From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 9 13:47:36 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id NAA29037; Wed, 9 Sep 1998 13:47:35 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Sep 6 23:27:56 1998 Message-Id: <35F350B1.626F@ameritech.net> Date: Sun, 06 Sep 1998 22:19:13 -0500 From: Hana Bizek Reply-To: hbizek@ameritech.net To: cube-lovers@ai.mit.edu Subject: Rubik's cube kingdom Hello, cube-lovers, the following miniessay talks about that realm of human knowledge, where the rubik's cube reigns supreme. The gates of this kingdom are open to anybody, but only those who love the cube, venture beyond its gates. The rest of humanity are either unaware of it, or ignore its existence. Those folks are truly missing a lot. After finishing {if indeed you choose to finish} this epistle, you may want to do several things: a} emit a few chuckles thinking what an insane idea this is b} pause to think about the contents, debating whether all this is worth your precious time c} email your ideas, commenrts, etc, to me at hbizek@ameritech.net Thank you. WELCOME TO THE RUBIK'S CUBE KINGDOM. This is not a kingdom of people, it is a kingdom of ideas. Its king is the Rubik's cube. We pay homage to our king by trying to learn more about him and see if this knowledge could be extended to other areas of human pursuit. I am now going to tell you what I think those areas are and point to a couple of web sites where results may be found. This is by no means a finite list. As new ideas occur to all of us, they should be added to the kingdom. The Rubik's cube kingdom is there for anyone to benefit from, just as any other field. Please feel welcome to sample and browse. Initially, of course, one should master the Rubik's cube solution. The ability to solve one side should be the absolute minimum. It is far better and morre satisfying to be able to solve the cube completely, that is, get all six sides a solid color. I have seen numerous solution algorithms on the web. Someone might consider collecting those varied algorithms in a handy volume for cube lovers and others to use. I have seen some solutions on the web, in which you are presented with a solved cube in a little square field. You are instructed to press 's' to scramble the cube and 'r' to restore {i. e. solve} it. That is not solving the cube! You have to understand the steps of the solution algorithm. The areas where the cube has any impact are art, mathematics and science. Let me look at art first. Quite recently there was a small item in TIME magazine stating that the cube has entered Hollywood and is the subject of some movies. I heard that there are also songs about the cube. By the above definition those human expressions too belong in the Rubik's cube kingdom. However, I am going to zero on two aspects: pictures and sculptures. At http://www.wunderland.com/WTS/Jake/CubeArt one may see creations by the people at Wunderland company {the spelling is correct} that depict mostly 2-dimensional picture-like creations from a set of many Rubik's cubes. They just show the pictures, thwy do not describe their method in any book, as far as I can determine {of course, I can be wrong}. But from my observation it seems to me they need to be worried about continuity from one side of the cube to the one side of the next cube, which is not too complicated. The sculptures are 3-dimensional structures and require some symmetry on all the faces of the cube, simultaneously. In this case, the complete solution of the cube is a must. The required algorithm is described in the book, "Mathematics of the Rubik's cube design," written by me and published last year. As far as I know, I am the author of these designs, described in my previous postings to the cube lovers. I have stated a web site where three of these designs may be seen. I repeat it here for completeness: http://www.ssie.binghamton.edu/~jirif/hana1.html Next, I am going to talk about science. I have to warn you: those ideas are, as far as I know, unknown and undeveloped, as is, indeed, the design Problem itself. First on the agenda is fractals and fractal design prototypes. Some of the designs in my book, such as the Menger sponge, are such fractals. One can think of the Rubik's cube as a three dimensionl version of a Cantor set, which is a {one dimensional} line. Actually these fractals are neither three nor one dimensional; they have fractional dimensions. But the 0th iteration are. One can formulate rules for geometrical fractal iteration. Remember that iteration preserves fractal dimension. By the same token, for integer dimension, it really doesn't matter if you subdivide by m or 10000n; the dimension is always the same. Between the integer dimension there are fractals of fractional dimension. These fractals can be reached by breaking up of integer dimension or by some other manipulation. Fractal design made from cubes suggest one such manipulation, as witnessed by box fractal. But this is supposed to be a miniessay, nor a book. Another idea consists of computerizing the design algorithm, in some comprehensive way so that the Rubik's cubes are used as 3-dimensional cellular automata. It would be deliciously complicated game, perhaps employing some of the patterns in the design theory. Finally there is the question of what happens if internal combined faces of a design that touch are colored the same? I will let you figure that one out. It is not too hard. Personally, I had much more difficulty to properly formulate the question than to provide the answer. Finally, there is math. In this respect the ideas were formulated partially by mathematicians in the 1980s during the heyday of the cube, partially by cube lovers today. I would like to include my book in this category. At least its title indicates that there is some math, if that has to be the only reason. And, as every cube lover knows, all possible elements of a Rubik's cube form a mathematical group. A visitor to the Rubik's cube kingdom will surely encounter some joys of group theory on his travels. Any objections, criticism, etc are welcome. Free speach prevails in the Rubik's cube kingdom. Suppress free speach and not much is left. Hana [ Moderator's note: I am somewhat concerned at the low information-to-woowoo ratio of Hana's "miniessay", but I think there are enough real ideas there that I've passed it on to the list. I must, however, note that while "free speach" may prevail on the Internet, the contents of the cube-lovers mailing list is subject to editing for topicality, format, sensibility, and content. Which is to say that if the silliness level gets too high, you may have to find somewhere else to make your "kingdom". I encourage guidance from the readership on where to draw the line; send your opinions to cube-lovers-request@ai.mit.edu. --Dan ]