Rubik's Cube/Using Pictures, Diagrams, Notation, and Abbreviations
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Ray Calvin Baker 19:28, 18 August 2011 (UTC)
| HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE | | (Preliminary April 20, 2007 version) | | by Mr. Ray Calvin Baker | | FREE Public Domain Educational Material | | | | Chapter Two - - - - - - - - - - Using Pictures, Diagrams, Notation, and Abbreviations | | | | I am using a monospaced font on my word processor in order to keep more control of the | | spacing and the formatting of the characters. When I was young, I used to enjoy trying to | | make pictures on a typewriter. So, let's see if I can make a picture of Rubik's Cube. (The | | limitations of a typewriter sometimes mean that I can't quite draw a line segment exactly | | the way I would like to. You may trace my drawings with a pencil, making those line | | segments exactly correct, if you want to.) | | | | __ * __ The BACK side | | The LEFT side __ -- -- __ is hidden | | is hidden __ * __ __ * __ here | | here ... __ -- -- __ __ -- -- __ ... | | __ * __ __ * __ __ * __ For your | | __ -- -- __ __ -- TOP -- __ __ -- -- __ information, | | * __ __ * __ side __ * __ __ * this is an | | | -- __ __ -- -- __ __ -- -- __ __ -- | "isometric" | | | * __ __ * __ __ * | diagram. | | | | -- __ __ -- This is -- __ __ -- | | | | | | * __the FRT position_ * This is | | I will say | | | | | -- __ __ -- | a place | | more about | | * __ | | * | for an | __ * what that | | | -- __ | | | | edge | __ -- | means later, | | | * __ | | | cubie __ * This | on diagram | | | | -- __ | | | __ -- | is the | 2-21. | | | | * __ | __ * | KR | | | | | | -- __ | __ -- | | position | | | * __ | FRONT | * | RIGHT | __ * For now, | | | -- __ | side | | | side | __ -- | it is just | | | * __ | | | __ * | a way to | | | | -- __ | | | __ -- | | "picture" | | | | * __ | __ * | | a 3-D object | | | | | -- __ | __ -- | | | on a 2-D | | * __ | | This is * | | __ * surface. | | -- __ | | a place | | | __ -- | | Diagram * __ | for a | | __ * | | of an intact -- __ | corner | | __ -- The | | Rubik's Cube, * __ cubie | __ * BOTTOM side | | showing how some parts -- __ | __ -- ... is hidden under here | | are named. * somewhere | | | | DIAGRAM 2-1. An Intact Rubik's Cube, Showing How Some Parts Are Named | | | | Notice especially the parts of the Cube labelled, "TOP side", "FRONT side", and "RIGHT | | side". These will be your "fixed landmarks", at least until you turn the entire Cube, to | | expose some "new" sides. | | | | Yes, it is an awkward thing that at least three faces of the Cube are always hidden, when- | | ever I try to make an accurate picture. Even if I try to make a drawing with parts of the | | Cube removed so that we can see "the innards", I still can't show everything all at once. | | However, I can still try to make conceptual DIAGRAMS that distort some details so that I | | can emphasize others. You will see some of these diagrams in later chapters. | | | | __ * __ __ * __ | | __ -- The TOP -- __ __ -- Corner -- __ | | * __ side rotates __ * _cubies have studs_ * | | | -- __ on__ -- | -- __that_ -- | | | __ * __ the * axle. __ * __ *fit into | | | __ -- __ -- __ | __ --____ \ |grooves | | | __ * __ __ -- | | * | | \ \ |in the | | | __ -- Edge * __* __| | Axle | | \ | |edge | | | * __cubies fit |___/ -| | | | | |cubies.__ * | | | -- __ __ * __| __ -- | * _ |__ * __ | __ -- | | | | between * __ -- __ /The\ _- -- __ * | | | | the core | -- __ __ * Central * __ __ -- | | | | | and the | * | Core looks| * | | | | | corner | | |__ - - __| | | | | | * __cubies.| | | | like a| | | | __ * | | | -- __ | | | |_jack._| | | | __ -- | | | | * __ | | -- --- -- | | __ * | | | | | -- __ | __ * __ ___ __ * __ | __ -- | | | | | | * | | * | | | | | | | | | | | | | | * __ | | \ __ * __ / | | __ * | | -- __ | | | | | __ -- | | Diagram of * __ | __ * __ | __ * | | a partially -- __ | __ -- -- __ | __ -- | | disassembled Cube * . * | | | | Another view of some parts of a Cube which are usually hidden... | | | | DIAGRAM 2-2. Diagram of a Partially Disassembled Cube | | | | +-----------+ Another | | | TOP | perspective | | +-----------+ of a | | | | partially | | | | ... another axle disassembled | | +---+ | | Cube | | | L | +-----------+-----------+ and some | | | E | | | | of its | | | F |-------| FRONT | FR edge | component | | | T | Axle | side | cubie | Cubies | | | |-------| | in place | | | +---+ | | | Frontal | | | | +-----------+-----------+ View | | +---+ | | | | | | : | | | BFR | | | : | | | corner | | | Rotated +-----------| cubie | | | slightly, | BOTTOM | in place | | | to show +-----------+-----------+ | | how the | | center | | square +-----------+ +-----+-----+ | | rotates on | | Looking at the | | | | | its axle. | | BACK RIGHT edge +-----------+ | | | | | cubie from the | | | | | +-- _ | FRONT +-----------+ | | | | \ | | | | | | | Stud \--------+ +-----+-----+ | | | | | | +-------+ +-----------+-----------+ | | Looking at the ||||||||||||| | | | Looking at the RIGHT TOP edge +-----------| | | | BACK RIGHT TOP cubie from the This +- _| | | | corner cubie FRONT. This | -_ | | | from the FRONT. edge | \ | | | Shaded parts rubs | \------+ | | show how the against.| | |///| | | axles and an axle.+---------+ |///| | | face plates This edge |///| | | clamp an also rubs |///| | | edge cubie against an axle, |///| | | in place +---+ | | | | DIAGRAM 2-3. Another Perspective of a Partially Disassembled Cube | | | | If you are desperate to see for yourself what the "innards" of the Cube really look like, | | or if you want to make your Cube look like it is fresh out of the store, here is a way to | | physically disassemble your Cube. (I call this "the Physicist's method".) It is much less | | damaging to your Cube than the Chemist's method, but it may make your Cube weak, wobbly, | | and stretched out of shape if you try this very often. | | | | Turn the top layer of the Cube 45 degrees. Slip a butter knife (NOT a sharp knife!) between | | the bottom edge of an edge cubie on the top layer and the edge cubie it rests upon. Pry up | | gently, and the cubie above the butter knife should pop out. Two corner cubies can then be | | removed from the top layer, and further disassembly should be easy, simple, and obvious. | | | | To reassemble your Cube, follow these steps in reverse order (although you will not need to | | use a butter knife blade to re-insert the final cubie). Use common sense to restore your | | Cube to its (almost) pristine condition, or be bold and explore one of the eleven other | | ways to re-assemble the Cube. (However, if you try one of these eleven alternative orbits, | | you will never be able to solve your Cube using only mathematical moves, until you | | disassemble the Cube and put it back together "properly".) | | | | 1. The _______________________________________________ | | TOP | | | | | | / layer | | | | | | / has been |_______________|_______________|_______________| | | | turned | | TOP | Pry this | | | | 45 | | side | cubie out. | | | \ degrees |_______________|_______________|_______________| | | \ _ | | | | ___________________ | | --> | | | | <---/ | | _|_______________|_______________|_______________|_ / 2. Slide the blade | | __ -- | | | | --| of a butter knife | | * __ | | FRONT side of | | | through here, | | | -- _| | FRONT TOP | |_ -- \ then pry up. | | | | | edge cubie | | \___________________ | | | | | | | | | | | |_______________|_______________|_______________| | | | | | FRONT | -- __ __ -- | RIGHT | | | | * __ | side | * | side | __ * | | | -- __ | center | | | center | __ -- | | | | * __cubie | | | cubie __ * | | | | | -- __ | | | __ -- | | | | | | * __ | __ * | | | | | | | -- __ | __ -- | | | | | * __ | | * | | __ * | | -- __ | | | | | __ -- | | * __ | | | __ * | | -- __ | | | __ -- | | * __ | __ * | | -- __ | __ -- | | * | | | | DIAGRAM 2-4. The Physicist's Method -- How to Disassemble the Cube | | | | _ | | __ --- | Prying out the | | +-----------+-----------+-----------+ TOP RIGHT | | | | FRONT | | edge cubie ... | | | | side of | | | | | | FRONT TOP | || | | | | edge | |_| | | | | cubie | |-- __ ... using the flat, dull | | +-------+-------+-------+-------+-------+-------+-- __ blade of a | | | | | | | | | -- __ butter knife. | | | | FRONT | | | RIGHT | | -- __ __ | | | | side | | | side | | -/ -- __ | | | | center| | | center| | \ _ -- __ | | | | cubie | | | cubie | | -- __ -- | | +-------+-------+-------+-------+-------+-------+ -- __ | | | | | | | | | -- | | | | | | | | | | | | | | | | | | | | | | | | | | | Another view -- how to | | | | | | | | | disassemble the Cube | | +-------+-------+-------+-------+-------+-------+ | | | | DIAGRAM 2-5. Another View -- How to Disassemble the Cube | | | | It may be useful to have some standard abbreviations. I propose these: | | | | For indicating locations and cubies: For indicating colors: | | use B for BOTTOM use b for BLUE | | use F for FRONT use g for GREEN | | use K for BACK use k for BLACK | | use L for LEFT use o for ORANGE | | use R for RIGHT use p for PURPLE | | use T for TOP use r for RED | | use w for WHITE | | use y for YELLOW | | | | To avoid confusion with "B" for "BOTTOM", I have used "K" for "BACK". To avoid confusion | | with "b" for "blue", I have used "k" for "black". "BACK" is the only key word which | | contains the letter "K", and "black" is the only key word which contains the letter "k". | | | | I have arranged these key words alphabetically. I will try to be fairly consistent in how I | | use these words and their abbreviations. (Yes, there are only six colors on any one | | individual good Cube. But different Cubes may have different colors, so we may need some | | extra abbreviations for the extra colors.) Since I cannot see your Cube as I write this | | book, I will not be able to use the lower case letters as abbreviations for the colors on | | your Cube. | | | | It is often important to distinguish between locations on a Cube, independently of the | | cubie which occupies that location. It makes sense to say, "The orange, purple, and yellow | | (opy) cubie is at the FRONT RIGHT TOP (FRT) location. I want to move that cubie to the | | FRONT RIGHT BOTTOM (FRB) location." Of course, you must move a lot of other cubies as well | | when you move the "opy" cubie. | | | | I will try to use CAPITAL LETTERS to abbreviate locations, and lower case letters to | | abbreviate colors. Actually, since I don't know what colors your cube is, I won't be | | referring to colors much at all. But feel free to use colors in your personal notes. It's | | much easier and more reliable than trying always to refer to the position and orientation | | of each cubie, as I must do as I write this book. | | | | Corner locations can be uniquely identified by using three of the key words, for example, | | BACK FRONT TOP. Edge locations can be uniquely identified by using two of the key words, | | for eample BOTTOM LEFT, A face can be uniquely determined by using only one of the key | | words, for example, BOTTOM. By the way, you should remember that the face cubie -- the | | central square of any side -- can only rotate in its place, so it can often be used as a | | "landmark" to identify a face of the Cube. | | | | I sometimes use letters and numbers as arbitrary symbols to identify the various faces of | | cubies when I try to describe things to do, such as interchanging two corner cubies. (These | | are often called OPERATORS or OPERATIONS.) | | | | You probably know that the simplest operations which can be done on a Rubik's Cube are to | | turn one layer of the cube 90 degrees. (Turning the entire Cube is sometimes done for | | special reasons, but that particular operation does not change the essential position or | | orientation of the cubies with respect to each other.) | | | | Suppose we have an arbitrarily labeled Cube. (You may use small Post-It notes if you want | | to label your Cube this way. And you can peel off Post-It notes easily, and reuse them.) | | | | I am going to depart from alphabetic sequence to show you "FRONT" before I show you | | "BOTTOM". If you hold your Cube so that it looks somewhat like the diagram, you can see the | | "FRONT", but the "BOTTOM" is hidden. It helps to be able to see clearly what you are going | | to do, especially when you are learning to do something new (and possibly puzzling). | | | | We can rotate the FRONT face clockwise, an operation abbreviated "Fv", and pronounced, | | "FRONT clockwise". We can rotate the FRONT face counterclockwise, abbreviated "F^", and | | pronounced, "FRONT counterclockwise". You may want to draw a curved arrow around the symbol | | for the face of the Cube to emphasize that these notations mean "rotate clockwise" and | | "rotate counterclockwise". | | | | _ a _ _ a _ _ a _ These are supposed to be | | b _ _ c ) e _ _ c b _ _ c tiny little isometric | | | F d | Fv | F b | F^ | F d | drawings of a Cube, with | | e _ | _ f g _ | _ f ) e _ | _ f the corner cubies labeled | | g d g with lower case letters. | | "before Fv" "after Fv" | | "before F^" "after F^" I can't draw the curved | | arrows very well with a | | typewriter, so I will | | usually just type "Fv" or "F^". | | | | DIAGRAM 2-6. FRONT Clockwise Move and FRONT Counterclockwise Move | | | | Since we can usually see the FRONT side, there should be no confusion here. | | | | I want you to think of "v" and "^" as little arrows that show you how to rotate a face of | | the Cube -- either clockwise or counterclockwise. This should work nicely when you can see | | the side of the Cube you want to turn. | | | | I want to show you another simple move -- rotating a face of the Cube by 180 degrees. You | | can do this to any face of the Cube. It doesn't matter whether you turn 180 degrees | | clockwise or counterclockwise -- the result is the same! | | | | _ a _ _ a _ _ a _ | | b _ _ c e _ _ c g _ _ c | | | F d | Fv | F b | Fv | F e | | | e _ | _ f g _ | _ f (again) d _ | _ f | | g --> d --> b | | Either way, | | _ a _ _ a _ clockwise or | | b _ _ c g _ _ c counterclockwise, | | | F d | F2 | F e | the results | | e _ | _ f (Think, "FRONT twice".) d _ | _ f are exactly | | g ------------------------------> b the same! | | | | _ a _ _ a _ _ a _ | | b _ _ c d _ _ c g _ _ c | | | F d | F^ | F g | F^ | F e | | | e _ | _ f b _ | _ f (again) d _ | _ f | | g --> e --> b | | | | DIAGRAM 2-7. The FRONT Twice Move | | | | I abbreviate this operation on the FRONT of the Cube as "F2" (think of this move as "FRONT | | twice".). If you turn the BOTTOM, BACK, LEFT, RIGHT, or TOP, the abbreviations are "B2", | | "K2", "L2", "R2", and "T2". (You can guess that I would call these, "BOTTOM twice", "BACK | | twice", "LEFT twice", "RIGHT twice", and "TOP twice".) | | | |---------------------------------------------------------------------------------------------| | | | If the face of the Cube is hidden (like the "BOOTOM", "BACK", and "LEFT" sides), things can | | seem somewhat confusing. So, I intend to introduce a different notation to show how to move | | the parts of the Cube you CAN see. If you feel you must use this alternative notation, be | | sure to draw it differently from the symbols "Bv" and "B^", which indicate "rotate | | clockwise (when looking at the face)" and "rotate counterclockwise (when looking at the | | face)". I use straight line segments with arrow heads instead of curved arrows, so I can | | tell the difference. | | | | We can rotate the BOTTOM face clockwise, an operation abbreviated "Bv", and pronounced as | | "BOTTOM clockwise".. The alternate notation for this is: | | | | ) | | _ a _ Bv _ a _ | | b _ _ c or b _ _ c (The BOTTOM | | | d | | d | face is | | e _ | _ f - _ B _ - > h _ | _ g usually | | g - > - e hidden.) | | "Before" "After" | | | | DIAGRAM 2-8. BOTTOM Clockwise Move | | | | This is messy to draw on a typewriter, though I hope it suggests to you how the "BOTTOM" | | cubies move. I hope you can learn to live with the "Bv" notation. | | | | We can rotate the BOTTOM face counterclockwise, abbreviated "B^". (I call this "BOTTOM | | counterclockwise",) The alternate notation for this is: | | | | _ a _ B^ _ a _ | | b _ _ c or b _ _ c (The BOTTOM | | | d | | d | face is | | e _ | _ f < - _ B _ - g _ | _ h usually | | g - < - f hidden.) | | "Before" "After" | | | | DIAGRAM 2-9. BOTTOM Counterclockwise Move | | | | You may have noticed that, since the BOTTOM face is usually hidden, we are looking at it | | from the "wrong" side. You may turn the Cube in your hand temporarily to "peek" and see | | that we are turning the BOTTOM layer the way we say, when we are actually looking at that | | layer from the "correct" direction. | | | | _ a _ _ a _ _ a _ Every operation has | | b _ _ c b _ _ c Note that b _ _ c an "inverse" | | | d | Bv | d | B^ | d | operation which | | e _ | _ f h _ | _ g restores e _ | _ f "undoes" it; the | | g e the Cube to g combination | | its starting Bv -> B^ | | position. is an example of a | | "do nothing" identity | | operation. | | | | DIAGRAM 2-10. BOTTOM Clockwise and BOTTOM Counterclockwise Are Invers Operations | | | | OK. You got confused. I'll try again to show you the BOTTOM side, turning it first | | clockwise, then counterclockwise. | | | | _ a _ Rotate the _ d _ _ d _ _ d _ | | b _ T _ c entire Cube b F | R c b F | R c b F | R c | | ---| F d R |--- to peek ---| _ g _ |--- Bv | _ e _ | B^ | _ g _ | | | e _ | _ f at the e _ B _ f h _ B _ g e _ B _ f | | g BOTTOM side h f h | | | | DIAGRAM 2-11. Peeking at the BOTTOM Side to See Bv and B^ More Clearly | | | | Don't forget to rotate the entire Cube back to its original position, so you can see the | | TOP side, the FRONT side, and the RIGHT side, as I will usually draw them. | | | |---------------------------------------------------------------------------------------------| | | | We can rotate the BACK face clockwise, an operation abbreviated "Kv". (Think, "BACK | | clockwise"!) We can rotate the BACK face counterclockwise, abbreviated "K^". (Think, "BACK | | counterclockwise".) Again, I will also ehow the alternate notation for this: | | | | Kv K^ | | _ a _ or _ c _ or _ a _ | | b _ _ c < - _ b _ _ f - _ b _ _ c (The BACK | | | d | - | d | - > | d | face is | | e _ | _ f K e _ | _ h K e _ | _ f usually | | g ^ g | g hidden.) | | | v | | | | DIAGRAM 2-12. BACK Clockwise Move and BACK Countertclockwise Move | | | | The BACK side is viewed here from the "wrong" direction, so don't get confused! | | | |---------------------------------------------------------------------------------------------| | | | We can rotate the LEFT face clockwise, an operation abbreviated "Lv". We can rotate the | | LEFT face counterclockwise, abbreviated "L^". Again, I will also ehow the alternate | | notation for this: | | | | Lv L^ | | _ a _ or _ h _ or _ a _ | | b _ _ c _ - a _ _ c _ - > a _ _ c (The LEFT | | | d | < - | d | - | d | face is | | e _ | _ f L b _ | _ f L e _ | _ f usually | | g | g ^ g hidden.) | | v | | | | | DIAGRAM 2-13. LEFT Clockwise Move and LEFT Counterclockwise Move | | | | The LEFT side is viewed here from the "wrong" direction, so don't get confused! | | | |---------------------------------------------------------------------------------------------| | | | We can rotate the RIGHT face clockwise, an operation abbreviated "Rv". We can rotate the | | RIGHT face counterclockwise, abbreviated "R^". | | | | _ a _ _ a _ _ a _ | | b _ _ c b _ _ d b _ _ c | | | d R | Rv | g R | R^ | d R | | | e _ | _ f e _ | _ c e _ | _ f | | g f g | | "before Rv" "after Rv" | | "before R^" "after R^" | | | | DIAGRAM 2-14. RIGHT Clockwise Move and RIGHT Counterclockwise Move | | | | Since we can usually see the RIGHT side, there should be no confusion here. | | | |---------------------------------------------------------------------------------------------| | | | We can rotate the TOP face clockwise, an operation abbreviated "Tv". We can rotate the TOP | | face counterclockwise, abbreviated "T^". | | | | _ a _ _ b _ _ a _ | | b _ T _ c d _ T _ a b _ T _ c | | | d | Tv | c | T^ | d | | | e _ | _ f e _ | _ f e _ | _ f | | g g g | | "before Tv" "after Tv" | | "before T^" "after T^" | | | | DIAGRAM 2-15. TOP Clockwise Move and TOP Counterclockwise Move | | | | Since we can usually see the TOP side, there should be no confusion here. | | | |---------------------------------------------------------------------------------------------| | | | If it's any consolation to you, we will usually be working with the TOP side, the FRONT | | side, and the RIGHT side. | | | | Like Shrek, onions, parfaits, and fancy cakes, the Cube has LAYERS! Here is a diagram to | | help you visualize the Cube in "layers". | | | | _ * _ _ * _ _ * _ | | _ * _ - _ _ - _ * _ _ - - _ | | _ * _ - _ - _ _ - _ - _ * _ _ - _ * _ - _ | | * _ - _ - _ _ * * _ _ - _ - _ * * _ * _TOP_ * _ * | | | - _ - _ _ * | | * _ _ - _ - | | - _ * _ - | | | | - _ _ * | | | | * _ _ - | | - _ _ - | | | | * | | | | | | * | * _ * _ * | | | * _ | | | | | | | | _ * | | - _ | _ - | | | | | * | | | | | | | | * | | | - _ | _ - | | | | | F | | | | | | | | | | R | | * _ * _ * | | | * _ | | | | | | | | | | _ * | | - _ | _ - | | | | * | | | | | | | | * | | - _ | _ - | | | * _ | | | _ * * _ | | | _ * * _ * _ * | | - _ | | _ * * _ | | _ * - _ | _ - | | - _ | _ * * _ | _ * - _ | _ - | | * * * | | 3 layers between BACK 3 layers between LEFT 3 layers between BOTTOM | | and FRONT and RIGHT and TOP | | | | DIAGRAM 2-16A. DIAGRAM 2-16B. DIAGRAM 2-16C. | | | | DIAGRAM 2-16. Three Types of Layers | | | | All of the moves described so far in this chapter (B^, B2, Bv, F^, F2, Fv, K^, K2, Kv, L^, | | L2, Lv, R^, R2, Rv, T^, T2, and Tv) have involved just one single layer -- one layer moves; | | the other two layers stay fixed in space. During all of these moves, the center squares of | | the Cube remain fixed in space. If you wanted to emphasize that you were moving just one | | layer, you could write, "1B^", "1B2", "1Bv", "1F^", "1F2", "1Fv", "1K^", "1K2", "1Kv", | | "1L^", "1L2", "1Lv", "1R^", "1R2", "1Rv", "1T^", "1T2", and "1Tv". But, just as in algebra | | there is no real need to write "1x" when "x" will do, so there is no urgent need to write | | "1B^", since "B^" will do just as well. | | | | However, there have been some hints that other moves are possible. (Diagram 2-11 was a very | | temporary "sneak peek" at the BOTTOM of the Cube. We immediately returned the Cube to its | | normal orientation after the "peek".) Indeed there are some other important moves. | | | | From time to time, it may become necessary to rotate the entire Cube. I can indicate these | | moves as 3F^, 3F2, 3Fv, 3R^, 3R2, 3Rv, 3T^, 3T2, and 3Tv. Rotate all three layers (instead | | of just one layer) and you have rotated the entire Cube. I think of these moves as "three | | layer FRONT counterclockwise", "three layer FRONT twice". "three layer FRONT clockwise", | | and so on. | | | | Do you see why I do not need to mention 3B^, 3B2, 3Bv, 3K^, 3K2, 3Kv, 3L^, 3L2, or 3Lv? | | (Partial answer: 3B^ = 3Tv, and 3L2 = 3R2.) It should be obvious to you that each rotation | | of the entire Cube causes the center square of four sides to move (and each rotation causes | | the center square of the remaining two sides to rotate). If you move all three layers, you | | do need to write the "3", unless you can work out your own complete system of notation. | | Then, you would be writing the book, and I would be reading it! | | | | Here is a sequence of rotations of the entire Cube which will cause each of the six sides | | to enjoy a moment on TOP. (Each side also takes a turn on the BOTTOM.) | | | | _ * _ _ * _ _ * _ _ * _ _ * _ _ * _ | | * _ t _ * 3Rv * _ f _ * 3Fv * _ l _ * 3Fv * _ k _ * 3Fv * _ r _ * 3Rv * _ b _ * | | | f * r | | b * r | | b * f | | b * l | | b * k | | l * k | | | * _ | _ | --> * _ | _ | --> * _ | _ * --> * _ | _ * --> * _ | _ * --> * _ | _ * | | * * * * * * | | | | DIAGRAM 2-17. Each Face of the Cube Takes a Moment on TOP | | | | It may be helpful to practice other ways to rotate the Cube. Here we return it to its | | original postion. | | | | _ * _ _ * _ _ * _ _ * _ | | * _ b _ * 3F^ * _ k _ * 3F^ * _ t _ * 3Tv * _ t _ * | | | l * k | | l * t | | l * f | | f * r | | | * _ | _ * --> * _ | _ * --> * _ | _ * --> * _ | _ * | | * * * * | | | | DIAGRAM 2-18. We Return the Cube to its Original Orientation | | | | If you want to get really fluent in moving your Cube, try these rotations of 180 degrees | | about an axis through the midpoint of two opposite edges. (There are six ways to rotate a | | Cube in this fashion. You may recall diagram 2-11 , where I showed you a "sneak peek" of | | the BOTTOM of the Cube. I do not intend to make any further use of these methods, or to | | develop a notation for them. But you can if you want to!) | | | | \ | | __ * __ \ __ * __ | | __ --- --- __ __ -O- --- __ | | * __ TOP __ * * __ TOP __ * | | | --- __ __ --- | | --- __ __ --- | | | | * | | * | | | ---O | O--- | | | | | | FRONT | RIGHT | | FRONT | RIGHT | | | | | | | | | | | * __ | __ * * __ | __ * | | --- __ | __ -- --- __ | __ -O- | | * * \ | | \ | | An axis through An axis through | | the centers of the the centers of the | | FRONT LEFT and LEFT TOP and | | BACK RIGHT BOTTOM RIGHT | | edge cubies. cubies. | | | | DIAGRAM 2-19A. DIAGRAM 2-19B. | | | | / | | __ * __ / __ * __ | | __ --- -O- __ --- __ __ --- --- __ (The place | | * __ TOP __ * --- __ TOP __ * where the | | | --- __ __ --- | | O- __ __ --- | axis comes | | | * | | * | out is hidden | | | | | | | | behind and | | | FRONT | RIGHT | | FRONT | RIGHT | underneath | | | | | | | ( ) | the Cube.) | | * __ | __ * * __ | __ * __ | | -O- __ | __ -- --- __ | __ --- --- | | / * * | | / | | An axis through An axis through | | the centers of the the centers of the | | BACK TOP and FRONT TOP and | | BOTTOM FRONT BOTTOM BACK | | edge cubies. edge cubies. | | | | DIAGRAM 2-19C. DIAGRAM 2-19D. | | | | | | | | | | __ * __ __ * __ | | __ --- --- __ __ --- __ --- --- __ | | (The place * __ TOP __ --- * __ TOP __ * (The place | | where the | --- __ __ -O | | --- __( )__ --- | where the | | axis comes | * | | * | axis comes | | out is | | | | | | out is | | hidden.) | FRONT | RIGHT | | FRONT | RIGHT | hidden.) | | | ( ) | | | O | | | __ * __ | __ * * __ | __ * | | --- --- __ | __ -- --- __ | __ --- | | * | | | | | | An axis through An axis through | | the centers of the the centers of the | | RIGHT TOP and FRONT RIGHT and | | BOTTOM LEFT BACK LEFT | | edge cubies. edge cubies. | | | | DIAGRAM 2-19E. DIAGRAM 2-19F. | | | | DIAGRAM 2-19. Six More Ways to Rotate the Entire Cube | | | | Can you find sequences of three-layer moves, using our usual notation, which accomplish the | | same results? | | | | There are also four ways to rotate the Cube by 120 degrees, about an axis through opposite | | corners of the Cube. You can rotate either clockwise or counterclockwise. (Again, I do not | | intend to make much use of these methods -- but you can!) | | | | | (You are | | _ | _ looking | | - _ _ * _ _ * _ _ * _ _ - _ * _ straight | | * _ T _ * * _ T _ * * _ T _ * * _ T _ * along the | | | F * R | | F * R | | F * R | | F(*)R | axis -- it | | * _ | _ * _ * _ | _ * _ * _ | _ * * _ | _ * goes through | | * - _ * _ - * * the center | | | of the Cube.) | | | | | An axis through An axis through An axis through An axis through | | centers of centers of centers of centers of | | FRONT LEFT TOP and BACK LEFT TOP and BACK RIGHT TOP and FRONT RIGHT TOP and | | BACK RIGHT BOTTOM FRONT RIGHT BOTTOM FRONT LEFT BOTTOM BACK LEFT | | cubies. cubies. cubies. cubies. | | | | DIAGRAM 2-20A. DIAGRAM 2-20B. DIAGRAM 2-20C. DIAGRAM 2-20D. | | | | DIAGRAM 2-20. Four More Ways to Rotate the Entire Cube | | | | Again, can you find sequences of moves, using our usual notation, which accomplish the same | | results? | | | | Just as a hint of what's coming up, Chapter Eight, "Moving Edge Cubies" and Chapter Nine, | | "Rubik's Maneuver -- How to Flip Two Edge Cubies", will move two layers (and keep the third | | layer fixed in space). These moves will be called "slice up" and "slice down". They are | | useful for moving edge cubies around. Like the "three layer" moves, they also cause the | | center square of several sides to move around. | | | | Always be careful when you try to use the two-layer moves and the three-layer moves. The | | cubies move, but the locations (FRONT, RIGHT, TOP, etc.) do not. When I show you an | | isometric diagram of the Cube like this... | | | | _ * _ | | _ - ... - _ | | _ - this side - _ | | * _is always called _ * This way of representing a cube on a two-dimensional | | | - the TOP side. - | surface is called an "isometric" view, or isometric | | | - _ _ - | projection. If drawn with drafting instruments | | | ... this * ... this | (instead of a typewriter), a diagram like this has | | | side is | side is | equal measurements along each of the three main | | | alwayss | always | directions (iso = same, metric = measure). | | | called | called | | | | the | the | ^ height | | | FRONT | RIGHT | | The three main directions | | * _side. | side. _ * < _ | _ > of an isometric drawing | | - _ | _ - - _ | _ - | | - _ | _ - width * depth | | * | | | | DIAGRAM 2-21. An Explanation of an Isometric Drawing With a Cube | | | | As you know, a cube is really a three-dimensional shape. My two-dimensional diagrams cannot | | really do justice to that essential fact. (Here's another fact about dimensions -- when you | | move part of your Cube, you are actually explaoring a shape with FOUR dimensions! At | | different moments of TIME, different parts of your Cube are in different places. It would | | take (at least) four different numbers to describe truly and completely the position of | | each particle of your Cube.) | | | *---------------------------------------------------------------------------------------------*
Ray Calvin Baker 15:03, 29 October 2011 (UTC)