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......
^,..,^
