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