# 24 Worlds of Pattern Cubing



## Math Bear (Feb 26, 2016)

Greetings everyone! Most people go into the Rubik's Cube with the idea of eventually solving it. They sweat blood for a considerable while then, if they are persistent, they finally succeed but there they stop. Some may go on a little to try to learn a few "pretty patterns. There is however a world of Power Cubing that goes far beyond merely solving it. The largest such group is the speed cubers and they have accomplished a great deal and achieved truly impressive records. A second Power Cubing group are the theoretical cubers who use the tools of group theory and abstract algebra to analyse the nature of the Rubik's Cube. We are all aware of the important accomplishments that professional mathematicians have made about the cube, A third group are the super cubers who work with larger or unusual versions of the cube and have made fascinating and important discoveries about the relatives of the classic Rubik's Cube. There may be others that i have not thought of. The field of Super Cubing constantly grows and expands.

Now I am proposing a new field of Super Cubing based on what I call Pattern Cubing. Many people have researched "pretty patterns" but I have comprehensively developed the whole field to remarkable levels of subtlety and complexity with the purpose of creating a new and wonderful sport of endless fascination that you can explore with your eyes and your fingers on a classic 3X3 cube. i have been studying and exploring Pattern Cubing for several years now and I am still finding new things, indeed I made a couple of important new discoveries just today. I am also aware there is much more to be done. I will leave it to others to experiment with alternate or higher order cubes and would be very grateful if those specialising in Super Cubing would be kind enough to share their discoveries on here. I am primarily interested in having fun rather than performing serious research, but if any of you math geek theoretical cubers find something interesting I would be delighted if you would contribute your insights. But I want to emphasize the fun aspect of exploring the many secrets your seemingly humdrum standard cube has been hiding from you all this time. I highly recommend that you use the marvellous New Rubik's Speed Cube as your instrument of exploration. Not only is silky smooth and table to the touch, it allows you to easily remove and exchange center cubies, an absolutely essential operation if you are to fully explore and understand the vast world of cube patterns. I keep 6 of these cubes on my bedside table. They currently are showing 6 different patterns of the serpentine symmetry and with the same basic color arrangement, with white in blue, red in yellow, and green in orange. Tomorrow, i will transform them into the reflection symmetry, which can be easily done typically from the serpentine symmetry. I will keep the same patterns though: Eyes, Checkerboard, Crosses, Snakes. Rings and Ribbons. I can also transform all 6 patterns into the rotary and flipped symmetries. There are many other patterns and also other symmetries to explore. Out of the 24 orbits of the3X3 standard cube, both Normal and Mirror, I think I have explored patterns in about ten of them. I think there is much more yet to investigate and I would like to interest other power cubers in making new discoveries. 

The basic type of patterns I have researched the most are what I call "classical patterns". These are based on the various rotations of a cube in 3D space including the "improper rotations", i.e, a rotation and a reflection, For the mathematically inclined, the classical patterns are describable by the subgroup of O(3) restricted to orthogonal rotations but including the improper rotations and applied to a cube in E3. These include a "mirror" or reflection element added onto a proper rotation. This generates five basic classical symmetry families involving 6 faces of the cube and three involving 4 equatorial faces of the cube. Of these 8 classical symmetries, Four are generated by "proper" orthogonal rotations and four by improper rotations, Besides belonging to these symmetry families, classical patterns must also have the same geometric pattern on all 6 faces of the cube and have only 2 colors on any given face to make the pattern. Each cube pattern in any particular symmetry family usually has multiple possible color permutations (except for some of the mirror/reflection patterns). Some patterns are simple but others are compound and composed of 2 simple patterns. These are the sub-patterns of that particular compound pattern, An example is the 6-way Cross patterns. Each is composed of a 6-way Eyes and Checkerboard pattern of the same symmetry family and color arrangement. If you count the color permutations, you have 87 diifferent possible 6-way Eyes, Checkerboard and Cross Patterns and 45 possible 4-way patterns of this kind. It would be impressive to reproduce all 132 of these patterns, perhaps in a room full of cubers with steady and swift fingers and sure eyes!

I have developed a geometric language to describe the patterns. This is designed to facilitate experienced cubers in exploring Pattern Cubing and not only math people. You should be able to make these discoveries by hands and eyes applied to the cube, You should also be able to geometrically mutate patterns at will with simple algorithms and freely create or transmute patterns stage by stage with you being able to see clearly what you are doing at each stage. I do not really approve of long complicated sequences of moves often generated by a computer that make no sense until you get to the end where, if you are lucky, the pattern pops out. You should have full control over your cube at all times and be able to play it like a musical instrument. I also cannot approve of using the language of abstract algebra to describe patterns in order to use group theory software to find a pattern and create an algorithm to generate it. That sort of thing properly belongs to Theoretical Cubing. In Pattern cubing, you should never need a computer to create a pattern, Your eyes and hands and mind are computer enough. That being said, there are times when it is worthwhile to consult with a mathematician friend to explore the question of whether or not you have found all the patterns of a particular type or whether patterns of a hypothetical type actually exist. But you should only use a computer to verify your intuitions or establish existence. Do not use the computer to generate the actual pattern or a method to generate it, In Pattern Cubing this is cheating and spoils all the fun. Once you know that something exists, you can find it yourself. I intend to share with you a number of simple, basic algorithms that will give you geometric control of the cube and let you create the patterns yourself and better understand your cube and its patterns. We are directly exploring the deep geometric symmetries of the Rubik's Cube with simple tools applied with sophisticated understanding. There are many opportunities for Speed Cubers to find challenges too, how long can it take to permutate through all the colors on a particular pattern? How long to transform a number of patterns in one symmetry family to another? can you find ways to change one pattern into another keeping the same color arrangement and or perhaps symmetry family? To explore Classical patterns I have needed to use 4 orbits of the cube except for one very unusual pattern that requires a different orbit than the others. Anyway, it is handy to have 4 speed cubes at hand configured to the 4 orbits. This does not require disassembling the cube but instead just popping out 2 to 4 centers and putting them back in in a different order. I will explain about this later, Dont do this promiscuously or you will ruin the symmetry of your cube. 


Finally, I expect everybody who contributes on here to maintain a friendly encouraging tone. I will ask the moderators to deal with hostility, belittling, insults and heavy sarcasm or personal attacks or any other form of trolling. Please keep the tone positive or go to another thread.

Sincerely,

Math Bear

P.S, i know I have a lot additional to explain. This will appear soon in new messages. ^,..,^


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## Joel2274 (Feb 26, 2016)

This sounds really cool! Can't wait to see more of it


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## Math Bear (Feb 27, 2016)

I need to find a way to show Rubik's Cubes upon this thread. Can anybody help?


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## unsolved (Feb 27, 2016)

Math Bear said:


> I need to find a way to show Rubik's Cubes upon this thread. Can anybody help?



Yep. Right here:

http://cube.crider.co.uk/visualcube.php


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## Math Bear (Feb 27, 2016)

In the absence of pics, i will try to describe my basic setup verbally. 
First of all, since I treat centers as moveable elements, I define the faces of the cube by the corners. The Classic patterns involve at most 2 corners being exchanged or twisted so they make good indicators. This may not apply to patterns with non-Classical symmetry. In some cases there is much more action in the corners and it is better to use the centers to define the faces but i won't be dealing with situations like that for quite a while. Left right parity plays a big role in Pattern Cubing and it is best to consider it up front. There are 3 different kinds of cubies and they have different properties with respect to parity and it is important to be aware of this difference. The edge cubies are indistinguishable with respect to their mirror images and can be easily moved without worrying about parity effects, even staying within the Prime orbit, The classic checkerboard design which everybody learns early on has reflection symmetry between all 3 opposite pairs of faces yet is easily accomplished within the standard orbit by three half turn center slice moves. This well known pattern is an excellent example of what I call "Reflection Symmetry". In the Prime Orbit it can only be accomplished with edge cubies.
Then there are the corner cubies. Each one is asymmetric with respect to its mirror image. The reflected versions of corner cubies are different cubies from what you started with. Thus, you cannot apply reflection symmetry to corner cubies. It is forbidden. Then there are the centers. Although they are also mirror symmetric they maintain a fixed relationship with each other, and the pattern formed by the 6 centers is mirror image asymmetric, However, no new cubies are introduced so it isn't forbidden. You can apply reflection symmetry of the centers to any cube orbit by exchanging a pair of centers on aopposite faces. This creates a "Mirror" orbit with fundamentally different symmetries in the patterns, usually Reflection or Serpentine symmetry. (more on this later). Since there are 12 "Normal orbits to the Rubik's Cube, The mirroring operation creates 12 new Mirror orbits, thus the "24 Worlds of Pattern Cubing" mentioned in the title to this thread. The Mirror orbits are no trivial afterthought to my methods, they are quite essential to a systematic understanding of the cube patterns. Where symmetries are derived from "proper" rotations of the cube, the Mirror equivalents are the "improper" rotations.I have actually made use of about 8 out of the 24 orbits in creating patterns. I expect that there are many more patterns to be discovered in the others. The Classical patterns involve (with one exception) four orbits which I will describe here: the first is the Prime orbit which contains the ground state of the Rubik's Cube and all its usual patterns and configurations. This orbit is characterized by: 1, an even number of pair exchanges; 2,an even number of flipped edges; 3, the total amount of twist in the corner cubies must add up to an integer; and 4, the symmetry of the centers is Normal and not Mirror. The next orbit is the Odd Pair orbit, in which an odd number of pair exchanges have occurred. Otherwise, it is like the Prime orbit. Then there is the Prime Mirror orbit which is made by exchanging two centers from opposite faces. The other properties are of course like the Prime (or Normal Prime) orbit. the final orbit we will be working with here is the Odd Pair Mirror orbit, It has the same properties with respect to flipped edges and twisted corners as the others. I will not deal with other orbits until much later. 

It is possible to create the other orbits by just moving the centers around without having to disassemble the cube. On the new speed cube it is easy to pry out and snap in centers with a small screwdriver, just be cautious not to damage the centers. I have already indicated how to put your cube in the Prime Mirror orbit by exchanging two opposite centers. To make the other two, you need to move 4 centers lying in the same slice, I usually pick the horizontal slice. You can advance all 4 centers in the horizontal slice by a quarter turn. it doesn't matter which way, to put the cube in the Odd Pair orbit, Or, you can divide the 4 centers into two groups of adjacent centers that are opposite to the other group then exchange the 2 pairs of adjacent centers in each group, This creates the OddPair Mirror orbit. If you have multiple cubes, try putting 4 of them into the four orbits I have mentioned. I will save the discussion of the five basic 6-way pattern symmetries and the three 4-way pattern symmetries for next time.

Cheers! ^,..,^


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## Math Bear (Feb 29, 2016)

*Axes and Cubies*

To understand symmetry families, we need to talk about the various axes the cube can rotate around and the ways cubies can relate to each other. There are three kinds of cubies. The simplest, with one facelet are the centers. These are either opposite each other or adjacent, with one center opposite and 4 adjacent. A line through the center of the two centers passes through the center of the cube and is thus a rotation axis, the center axis. There are 3 of these. 

The next cubie is the edge cubie. There are 12 of these. It has two facelets connected to a ridge. Two edge cubies can be opposite, meaning they are on opposite sides of a center slice and a line through the middles of the two ridges will pass through the center of the cube. These are the edge axes and you have 6 of these. Edge cubies can also be adjacent, which means they are 90 degrees with respect to a center, faciing, meaning on the same face and across from each other with a center in between, or they are skew and neither on the same face nor on the same slice. Two edge cubies located on the same slice are either opposite or facing. 

The last kind of cubies are the corner cubies. They have 3 faces and are mirror image asymmetric and there are 8 of them. Corner cubes are opposite if they have no facelet colors in common and are as far away from each other as possible. A line connecting the point of the one corner cubie to the point of the other passes through the center of the cube and there are thus axes. There are 4 of these and they are naturally called corner axes. Two corners are adjacent if they are neighbors on the same face, They have 2 color facelets in common, Two corners are diagonal if the are located opposite to each other on the same face. They have only one color facelet in common. Remember that for classic patterns, we define the corners as fixed and the centers as moveable. This still holds even if we reorient or switch a single pair of corners. The corners define the colors of each face.


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## Math Bear (Feb 29, 2016)

The simplest symmetry is the Reflection symmetry. It is exchanging edge and or center cubies between the three pairs of opposite faces. There are no color permutations as such, but the more sophisticated patterns may have more than one way of forming the pattern. The Checkerboard pattern belongs to the Prime orbit but most of the simpler patterns belong to the Mirror (= Mirror Even Pair) orbit, Some of the more sophisticated patterns are Mirror Odd Pair orbit. 

Here is the Checkerboard pattern, the only one that belongs to the Reflection symmetry, though many others are possible (28 in fact) in other symmetries. 

Here is the Reflection 6-way Eyes pattern belonging to the Mirror orbit.  Together, these create the reflection version of the 6-way ?Crosses pattern, also Mirror Orbit.


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## leeo (Jul 8, 2016)

Here are the 24 routines that change just the orientation of the center spots. I name the orientation based on the Speffz name of the solved UF edge that would match the center faces at the UF position. The routines only change the "center spots", and can be loaded into Cube Explorer. Cube Explorer does not directly support whole-cube moves (x, y, or z), so the name includes the whole-cube moves that you need to append on the right to the routine given to give the effect that just appears to move the center spots.

Half of the targets have a corner-edge parity that I resolve by exchanging the UB and UL edges (speffz A_ and D_).

```
D B2 F2 D U' L2 R2 U'  (8f*)  //A y2
D F2 D' B' F L' D U' F2 R2 F' L R'  (13f*)  //B(AD_) y
U U'  (0f*)  //C
D F2 D' B' F L F2 L2 D' U B L' R  (13f*)  //D(AD_) y'
B D U' L' R B F' U'  (8f*)  //E x' z
L F' U2 B2 L' R D' B F' R' B2 L U'  (13f*)  //F(AD_) z
F' D' U L R' B F' U  (8f*)  //G x z
R U' B2 D B' F L' B2 L2 D U' B F2 U'  (14f*)  //H(AD_) x2 z
D F2 U' L' R F' D U' L R U2 R' D U'  (14f*)  //I(AD_) x y2
L D U' B F' L R' U'  (8f*)  //J x y
F' U L2 D' L R' B' D U' L2 F2 R' U  (13f*)  //K(AD_) x
R' D' U B' F L R' U  (8f*)  //L x y'
B' D' U L' R B' F U  (8f*)  //M x' z'
L F' U2 F2 L R' U' B F' L R2 F2 R U'  (14f*)  //N(AD_) x2 z'
F D U' L R' B' F U'  (8f*)  //O x z'
R U' B2 D B' F L' B2 R2 D' U F' U'  (13f*)  //P(_AB) z'
U' L' R B U2 B2 L' R D L R' B U2  (13f*)  //Q(AD_) x'
L' D' U B F' L' R U  (8f*)  //R x' y'
F' U L2 D' L R' B L2 B2 D' U L' R2 U  (14f*)  //S(_AB) x' y2
R D U' B' F L' R U'  (8f*)  //T x' y
U2 B F' L2 R2 B F' U2  (8f*)  //U z2
U2 B2 D' B F' R B2 L2 D U' B L R' U'  (14f*)  //V(AD_) x2 y
U2 L R' B2 F2 L R' U2  (8f*)  //W x2
F R' F2 L D' F L R' D' F U' L2 D R'  (14f*)  //X(AD_) x2 y'
```
Just copy the above code to a text file and save it; then in Cube Explorer apply "Load maneuvers" and select the text file you just saved.


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