (Feb 22, 2015)

Math Bear said:


> For the record, what I meant by "fundamentally the same basic symmetry, I meant that you have 4 tetrahedrally arranged corners connected by diagonals. The other 4 corners, also diagonally arranged are each surrounded by their neighboring edge cubies, arranged so the color faces of the corner and edges match up. This is what I meant by the "same basic symmetry". I recognise that you can exchange the corners and their attendent edge cubies in various creative ways around the 4 points of the tetrahedron but I see no way you can depart from this basic pattern in creating any 6-way TriColor Diagonal pattern.



So we talked again about different things, because you use the word symmetry in quite unusual way here and in a way not consistent with the meaning in other parts of your posts and not at all in a way it is used in the mathematical sense.



Math Bear said:


> The flip symmetry is C2, the rotation symmetry is C3 and the center turn symmetry is C4. I understand "S4" as referring to the group of the 24 possible permutations of four objects in a row. I cannot visualise what you are talking about. Please stop using jargon.



The advantage of using jargon is that there is a precise meaning for each term. Its not only an advantage for a reader to whom you would like to express your ideas, it is also an advantage for yourself because it guides you to clarify and sort your ideas.
Nobody except you uses the terms "flip symmetry", "rotation symmetry" and "center turn symmetry", "twirl symmetry", "swirl symmetry". What you do not seem to notice that it is you who created your own jargon and worse in some cases you use the same terms but in a different meaning than usual. This makes it a communication about your statements almost impossible. 



Math Bear said:


> I told you right from the very beginning that I was developing a GEOMETRIC language to describe the Rubik's cube patterns. You insist on using a jargon based on abstract algebra that is meaningless to the great majority of viewers on here.



There are many visualizations of the symmetries of a cube and the corresponding "point group" which are very geometrical but you seem to close your eyes to this fact. Group theory and geometry are no contradiction, they complement each other. In the description of the symmetries on my homepage I do not use abstract algebra at all, the symmetries are visualized by pictograms which show what C2, S4 etc. mean. Maybe you have the opinion that your "geometric language" is clearer than these pictograms but do you really think that this holds for most other persons interested in the subject?


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## Artic (Feb 22, 2015)

He keeps insisting on using a vague "geometric language" to describe his ideas, but it's more confusing than the standard language already used to describe cube states. Herbert Kociemba's site is very clear to me, and anyone with a little humility and patience would easily understand its content. 

I don't know, but it seems to me he is trying to reinvent the wheel using chopsticks and bubble gum. The end result is something messy and esoteric that I, as well as others, find confusing and incomprehensible.


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## Math Bear (Feb 23, 2015)

obelisk477 said:


> Idk about any of this mess, but you should know that your language is as confusing to me as (if not more confusing than) what HK is saying. So stop pretending like you're communicating with the masses




Harrumph!

That is because you probably have trouble reading more than one paragraph at a time, otherwise you would have gotten some understanding out of what I wrote, especially if you were to pick up a cube and try out some of the ideas. They were carefully phrased but only a genuinely experienced and expert cubist could be expected to understand them, they are very emphatically not for beginners! You are being a troll. If you have nothing constructive to add to the discussion, then stay out of it and stop making a vulgar display of your abysmal ignorance of the thread that is being discussed. 

I am not interested in communiticating anything at all to the masses. The Rubik's cube is beyond the understanding of the great majority of the masses. I am only interested in communicating with those elite few who are capable of understanding the cube and sincerely interested in it. 

Are you one of those?


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## obelisk477 (Feb 23, 2015)

blah said:


> bearly



[emoji106]


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## qqwref (Feb 23, 2015)

Math Bear's posts are looking more and more like Time Cube as this thread goes on.


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## obelisk477 (Feb 23, 2015)

qqwref said:


> Math Bear's posts are looking more and more like Time Cube as this thread goes on.



I just had that thought but I couldn't remember what the website was anymore. I really need to bookmark that page


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## brian724080 (Feb 23, 2015)

Math Bear said:


> That is because you probably have trouble reading more than one paragraph at a time, otherwise you would have gotten some understanding out of what I wrote, especially if you were to pick up a cube and try out some of the ideas. They were carefully phrased but only a genuinely experienced and expert cubist could be expected to understand them, they are very emphatically not for beginners! You are being a troll. If you have nothing constructive to add to the discussion, then stay out of it and stop making a vulgar display of your abysmal ignorance of the thread that is being discussed.
> 
> I am not interested in communiticating anything at all to the masses. The Rubik's cube is beyond the understanding of the great majority of the masses. I am only interested in communicating with those elite few who are capable of understanding the cube and sincerely interested in it.
> 
> Are you one of those?



Not sure if you're joking here. The "elite few who are capable of understanding the cube and are sincerely interested in it" actually have a system of notation and other conventions for the cube that you do not follow in your posts. If you read Kociemba's response, you would have found that nobody uses the terminology that you have created. I'm sure that Kociemba knows a lot about patterns, and his posts make perfect sense to the average cuber because he isn't using some vaguely defined geometric term to describe patterns of the cube.

Also, you should seriously use Cube Explorer instead of finding patterns by hand. Most importantly, don't use the Rubik's speed cubes. There are so many better speed cubes that you can get for a fraction of the money that you spend on the cubes from the official Rubik's website. (Edit: apparently blah mentioned this previously, sorry for repeating)



qqwref said:


> Math Bear's posts are looking more and more like Time Cube as this thread goes on.



That is a very funny website indeed


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## ender9994 (Feb 23, 2015)

I'm starting to think Gaétan Guimond started another account without us knowing....


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## AlphaSheep (Feb 23, 2015)

I've already brought this up, but one big thing that makes Herbert Kociemba's site so easy to understand, but makes this discussion difficult to follow for the layperson is that this discussion has a severe lack of pictures... Visual cube makes showing examples of patterns so much easier.



You don't even need to know the notation or the moves needed to reach a state because you can just define the cube in terms of facelet colours!


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## Sin-H (Feb 23, 2015)

blah said:


> I'm not a physicist, but I'm 99% certain that physicists really don't want you posing as one of them. Of course, I can never be 100% certain because of the Heisenberg Uncertainty Principle. If you're unfamiliar with this principle, I challenge you to read about it on the Wikipedia page as homework! ^,..,^



I'm a theoretical physicist with some background in QFT and I'm mostly like: "What is this I don't even". I am reading lots of buzzwords with lots of vague descriptions and I don't understand what's going on (But maybe if I take a few hours it will work out, I don't know).

However, blah,



blah said:


> No. SU(2) is S^3, i.e., the 3-sphere, i.e., the unit quaternions. The "rotations of a solid object in 3D space" is not an algebra and it's not SU(2); it's a group and it's SO(3). Actually, you can find out all this just by reading the Wikipedia page for SU(n) ^,..,^



you might want to take into account that SU(2) is, topologically, a double cover of SO(3), and that's why theoretical physicists use it as the spin group in 3D (this also appears to be connected to the fact that you need to rotate fermions by 4pi before they look the same again, and not by 2pi). In that sense, yes, SU(2) can be practically taken as 3D rotations. For a physicist, this doesn't sound as wrong as it probably does for you.


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## Math Bear (Feb 24, 2015)

View attachment 4953Blah, you have inserted yourself into a discussion you have not carefully read. You also do not seem to understand the broader and deeper implications of mathematics and are spreading misinformation. Your complaints about the Rubik's cube group being discrete are silly. How can you not know that continuous groups can have discrete subgroups? Where did you see me saying that the Rubik's cube group is equivalent to SU(2)? But what makes you think SU(2) cannot have discrete subgroups? As for the comment about SU(2) versus SU(3), the first can be modeled in a space of 3 dimensions and easy to visualise as the space of the proper rotations of a 3D object. SU(3) requires a space of 8 dimensions to properly model it. Are you honestly going to insult my itelligence by claiming that it is just as easy to work with a space of 8 dimensions as one of 3 dimensions? Are you good at solving Rubik's cubes in 8 dimensions? 

I like playing with 4 dimensions too, but I know how to show it to others and work out problems in Euclidean 4 Dimensial geometry that others can I see and grasp. I wonder if you can do the same? For example, one important fact about 4D geometry is that two perpendicular planes intersect in a single point in 4D space. Can you convincingly demonstrate this to high school students? This is important in the Mandelbrot Set as you can define another fractal on any point in that set that corresponds to the plane perpendicular at that point in the larger 4D space containing the Mandelbrodt Set. . These fractals are called "julia Sets". 

As for S4, please enter "S4 group" into Google and see what comes up. You will see that it is shorthand for "Symmetric Group order 4" and my definition off it given above is accurate. If you guys are using it in a non-standard way, then by definition you are using jargon. You should call it something else if you specifically mean the rotational symetries of a cube, even if it is isomorphic to the conventional group S4. 

Failure to trecognise the geometrical implications of your algebraic descriptions is what I am complaining about. Can you properly describe the group of rotations of a cube and its subgroups in geometric terms? this is what is so sorely needed. the most basic symmetries involving all 6 faces I call "Rotation" and "Flip". if you deal with symmetries involving 4 equitorial faces of the cube, you get "1/4 turn', and "1/2 turn" symmetries. Can you define all of these in your system of notation so that the definitions are geometrically relevant? These are not enough for the rubik's cube though. Patterns involving only edge cubies can violate parity restrictions on the rotational symmetries of the cube. Therefore, I need to add Mirror symmetries to my basic list of cube symmetries. "Reflection" is accomplished by inverting all three axes of the cube. This symmetry is shown by the classic Checkerboard pattern, you get by doing a half slice on all three middle layers of the cube. The other two Mirror symmetries are parity transforms of the Rotation and 1/4 Turn symmetries respectively. They are called "Swirl" and "Twirl". The last symmetry is 4-way and is made by applying the mirror transformation to the 6-way Flip symmetry. I call this new 4-way symmetry "Adjacent" as it creates two pairs of adjacent faces. thus we have 5 types of 6-way symmetries, 2 of which are Normal and 3 of which are Mirror. There are 3 groups of 4-way symmetries, two are Normal and one is Mirror. All 8 types of symmetries occur with the edge only patterns (such as Checkerboard) on a Normal cube. If a pattern also involves center cubies, then the Mirror patterns are only accessible on a cube that you have turned into a Mirror cube by exchanging two oppposite centers. A secondary problem is whether a pattern involves an even number or a an odd number of pair exchanges. Thus I need the Fermion orbit as well as the Boson Prime order. Most basic patterns occur in Boson/Fermion pairs. This distinction is important with the Mirror patterns too, thus I have to reckon with Mirror Boson and Mirror Fermion orbit patterns. Occasionally other orbits are involved too.

My challenge to you is that I do not believe that your purely abstract algebra approach to describing the symmetries of the cube is adequate. This is not a matter of mathematical theory, but one of linguistic description (and I have a M.A. in Linguistics too). You need to create a customized vocabulary for the Rubik's cube that expresses geometrical relationships as well as algebraic ones. And it should be straight-forward enough to be understandable to any reasonably intelligent cubist who puts forth the effort to master it. If you come up with a decent descriptive language, I will gladly abandon my idiosyncratic system for it but it must describe the geometry of cube patterns!

sincerely,

Math Bear


P.S. Blah, just in case you didnt know, SU(2) stands for "Special Unitary Group order 2". The corresponding Lie algebra is denoted "su(2)". Can you see the difference? I was using the term "algebra informally as a synonym for "group". Any group describes a basic but complete mathematical system of some sort which is what I had in mind. As I have stated before I am not trying to talk to the erudite but to ordinary, non-Sheldon-Cooper type mortals. Most assuredly, I was not thinking of Lie algebras. I am uncomfortable with their lack of asspciativity. It is easy to dispense with commutivity but I think non-associative systems are always at least a bit weird.


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## qqwref (Feb 24, 2015)

Math Bear said:


> and I have a M.A. in Linguistics too


Why do I find this hard to believe?


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## Sin-H (Feb 24, 2015)

Math Bear said:


> P.S. Blah, just in case you didnt know, SU(2) stands for "Special Unitary Group order 2". The corresponding Lie algebra is denoted "su(2)". Can you see the difference?


Mathematicians do know the difference, it's some physicists who never bother to distinguish the Lie Groups from their Lie Algebras.



Math Bear said:


> I was using the term "algebra informally as a synonym for "group". Any group describes a basic but complete mathematical system of some sort which is what I had in mind. As I have stated before I am not trying to talk to the erudite but to ordinary, non-Sheldon-Cooper type mortals.


Don't blame others for not understanding you when you randomly exchange terms which have an accepted distinct meaning. And don't exchange these terms for "mortals" either. I don't think the chance is higher that they know what an algebra is than that they know what a group is, so why not use the correct word in the first place?
Also, what is a "complete" mathematical system? Do you mean completeness in the Banach space sense? In the Gödel sense? In the measure sense? ...



Math Bear said:


> Most assuredly, I was not thinking of Lie algebras. I am uncomfortable with their lack of asspciativity. It is easy to dispense with commutivity but I think non-associative systems are always at least a bit weird.


Why the hell would anyone expect the Lie bracket to be associative? That is not its job. It even looks weird: [[x,y],z] = - [z, [x,y]] can only be equal to [x,[y,z]] when [y, [z,x]] = - [[z,x],y] is zero, and then all the terms are zero. Else the Jacobi identity is violated. And the Jacobi identity is the really important part of Lie algebras.



blah said:


> Math Bear said:
> 
> 
> > But what makes you think SU(2) cannot have discrete subgroups?
> ...


Wait... it has discrete subgroups, hasn't it? Some of the point groups are discrete subgroups of SU(2) iirc.



blah said:


> Math Bear said:
> 
> 
> > If you want to get technical the Group SU(2) underlies both the Rubik's cube and most particle physics.
> ...


loltechnical. A quick description of what is meant here: Our standard model relies heavily on so-called Yang-Mills theories which are quantum field theories with a gauge group (describing the symmetries of the theory) of the form SU(N). That is what "underlies" means - our theories have an underlying symmetry which is described by the special unitary group. For instance, Quantum Electrodynamics (QED) is based on U(1) (and therefore "easy", because it is abelian), and Quantum Chromodynamics (QCD) which describes the strong interaction is based on SU(3). That is why we have one photon (The Lie algebra u(1) has one generator) and eight gluons (the Lie algebra su(3) has eight generators). However, I don't know of any really applicable SU(2) theory. The electroweak part of the standard model has gauge group U(1) x SU(2) which then gives 4 gauge bosons (B, W^0, W^1, W^2) and you obtain the photon and the W+-/Z^0 bosons by doing a change of basis. I think that SU(2) cannot be that easily separated.



blah said:


> Mathematicians tend to work with spaces of n dimensions, so it's just as easy to work with spaces of 8000 dimensions. But that's too abstract for you.


<3


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## Herbert Kociemba (Feb 24, 2015)

Math Bear said:


> As for S4, please enter "S4 group" into Google and see what comes up.


As for S4, please enter "S4 point group" into Google and see what comes up.



Math Bear said:


> Can you properly describe the group of rotations of a cube and its subgroups in geometric terms?


Are flip, swirl and twirl geometric terms?



Math Bear said:


> the most basic symmetries involving all 6 faces I call "Rotation" and "Flip". if you deal with symmetries involving 4 equitorial faces of the cube, you get "1/4 turn', and "1/2 turn" symmetries. Can you define all of these in your system of notation so that the definitions are geometrically relevant?


Why do you redefine the term "Rotation" which has a well defined meaning im mathematics and physics? I suppose you mean the 120 degree rotation around the long diagonal of the cube.
I suppose with "Flip" you mean the 180 degree rotation around an edge. I suppose with "1/4 turn" you mean a 90 degree rotation around a face and with "1/2 turn" a 180 degree rotation around a face.
You think "Rotation", "1/4 turn", "1/2 turn" and "flip" is better than C3, C4, C2(b) and C2(a) ? I must admit I do not like the names for the two different C2-types but else I see no advantage in introducing new names for the Schoenflies-symbols of point groups. Ok, you have a M.A. in linguistics but still I am not convinced about the "geometrical relevance" of your creations (to be honest, I do not know what you mean with "geometrically relevant")



Math Bear said:


> "Reflection" is accomplished by inverting all three axes of the cube. This symmetry is shown by the classic Checkerboard pattern, you get by doing a half slice on all three middle layers of the cube.


I suppose you mean the point Group Ci here, a reflection through the center of the cube. The classic Checkerboard pattern is a bad example for this symmetry, because this pattern has all symmetries the cube has.



Math Bear said:


> The other two Mirror symmetries are parity transforms of the Rotation and 1/4 Turn symmetries respectively. They are called "Swirl" and "Twirl".


These correspond to the point groups S6 and S4. Absolutely no need for new names.



Math Bear said:


> The last symmetry is 4-way and is made by applying the mirror transformation to the 6-way Flip symmetry. I call this new 4-way symmetry "Adjacent" as it creates two pairs of adjacent faces.


This is nothing else as a reflection through a plane which contains a face diagonal. It is called Cs(b), If the reflection plane is parallel to two faces it is called Cs(a). I miss this "2-way symmetry" in your classification of patttern symmetries.

For the description of the symmetries of patterns I also miss in your description the cases where a pattern has more than one of the basic cube symmetries . What symmetry type has the classic Checkerboard pattern for example in your jargon?


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## Stefan (Feb 24, 2015)

Herbert Kociemba said:


> I suppose you mean the point Group Ci here, a reflection through the center of the cube. The classic Checkerboard pattern is a bad example for this symmetry, because this pattern has all symmetries the cube has.



Maybe he means something like if you consider *just the edges*, then M2 E2 S2 shows that reflection.


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## Math Bear (Feb 24, 2015)

*General Comments and Introduction*

Hee,hee! Boy, somebody sure has sensitive toes! ^,..,^

There is no profit in endless ruminations full of pettifogging and nit-picking.

If you don't like my ideas about the Rubik's cube, that is your problem.

I am ending what is obviously turning into a pointless flamewar.

What unites us is our interest in the Rubik's cube. I am going to make further
contributions using my distinctive approach, too bad if you don't like it.

It occurs to me that I should add visuals to my discussion. I will work on it.

To all the budding young Sheldon Coopers on here, hey! lighten up!

The first goal is to have fun and enhance our understanding of the cube,

Peace and Good Vibes,

Math Bear

P.S. My educational record is I started out as a chem major, with an emphasis on Organic Chemistry.
I was very good at the book work but not the lab work, so I decided to go into something else.
I studied accounting for a year at a business college and worked at a bank for several years.
I got bored stiff and went back to the university. I majored in Asian Studies with Chinese Emphasis
and eventually got a B.A. I later entered graduate school and started work on a degree in Linguistics.
after I got my M.A. in linguistics, I taught 7 years as an ESL instructor, in the U.S. , Mexico, taiwan and Saudi Arabia.
I ended up teaching ESL in a special program on a military base for military students from Kuwait and the U.E.A. 
the program i was in abruptly ended when the commander of the base discovered that the directors of the program
were allowing rampant cheating on the test given to test English proficiency at the end of the course. i decided to get
into alternative certification and teach high school. According to my credits, I could teach English, ESL or Mathematics.
Although I had never majored in math, I was always deeply interested in it and I had 20 credit hours past Calculus,
roughly the equivalent of a B.A. in math. I was enrolled in a 20 hour course at the graduate school in my loca university to become a certified
math teacher and was hired by a high school as a math teacher and got my certification in secondary mathematics. 
I taught math for a number of years and then was accepted into a program to enable high school teachers to get a
Master's degree in math sponsored by the National science Foundation. They paid for 80% of all expenses and arranged 
for the classes to be held in the evening or on weekends and during summer school so our classes didn't conflict with our
work schedules, a major consideration for working teachers. After 3 years I got my M.A. i originally wanted to go on to a phD program
but they are currently doing horrific things to post grads and most people who have PhD and I was definitely too old to have any hope of
ever getting tenure or a research oriented position. Besides the school I wanted to go to would not let you in their program unless you dedicated yourself body and soul to Algebraic Topology, a subject I had no interest in so I gave it up. I have a strong interest in math education on the internet.
I have actually set up a school online in the virtual world of Second Life. This is located in Dalton1 next to Linden Village on the old continent.
If any of you are on Second life (does the thought scare you? ^,..,^ ) You can find my school by putting "The Math Bear Education Intiative" into
Search and teleporting there. I am MathBear Cyberschreiber on Second life and just leave me a message. If you like, I will meet you in-world
at my school or maybe at a sim associated with your university if it has a presence on Second Life. 

P.P.S. most high school math teachers of my acquantance do not really understand the mathematics they are teaching. They just present 
the material blindly. The students readily pick up on this and this is partly why they are so cynical about math courses. Math teachers with higher degrees ARE knowledgeable bout math and some are pretty sharp. Their students are a lot better motivated.


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## obelisk477 (Feb 24, 2015)

You are the definition of pedantic


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## Herbert Kociemba (Feb 24, 2015)

Math Bear said:


> If you don't like my ideas about the Rubik's cube, that is your problem.
> 
> I am ending what is obviously turning into a pointless flamewar.



If this is the only response I really regret to have spent hours of time trying to understand what you mean and translating it to a language which is used by a broader community to classify symmetries. And my question about the missing of the symmetry Cs(a) and multiple symmetries in your classification should not be interpreted by you as a flamewar but as a constructive criticism.

So I wish you good luck and more success than in this forum for your further research in the ivory tower.


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## Artic (Feb 24, 2015)

Herbert Kociemba said:


> If this is the only response I really regret to have spent hours of time trying to understand what you mean and translating it to a language which is used by a broader community to classify symmetries. And my question about the missing of the symmetry Cs(a) and multiple symmetries in your classification should not be interpreted by you as a flamewar but as a constructive criticism.
> 
> So I wish you good luck and more success than in this forum for your further research in the _*ivory tower*_.



LOL LOL. And it's sad that the OP ruined what could have been an interesting discussion through his arrogance and dismissal of previous work and research on cube states. He needed to show pictures right from the beginning so we understood what he was talking about, instead of meandering hazy mathematical language that introduced confusion into the whole discussion.


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## guysensei1 (Feb 25, 2015)

This is definitely going up on the Forum Awards 2015.


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## Math Bear (Feb 26, 2015)

*Not your Fault, Herbert*



Herbert Kociemba said:


> If this is the only response I really regret to have spent hours of time trying to understand what you mean and translating it to a language which is used by a broader community to classify symmetries. And my question about the missing of the symmetry Cs(a) and multiple symmetries in your classification should not be interpreted by you as a flamewar but as a constructive criticism.
> 
> So I wish you good luck and more success than in this forum for your further research in the ivory tower.




Anything involving the Rubik's Cube theory is necessarily going to be rather "ivory tower", though I do see possible EDUCATIONAL applications. My Dear Herbert, you are not responsible for the so called "flame war" and your contributions have been immensely valuable to me and I would welcome any more that you may wish to contribute. So far, you seem to be the only one on here who has any understanding of what I am doing and you have helped lead me to new ideas. Please don't be so sensitive. The development of correlations between the geometric understanding of the cube and the algebraic understanding is essential and needed to fully understand the cube in its full depth. but watch your language. I have no idea what you mean by "symmetry Cs(a)" unless you describe it to me so I can SEE it. This goes too for an awful lot of other people who are reading this thread. The use of such opaque notation makes it impossible to visualise the symmetry it is describing. This board is intended for non-academics so my criticisn of such cryptic descriptive language is valid. I will leave the ALGEBRAIC description of Rubik's cube patterns to you since you obviously have considerable expertise in this. What I am doing is different. I am developing a gEOMETRIC descriptive language for describing the patterns of the cube. I clearly need to try out those programs for creating image files of the cube though so far I havent been successful in figuring out how to embed images into my messages. I realise now it is not enough to merely discuss the geometry of Rubik's cube patterns, I have to show actual pictures of what I am talking about. I understand it is rather perverse to merely talk about geometry without showing actual iimages.

But let me give you an example of what I am talking about when I discuss the importance of the correlation benween geometric and lagebraic understanding of a subject. Think about Freshman Calculus. The subject of the calculus can be developed purely algebraically, using the idea of linear operators on functions and ideas of limits and convergence from Real Analysis, but this purely algebraic approach is not used to teach calculus on a basic level. instead, extensive use of gemetric ideas such as tangent and slope and area under a curve are used to provide vivid motivations for the basic ideas of calculus. You could hardly successfully teach the subject otherwise. 

Likewise, if a subject that is as highly geometric as Rubik's cube patterns is to be presented to non-mathematicians, then it should not be presented purely algebraically.

I will rework the pesentation of my ideas with images this time to clarify what I am doing. I am hopeful that if I do that, it will inspire the cubist community at large to explore these patterns and the underlying ideas. I would be delighted if you would then explain the resulting symmetries in the language of group theory and show how they correlate with the visible geometry. I am also hopeful that the analogy between the symmetries of the cube and those of the subatomc particles in contemorary theories of physics will inspire the cubists to take an interest in physics. I suppose that it is maybe a bit too much to hope for to expect that maybe some of the people in particle physics might take an interest in the cube but I will keep trying.

The hard theoretical work that eventually needs to be done is to explain these analogies that occur between the realtionships bween pattern symmetries on the cube and the symmetries of subatomic particles. This is well beyond my expertise. It will require a particle physicist with deep knowledge of the Standard Theory and its extensions such as Supersymmetry, Grand unified Theory and Mirror Matter. I believe that the analogies are related somehow to SU(2) and its equivalets and extensions. It would be great if a technical mathematical explanation in the language of abstract algebra could be developed. Ift hese correlations are non-trivial, then study of the Rubik's cube could be a significant part of the education of students of particle physicts. At least I like the idea! ^,..,^

I won't name names my good Herbert but you were definitely NOT one of those I was thinking about when I was complaining about there being a flame war. You have shown deep insight and made worthwhile contributions. I cringe at the thought of alienatng you!

But I do have to understand you and so shoud others. When you present your ideas using the language of group theory, please explain what this means in visual, geometric terms also. This will help myself and others to learn your system.

Sincerely,

Math Bear


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## deadmanlsh (Feb 26, 2015)

What you are trying to do now is to bridge two disciplines together - that of mathematics and physics or Rubik's Cubes and Particle Theory in particular.

However, what you must understand is that a *rudimentary* level of understanding has to be reached *BEFORE* you embark on such an endeavour. That includes the learning of common *conventions* and other possible information that is *NECESSARY* in communicating your ideas in a *clear, precise and understandable* manner. For example, you used your own turning notation. While this is definitely not forbidden nor excessively obfuscating, it would have been easier to *reach your target audience* by using the *conventional notations* to ease learning.

It also certainly helps to use proper paragraphing. Chunk related pieces of information together, but have a *distinct separation* between different ideas. Your posts are walls of text that are hard to read and even harder to comprehend. Moreover, you are the one presenting the information to the users of this site. The onus is upon you to *organise* and delineate your intend messages properly to your target audience.

Another problem that is extremely evident in your writing is the absence of clarity and precision when it comes to *definitions*. This is intricately related to the previous problem, as having walls of text with definitions that were already vague enough buried within mounds of unnecessary fluff is extremely detrimental to the readability of your posts. Your definitions of "twirl", "symmetry", "spin" and other self-produced jargon is forgettable at best and incomprehensible at worst. Define them somewhere. Separate them clearly. Provide examples. Be concise and specific. This is in contrast to Herbert Kociemba's website, where the different types of symmetry are lined out in *organised* and easily digestible *tables*. You often criticise the website for being too "ALGEBRAIC" and "opaque", whereas yours is "gEOMETRIC (sic)" and "descriptive". If you even bothered to check out prior work done by Herbert Kociemba, you'll realise that his is infinitely more geometrical and intuitive, with *pictures and animated cubes* to show you the different types of symmetries. Compare this to your posts, which only give solid written instructions to produce on the physical cube. I wonder which is more abstract when it comes to the actual usage of illustration? His pictures even label points, axes and lines to illustrate how the symmetry is produced. This all ties in to the first point I made as well, which is to *familiarise yourself with prior work done*.

Finally, if you are indeed an experienced physicist, do remember to define technical jargon used in physics. You claim to want to be easy to understand, yet expect readers to know the terminologies used in physics without even providing a definition for *key terms*. This can be exemplified no more strongly than your use of "SU(2)", "Grand Unified Theory" and "Supersymmetry". I understand that you did refer the readers to other websites to get an idea of what the terms describe, but it would have been best if you at the very least provided *sources of information* you personally selected in an *accessible* place. Also, do note the possible hypocrisy you are committing; you expect others to read up on what you are familiar with but refuse to look through the conventions and research done by the community.

Of course, that does not mean all the vitriol being flung around by the community was any more allowable or preferable, but it was ultimately a result of *avoidable* misunderstandings. This is not helped by the fact that you retaliated in a snobbish manner against them, when in fact both parties probably know a whole lot more than the other party in their respective disciplines. It was especially disappointing to see you calling others "Sheldon Coopers" and other insults of attitude and intelligence. I would say, if anything, that your response was reflective of a foolish snob that is not only all too snobbish, but also all too ignorant, while at the same time expecting your badly written posts to be easily understood by the community (reminds me of a certain name I just mentioned, I wonder what it was?).

All in all, I know you are interested in drawing parallels and analogies between the different fields. However, I have to comment that your methods were rather ineffective. Please familiarise yourself with Rubik's Cube a little more; please present your ideas in a more comprehensible and clear manner; please define your terms; please know your target audience and please do not think that you are a holder of arcane knowledge.

Finally, good luck with continuing on with this project. I hope to see quality content from you soon.


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## Herbert Kociemba (Feb 26, 2015)

Mathbear, it would greatly help if you have a closer look at this page where all the basic symmetries of a cube are classified and represented by pictograms and also at this page where all the possible symmetry types of a cube are explained. It seems that you have not done this yet because you find there for example also the "Cs(a)" symmetry subgroup and you say that you do not know what this means. I really prefer this instead of what you say is the "mirror transformation of the 1/2-turn symmetry" or maybe the "parity transform of the 1/2-turn symmetry". It did not find this symmetry in your classification.
The other point I think you miss is the the classification of patterns with several symmetries. If for example a pattern has C3 (your "Rotation") *and* C4 (your "1/4-turn") symmetry, the patterns has O-symmetry. I have have the impression you only use the "basic" symmetry types C2(a), C2(b),[Cs(a)] , Cs(b), Ci, C3, C4, S4 and S6. But please do not tell me you do not know what these symbols mean


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## Math Bear (Mar 3, 2015)

Deadmanish, your advice is helpful but a bit excessive, I prefer to develop my own style rather than follow the crowd.

Herbert, thank you for your consideration. I did check out your page. I have been involved off and on in the Rubik's cube since the original Scientific american article came out in the early 80's. It inspired me to buy a cube and it took me about 2 weeks to figure out how to solve it. I have seen that notation many times before but I prefer not to use it. I am a math teacher with many years experience and I have a very good idea of what alienates the average person with regard to mathematics and makes them give up. I am sure that sort of notation is a great convenience for those with a serious mathematical background but it is as cryptic as Elamite to even well-educated non-mathematical people. I agree that my over-reliance on verbal description is not very helpful either. I want to show examples of cubes with brief but non-technical simple geometric language to describe what I want people to see. The use of informal terms like twirl and swirl make much better mnemonics than for the average educated person than technical matheamtical notation, as long as people can clearly perceive the implied symmettry. I prefer to give directions for building patterns by telling them what basic symmetry to start from then what additional patterns and symmetry transformations to add and let them work out the final result for themselves. My scheme is designed to facilitate this as i feel it gives cubists better insight into the cube.

I might add that I am a devoted disciple of Dr. David Hestenes and his Geometric Algebra (think "real valued Clifford Algebra in a much more user-friendly format). I never truly understood Linear algebra before I studied his ideas and i have found out this is true for many others, especially those in Computer Science. He has developed a common language for theoretical physics and its associated mathematics, Dr. Hestenes innsists of geometric interprestation as essential to understanding mathematical physics and I feel it applies in many other areas as well. It is especially good for intuitive understanding. Anyway, his geometric approach inspires mine. Although you could say his language reformulates Clifford Algebra, it greatly expands its range and power and thus in a practical sense really is something new.

I think I over-emphasized SU(2). I was just thinking of it as a synonym of SO(3) and this led to confusion. In fact SO(3) is not completely adequate either as I make use of parity transfornmations (think "improper reflections") and so I should probably use O(3) as the basis for analyzing the Rubik's cube.

Rotations are interesting things. Perhaps i should say a few words about them. They are basically a 2 dimensional phenomenon and if you deal with higher dimensions you must identify them in terms of their plane of rotation which more or less describes the "direction" of rotation. Only in 3 dimensions can you have a linear "axis" of rotation that is basically defined in terms of a line perpendicular to the plane of rotation. The use of the idea of "plane of rotation" is more generally useful. It is also a good idea to think of "rotation" as something that is static or active. When static, it is a representation of an angle. If the idea of rotary motion is implied then it is active and has a definite direction, clockwise or counterclockwise. This approach also makes it easier to deal with rotations greater than 2Pi radians and is esssential in situations involving angular velocity. 

I know of 5 distinct ways of representing rotations. The most basic is turns, and the easiest to visualise. One turn = 360 degrees. Then there is of course the traditional degrees which is especially good for representing angles. A lot less well known is the gradiant system developed by the French. They are apparently the only ones using it, especially in the military. This system divides the circle into 400 grads rather than 360 degrees. 100 grads is a right angle. It has certain practical convences. You can easily find the quadrant for example, or an orthogonal angle by adding or subtracting 100. 

The next system is of course radians, familair to all. Because they incorporate the element of Pi in the angle measure, they simplify the form of trig functions and are especially useful for higher mathematics. They also allow for the trig functions to be defined as "trancedental" whereas this is not really true of the preceding systems since the value of a trig function of an angle that is a rational part of a full circle is necessarily algebraic (think (Cyclotomic Polynomials). 

The last system is not used practically but is sometimes used for theoretical explanations. This is "Arials". These are constructed like radians but instead of calculating the arc of circumference of the unit circle the angle subtends, you instead calculate the area of the sector of the circle defined by the same arc. If you imagine a unit vector sweeping around the unit circle for one full turn, it sweeps out an area of Pi arials (=360 degrees). This is of course the same as 2Pi radians so you might consider the difference between the two trivial. The difference is conceptual. Planar area turns out to be the most natural and simple way of expressing angle measure in higher dimensions. To give an example of why arials can be conceptually useful, consider that mysterious idea that so many students have trouble accepting, namely that in the particle physics concept of spin, you need to make two full turns to get back to your starting state. With other methods of measuring rotation this looks unfathomable but not with arials. For example, do you know how long a given date, say April 1, 2015 lasts upon the Earth? The answer is 48 hours, It takes 24 hours, starting from the International Date Line for the date to sweep over the Earth and another 24 hours to be swept off of it. Likewise, if you use the arial representation of rotation , it takes one full turn to sweep out the area of the unit circle and a second full turn to erase it completely again and get back to your intial state. This example might help students to visualise the concept and accept it.

Cheers!

Math Bear


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## AlphaSheep (Mar 3, 2015)

Math Bear said:


> I am sure that sort of notation is a great convenience for those with a serious mathematical background but it is as cryptic as Elamite to even well-educated non-mathematical people.



Please read this for notation to describe turns. I've met a couple of 7 year olds who have no problem understanding this notation. I mean, all it really takes is to know the words Left, Right, Up, Down, Front and Back, and then to know which directions are clockwise and anticlockwise. Nothing cryptic or mathematical about it at all.


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## Herbert Kociemba (Mar 3, 2015)

Hi Math Bear,

you still are interested in discussion about the symmetries of the cube? Your observations about the measurement of angles seem a bit off topic here.

Maybe you do not use the Cs(a) symmetry because it maps 4 faces to itself and exchanges only two faces. 
But is there any particular reason for this which is related to your particle stuff?
Can you *exactly *define the patterns which you permit (symmetries, number of colors on each face etc.) *at one place here *and also give the reason for these decisions in connection with particle physics? You really should be able to do this, in the moment the restrictions you give are scattered across many pages of your text and seem quite arbitrary on first sight.


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## qqwref (Mar 4, 2015)

Herbert Kociemba said:


> Your observations about the measurement of angles seem a bit off topic here.


I think considering the different units as equivalent up to a constant multiplicative factor is too abstract for him.


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## Math Bear (Mar 30, 2015)

*Grrreetings!*



qqwref said:


> I think considering the different units as equivalent up to a constant multiplicative factor is too abstract for him.



I have reasons for avoiding cryptic notation and abstract mathematics. I want to make the Rubik's Cube friendly to people who
are NOT math geeks! Let me give an example: As is famously well known among the math community, the set of real numbers under addition is isomorphic to the set of positive reals under multiplication using exponentials/logarithms as the mapping operations. But would you honestly tell lay people that multiplication is "essentially the same" as addition? You would just confuse the hell out of them as the practical difference between multuplication and addition is very large. What is true under theoretical mathematics is often not very helpful when dealing with practical situations. That is my general attitude and philosophy with respect to the Rubik's Cube. I am planning on putting up my basic non-theoretically expressed concepts with illustrations soon. I am still unable to embed images in my messages. I kind of wonder what is the big secret? I have been studying Geometric Algebra fairly intensely recently which is why I have not been on here. 
I hope I have inspired at least some of you to explore Rubik's Cube patterns and their symmetries, or at least impressed on at least some of you that the topic is not trivial but worthy of consideration.

Cheers and Good Vibes to you all!

Mathe Bear ^,..,^


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## unsolved (Apr 2, 2015)

Math Bear said:


> But would you honestly tell lay people that multiplication is "essentially the same" as addition?



Multiplication is repeat addition.
Division is repeat subtraction.


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