Magic Cube 4D Copyright 1994 - 2003 by Melinda Green, Don Hatch and E. Jay Berkenbilt MagicCube4D is a four-dimensional Rubik's cube. It is an exact analogy in four dimensions to the original plastic three dimensional puzzle, but with some useful features - such as a "reset" button - which the original puzzle lacked. Q: How can it really be a four dimensional object when there are no such things as 4D objects? A: It's true that there are only three spatial dimensions in this universe, but mathematically speaking, there is no problem at all extrapolating into an arbitrary number of dimensions. Computers are perfectly happy to model objects in higher dimensions. Q: Even if the computer can deal with 4D objects, how can it display one on on a 2D screen? A: Using the exact same mathematical techniques that are used to project 3D objects onto 2D screens, we "project" 4D objects into 3D. The resulting 3D objects can then be rendered with conventional graphics software onto the screen. Q: If you're projecting from 4D down to 2D, isn't it impossible to understand what you're seeing? A: The 3D objects can easily be understood by rotating them around on the screen. You can do that by clicking the left button of the mouse in the window and dragging the mouse around with the button held down. It is probably impossible for a human to ever truly understand 4D objects by examining their 3D "projections" with the same clarity that even a child easily understands the 3D nature of objects rendered on a computer screen. Even so, it is quite possible to gain a strong feeling for the 3D projections that result from some operations on 4D objects. Q: So what does it mean to make a 4D Rubik's cube? A: Every feature of the original puzzle has an analogy in four dimensions. For the rest of this document, those features will be in double quotes when we are talking about higher dimensional analogies. The little 2D colored stickers of the original puzzle are replaced by 3D little 3D colored boxes. The original 2D stickers on the face of a solved cube were arranged in a 3x3 square array. In the 4D version, the 3D "stickers" are arranged in 3x3x3 cubic "faces". Both puzzles are solved when all sticker on the same face are the same color, and both puzzles start in their solved states. Q: When running MagicCube4D, only one of the "faces" is really a cube. Why are the other "faces" distorted? A: The distortion is due to the perspective projection of the 4D "faces" into 3D. They are distorted for the same reason that the square faces of a 3D cube are distorted when projecting them onto a 2D screen or photograph. In a photograph of a 3D cube, only one of its faces can be truly square on the image. That is why only one of the eight "faces" of the 4D Magic Cube cube is truly cubic. Q: If there are eight "faces" in a 4D Rubik's cube, then why do I see only seven when I run MagicCube4D? A: Notice that you can never see all six faces of a 3D cube at the same time. The display in MagicCube4D is similar but different. The missing eighth "face" is really the one closest to the viewer in 4D, but the distortion of its projection into 3D turns it completely inside out. It could still be drawn, but it would overlap most of the other geometry. The view that MagicCube4D gives you is more analogous to looking into a box with the lid taken off. The cubic "face" in the center is the smallest because it's really the one furthest from the viewer, and is therefore analogous to the bottom of an open 3D box. Q: I can turn a real cube around so that I can see the hidden faces, can I do something similar to see the invisible eighth "face"? A: Yes. If you hold down the control key and click either mouse button on any part of a "face", the puzzle will "rotate" in 4D until that "face" is in the center. That "rotation" will bring the invisible face into the same position as the one you clicked on. The "face" on the opposite side of the puzzle will "rotate" out until it turns inside-out and becomes the invisible "face". This "turning inside-out" motion is very typical of 4D "rotations". Notice that control-clicking either mouse button on the central "face" does nothing because it's already in the center. It's important to notice that rotations never affect the state of the puzzle, they just let you look at the same puzzle from different angles. So "rotating" a solved puzzle (in 3D or 4D) will always leave it in its solved position. Only twists will affect the state of the puzzles. Q: So what does it mean to "twist" on a 4D Rubik's cube? A: People generally think of twists in 3D as turning something about an axis. It's just a quirk of three dimensions that that makes any sense, and is no help in the general case. It's better to think about a ninety degree twist as follows: take the face you want to twist and remove it from the larger object. Turn it around any way you like without flipping it over, and then put it back so that it fits exactly like it did before. On a 3D Rubik's cube, there are therefore only four possible ways to put the face back on. With a "face" of a 4D cube, it's like taking a cube out of a box, turning it any which way (but not turning it inside-out), and putting it back in its box. There are 24 different ways to do this. Q: How do you perform a "twist" on a "face" in your program? A: Notice that each 3x3x3 "face" can be thought of as 26 little "stickers" surrounding a 27th one. If you click on any of those outer "stickers", that whole face spins about the axis that goes through the center of that "sticker" and the central one. It spins until it's back in the same orientation that it started in. So if you click on a sticker which is in the center of one of that "face's" 2D face, it will take four twists before it is back where it started. Likewise, if you click on one of the corner "stickers", it will only take three 120 degree twists before it comes all the way around, and if you click on an edge "sticker", it will only take two 180 degree twists. Using the left mouse button twists counter-clockwise, and the right button twists clockwise. Q: OK, so clicking a "sticker" on a "face" twists that face into a new position without changing it, but why do some 3x3 slices of other faces spin or fly onto other faces? A: That is the scrambling (or unscrambling) effect of twisting a "face" on a 4D Rubik's cube. Notice that a twist on the original Rubik's cube doesn't change the state of the stickers on that face, but it does affect the state of adjacent faces. Notice also that the "faces" and "stickers" of the puzzle are separated from each other by gaps. In a real 4D Rubik's cube (if that makes any sense), all the "faces" and "stickers" would be slammed together. The view we present is simply an exploded version of the real 4D puzzle so that you can see the internal state. It is a good idea to imagine how they would slam together, because adjacent "stickers" on adjacent "faces" are permanently stuck together just like pairs and triplets of stickers on the original 3D Rubik's cube are permanently stuck together on the outer 27 plastic parts. Q: Why do the resulting twist animations look so different when performed on one "face" as opposed to another? A: This is due again to the perspective distortion of the 4D object into 3D. It's best to practice twisting only on the central "face" for a while because none of the twists on that face cause any distortions. Once you know exactly what each click will do on the central "face", try the following exercise: 1: Perform a single twist on the center "face" of a reset puzzle. (Select the "Reset" item under the "Options" menu first if needed.) 2: "Rotate" that "face" so that it's no longer the central "face". (control-click on one of the non-central "faces".) 3: Try to untwist the twisted face with a single mouse click. The right "sticker" to click on will be the same one that you clicked on before, but now it's in a new position. You will also need to click with the other mouse button to make it twist in the opposite direction. Watch how it animates back into place. After trying this a few times you get a good sense of what is happening. It's also good at first to only try the 90 degree twists (i.e. clicking only on "sticker" at the centers of the 2D faces). Another useful exercise is to first perform a twist on one of the non-central "faces" of a reset puzzle, then "rotate" that "face" into the center (control-click it), and finally try to twist it back into the solved state from there. Q: Sometimes it twists in ways that I didn't want. What am I doing wrong? A: Because there are so many "stickers" packed close together, it is easy to be a little bit off and to accidentally click a different one behind the one you expected. it is very important to have the tip of the mouse pointer be exactly on top of the "sticker" you are trying to hit. It may help to click the "maximize" button on the window so that it expands to full-screen. If that doesn't help, then go back to practicing only on the central "face". You can always undo a move by hitting control-Z, or via the Undo or Take-Back menu options, or simply by "twisting" on the same "sticker" in the opposite direction. Q: How do I solve the puzzle? A: You first need to scramble it up, and then perform twists until all "stickers" of each "face" are the same color. To truly solve the puzzle, you must first select the "Full" item under the "Scramble" menu. The first time you try that, it will be a shocking mess. It's a truly difficult job to solve it from a full scramble. If you ever do succeed, you will be one of a *very* elite group of people. You will almost certainly need to have previously mastered the original Rubik's cube before you can hope to solve this one. Luckily, all of the skills you learned for the original puzzle will help you with this one. Q: If it's so hard to solve, then why should I even bother with it? A: You don't need to ever solve the full puzzle to enjoy it. One fun game is to choose less than the full scramble and try to twist it back to the solved state. First master solving it starting from one random twist. Then work up to two, three, and more. Each higher level that you actually solve even once makes your skills much more impressive. Another fun thing is to fully scramble the puzzle, and then use the Options/Solve menu item and then watch as the puzzle solves itself. Finally, it's a nice thing to simply have some experience manipulating a four dimensional object. We hope you enjoy MagicCube4D. If you ever do solve the full puzzle, then please save and send us your MagicCube4D.log file. Also, feel free to send us any comments or suggestions you might have about the program. Finally, please feel free to share this puzzle with anyone you like for their personal use. For all commercial purposes including using the program as free demos to help promote a company, product, or service, you must license it from us first. Melinda Green - melinda@superliminal.com Don Hatch - hatch@hadron.org Jay Berkenbilt - ejb@ql.org