To all the offended math purists, I will hold back and let you cool off. I did not want you guys to dominate the discussion and scare everybody else off.
I am aware of lists of symmetries but they overwhelmingly deal with the regular (Bosonic) orbit of the cube. I largely deal with the other orbits so it is not that useful to my purposes. The realtionships between particle physics and the Rubik's cube are not mathematically exact! Technically speaking, don;t expect the relationships to be isomorphic or homomorphic. They are instead analogous but quite striking for all of that. This lack of mathematical exactness makes group theory less useful than it might be otherwise. I will develop my own informal language for describing the symmetries, but it will be strongly geometric rather than algebraic. Mathematics has two foundations, geometry and algebra. They reflect the two most highly developed aptitudes of the human mind, vision and language. To reallly understand a primary mathematical system, you need to use both modes of understanding. Western mathematics developed its enormous power because it found a way to link the geometric and the algebraic modes through the use of the Cartesian coordinate system. " Mathematics without geometry is blind, mathematics without algebra is dumb"(not my words, anybody know whose?). I suppose if dolphins were ever to develop a system of mathematics they would probably base it on harmonics, to match the unique power of their hearing. Anyway, modern mathematics has become largely incomprehensible to the general public in part because there is a huge emphasis on the algebraic (linguistic) modes of description nad an often extreme avoidance of the geometric (visual) modes of description. The Rubik's cube is fundamentally a geometric object and you need to emphasize understanding it geometrically first. Thus, while I am establishing the basics of my ideas I intend to avoid Group Theory and Abstract Algebra and maybe save them for later, after the whole thing has been properly presented. That will take a while....)
Now, if you dissasemble the cube and reassemble it randomly, you will only have a 1 in 12 chance of ending up with a cube that you can solve all the way to the "start" configuration, This is because you are not completely free to rearrange the cubies but must obey certain restrictions. If you exchange pairs, you can only exchange an even number at a time. If you flip edges, you can only flip an even number at a time. If you twist corners, the total amount of twist can only add up to an integer. You violate these restrictions and mix and match the results you get the 12 orbits. If none of these restrictions is violated, you end up with the Prime orbit which is the type of cube you have in the Start configuration. I call this the Bosonic orbit because an even number of pair exchnages is analogous to a particle with integer spin, and thus can be regarded as a "boson". Each pair exchnage corresponds to 1/2 spin. The next orbit has one pair exchanged, and the total number of pair exchanges of any pattern in this orbit must be an odd number. Any pattern in this group corresponds to a Fermion and has half-integer total spin. The introduction to the Wiki article on Supersymmetry is easy enough for anybody to understand the gist and some later parts are technical enough to appeal to the math geeks. It would help if you were to play with patterns and in particular with correlating patterns in the Prime orbit with those in the Fermionic orbit. Try using two cubes for this, one in the Prime orbit and one in the Fermionic orbit. the next orbit is one flipped edge or an odd number of flipped edges.After playing around with this for a long time, I decided it is analogous to magnetic poles. All particles have both a "north" and "south" pole or pairs of such, No particle has only a "north" pole or only a "south" pole. Such a particle would be called a "magnetic monopole" and some theories (G.U.T.) suggest they may exist, though none have been found. I only have one pattern for this, a version of the 6way-C (or 6-way U) pattern that appears like any of the others but if you examine the color symmetry, it has a bizzare, twisted lop-sidded symmetry that you won't find on any standard list of Rubik's cube symmetries. it is possible to create a new orbit by combining the previous two into an orbit of cube patterns with an odd number of pair exchnages and an odd number of flipped edges. I only know one such pattern and it may be regarded as a "Fermionic Monopole" to draw an anology with particles. Please read the intro to the Wikipedia article on "magnetic monopole" for a better picture.
The next orbit involves the corner cubies. (note: this correspondence goes all the way back to the 1st Scientific American article on the Rubik's cube) If you count a clockwise twist of a corner as +1/3 and a counterclockwise twist as -1/3, then the total twist must add up to an integer, never a fraction. This is a close analogy to the restrictions on how quarks may be combined to form a particle. 1/3 or 2/3 electric charges are associated with each quark, but quarks must combine in a way that the total electric charge of the particle is an integer. The mathematics used to describe the strong force that governs quark interactions is different from that used for Sypersymmetry. I only know two patterns in this orbit but they are very striking. They are both bosons and involve rotational symmetry through the corners.) Some have theorized the existence of particles that would have an extra quark and thus fractional electric charge. Patterns based on a cube in the single twisted corner orbit (actually 2 orbits, depending on whether the twist is clockwise or counterclockwise) correspond to these particles. I call this orbit the "quark" orbit. I haven't found it necessary to distinguish between the clockwise and counterclockwise versions. The other orbits are the possible combinations of the orbitals involving flip, pair exchange and corner twist.
So far, I am working with the Prime, Fermionic, Monopole orbits (both Boson and Fermion), and the two trivially different Quark orbits. This is only half of the 12 possible orbits and I would be very interested if anybody finds patterns corresponding to any of the othe others. There is also a whole realm of 12 orbits belonging to the Mirror cube and I have some patterns I have discovered in the Mirror Boson and Mirror Fermion orbits. I will save the discussion on these until I get into the weighty subject of parity.
Now, In describing cube moves, I use the F,B,R,L,T,D system and a cubies position is defined in terms of the corners, not edges. It makes it a lot easier to understand the patterns if you assume the corners are fixed but the centers can move. To describe moves, I just use the letter corresponding to the face you want to turn and add + (clockwise), - (counterclockwise) and 2 (180 degree turn). The face letter should be capitalized. It can be preceded by a small leter "s" (slice), "a" (antislice) and "c" (turn the whole cube). Thus, turn the bottom face counterclockwise is "D-", a clockwise antislice turn parallel to the right side is "aR+" , turning the top face 180 degrees is "T2" and a 180 degree slice move through thevertical layer parallel to the Front side is "sF2". I am not sure who to credit this system to but it feels familiar. Anyway, it is easy to use.
The basic symmetries of a cube govern the basic patterns, These are: the rotational symmetries of rotating the cube through a corner to corner axis, either 120 degrees clockwise or counterclockwise. Since there are 4 such axes, there are 8 posibble corner rotations; If you turn the cube about an axis through a pair of centers, you get 3 possible rotations )1/4 turn, clockwise, 1/2 turn and 1/4 turn counterclockwise. Since you have 3 center axes, you get 9 possible rotations. I prefer to keep the three 1/2 turn patterns separate from the six 1/4 turn patterns as the 1/2 turn patterns can also be describled as reflections (4 way reflection) and are distinctly different. Finally, if you flip over a cube 180 degrees through an axis through 2 diagonally opposite edges in a center layer, you get flip symmetry. It results in top and bottom being exchnaged and 2 pairs of adjacent faces on the sides.
A simple basic pattern is one that involves only one kind of cubie, is the same on all 6 sides or on 4 horizontal adjacent sides (6-way and 4-way) and no more than 2 colors should appear on any side. A pattern meeting all these criteria but involving 2 different cubies on each side(both the same color) is a compound basic pattern. Such patterns can be decomposed into two simple patterns. Think of it as two subparticles composing a more complicated kind of particle.
if a pattern is not identical on all relevant sides or involves twisting or moving corners or has more than 2 colors, then it should be called a "complex" pattern.
The well-known 6-way Checkerboard pattern does not fall into any if the above categories as it is the result of a forbidden parity transformation of the cube. Each side is the mirror reflection of its opposite side. This is forbidden because a Rubik's cube is not identical to its mirror image(i.e. it has parity). However, edge cubies are identical with their mirror images and if a move involves nothing but edge cubies it can violate parity restrictions. I call such patterns "forbidden" for short but they are not forbidden for the edge cubies, but calling them that reminds us to be careful with such moves. This corresponds to the fact that particle interactions involving only the weak force can violate parity restrictions too. Issues involving parity are often subtle and deep and I will save them for later.
All simple basic patterns invoving the Prime (or Bosonic) orbit all have "partners" in the Fermionic (odd pair exchange) orbit. Bosonic patterns have either corner rotation (6-way) or 1/2 center turn (4-way) symmetry. Their Fermionic partners have either Flip (6-way) or 1/4 corner turn (4-way) symmetry. This is a deep correspondence to particle physics. SuperSymmetry theory assigns a "superpartner" to every basic particle of the Standard Theory. If the standard particle is a Fermion, its superpartner is a Boson and if the standard particle is a Boson, its superpartner is a Fermion. The simple basic patterns of the cuNow, if you dissasemble the cube and reassemble it randomly, you will only have a 1 in 12 chance of ending up with a cube that you can solve all the way to the "start" configuration, This is because you are not completely free to rearrange the cubies but must obey ccertain restrictions. If you excahnge pairs, you can only exchnage an even number at a time. If you flip edges, you can only flip an even number at a time. If you twist corners, the total amount of twist can only add up to an integer. You can mix these restrictions and the result is the 12 orbits. If none of these is restrictions is violated, you end up with the Prime orbit which is the type of cube you have in the Start configuration. I call this the Bosonic orbit because an even number of pair exchnages is analogous to a particle with integer spin, and thus can regarded as a "boson". The next orbit has one pair exchanged, and the total number of pair exchanges of any pattern in this orbit must be an odd number. Any pattern in this group corresponds to a Fermion and has half-integer total spin. The introduction to the Wiki article on Supersymmetry is easy enough for anybody to understand the gist and some later parts are technical enough to appeal to the math geeks. If you play with patterns and in particular with correlating patterns in the Prime orbit with those in the Fermionic family, It would help to use two cubes for this, one in the Prime orbit and one in the Fermionic orbit. the nest orbit is one flipped edge or an odd number of flipped edges.After playing around with this for a long time, I decided it is analogous to magnetic poles. All particles have both a "north" and "south" pole or pairs of such, No particle has only a "north" pole or only a "south" pole. Such a particle would be called a "magnetic monopole" and some theories (G.U.T.) suggest they may exist, though none have been found. I only have one pattern for this, a version of the 6way-C (or 6-way U) pattern that appears like any of the others but if you examine the color symmetry, it has a bizzare, twisted lop-sidded symmetry that you won't find on any standard list of Rubik's cube symmetries. it is possible to create a new orbit by combining the previous two into an orbit of cube patterns with an odd number of pair exchnaged and an odd number of flipped edges. I only know one such pattern and it may be regarded as a "Fermionic Monopole" to draw an anology with particles. Please read the intro to the Wikipedia article on magnetic monopole for a better picture.
The next orbit involves the corner cubies. if you count a clockwise twist of a corner as +1/3 and a counterclockwise twist as -1/3, then the total twist must add up to an integer, never a fraction. This is a close analogy to the restrictions on how quarks may be combined to form a particle. 1/3 or 2/3 electric charges are associated with each quark, but quarks must combine in a way that the total electric charge of the particle is an integer. The mathematics used to describe the strong force that governs quark interactions is different from that used for spin. I only know two patterns in this orbit but they are very striking. They are both bosons and involve rotational symmetry through the corners.) Some hvae theorized the existence of particles that would have an extra quark and thus fractional electric charge. Patterns based on a cube in the single twisted corner orbit (actually 2 orbits, depending on whether the twist is clockwise or counterclockwise) correspond to these particles. I call this orbit the "quark" orbital. I haven't found it necessary to distinguish between the clockwise and counterclockwise versions. The other orbits are the possible combinations of the orbitals involving flip, pair exchange and corner twist.
So far, I ma working with the Prime, Fermionic, Monopole orbits (both Boson and Fermion), and the two trivially different Quark orbits. This is only half of the 12 possible orbits and I would be very interested if anybody finds patterns corresponding to any of the othe others. There is also a whole realm of 12 orbits belonging to the Mirror cube and I have some patterns I have discovered in the Mirror Boson and Mirror Fermion orbits. I will save the discussion on these until I get into the weighty subject of parity.be form pairs in exactly this way. So do many complex patterns. The compund basic patterns are different. Usually, if one kind of cubie forms a Bosonic pattern then so does the other, Like wise, if one is Fermionic so is the other. Now two Bosons combine to form a Boson because even plus even equals even (number of pair exchnages). Likewise two Fermions combine to form a Boson as well because odd plus odd also equals even. Anyway, the combined particle in either case is a Fermion and both the basic partner and the superpartner belong to the Prime orbit. You can distinguish them by examining their component sub patterns. , Both bosons for a Standard pattern and both fermions for a Supersymmetric pattern. Again, this parallels particle physics where two fermions combine to become a boson.
All simple basic patterns invoving the Prime (or Bosonic) orbit all have "partners" in the Fermionic (odd pair exchange) orbit. Bosonic patterns have either corner rotation (6-way) or 1/2 center turn (4-way) symmetry. Their Fermionic partners have either Flip (6-way) or 1/4 corner turn (4-way) symmetry. This is a deep correspondence to particle physics. SuperSymmetry theory assigns a "superpartner" to every basic particle of the Standard Theory. If the standard particle is a Fermion, its superfermion is a Boson and if the oarticle is a Boson, it's superpartner is a Fermion. The simple basic patterns of the cube form pairs in exactly this way. So do many complex patterns. The compund patterns are different. Usually, if one kind of cubie forms a Bosonic pattern then so does the other, Like wise, if one is Fermionic so is the other. Now two Bosons combine to form a Boson because even plus even equals even (number of pair exchnages). likewise two Fermions combine to form a Boson as well bexause odd plus odd equals even. In Particle theory, two fermions likewise combine to form a boson. Thus, the supersymmetry pairs of the compound basic patterns all are bosons in the Prime orbit, You can tell which is Standard and which is Suppersymmetric by looking at the subpatterns.
Two common 6-way simple basic patterns in the Prime orbit (both bosons of course) are the rotational 6-way Eyes pattern and the Rotational 6-way Checkerboard pattern (not the Reflection Checkerboard!). Both patterns are in the corner rotation symetry family. their Fermionic counterparts are theFlip 6-way Eyes pattern theand yhe Flip 6-way Checkerboard, both possible only in the Fermionc orbit. Note you can create a compound basic 6-way Crosses pattern with rotational symmetry by combining the two bosons and of course its a boson. If you combine the two fermionic patterns you get the compound basic Flip 6-way Crosses pattern but this is also a boson though made up of two fermions. There are two 4-way crosses in the Prime orbit. One has Center Half Turn symmetry and the other Center 1/4 turn symmetry and of course both are bosons. The 1/2Turn pattern is the Standard cube particle because it is made up of two bosons, both of what should be familiar but the 1/4Turn pattern is composed of two fermions that you are probably not familiar with.
For "home work" I recommend putting your cube into the one-pair-exchange or fermionic orbit and try creating all of the fermionic patterns I have described above. It will really help clarify for you what I am talking about. you might try finding some of the fermionic partners of the prime orbit complex patterns. if you really feel ambitious. They are not always easy to find. Try finding the 3rd type of the Prime orbit 6-way Checkerboard and the 2nd type of the Fermionic 6-way Checkerboard pattern.
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Cheers,
Math Bear ^,..,^
