### My Cube Insanity

Long, long ago, during those salad days, when I was finishing my dissertation, a friend entered my office at the university, and placed an unsolved Rubik's cube in front of me, commanding: "Solve this!" And that is how my personal cube mania started.

Now it would be suitable to say that I picked up this cubical thing, and to the amazement of my friend, solved it in five seconds. Maybe it would be suitable, but it would be a rotten lie. The truth is, that at that time I wasn't even able to master one side. I stared at the cube and said to myself, boy, will I EVER be able to put it together.... And I wasn't able to put it together. Another friend of mine taught me his algorithm, which HE has formulated. I remember, when, in about a month, I solved the cube for the first time. I was again sitting in my office, a piece of paper with my friend's algorithm in front of me, and a solved Rubik's cube in my hand and whispered: "I did it..., I did it..."

This was a time of cube mania. People bought the cube only to discover, that its solution is not as simple as it should be. Books were written about the cube. World-class competitions were held, as to who can solve the cube the fastest. The champion of what was then Czechoslovakia in "speed cubing" was J. Fridrich. J. Fridrich was placed tenth in the world. Much, much later we got to know each other on the Internet, and still later we met personally, when J. Fridrich came here for a conference. In effect J. Fridrich was the first to introduce my designs on the web in "Fridrich's speed cubing page."

Besides books on solution algorithms and scientific/mathematical treatises, there were books on how to destroy the cube. Those books were motivated by people's frustration with the relative complexity of the solution algorithms. These authors stated that the Rubik's cube is a Hungarian horror, which can be solved only by a chosen few (this is a sheer nonsense... after all, I myself can solve it) and the rest of humanity should not even try. I was browsing through one such paperback, when a cartoon of a bricklayer building a wall of messed up Rubik's cubes caught my eye. Suddenly an idea struck me; why not build such a wall, but instead of using messed-up cubes, use patterned cubes and create an over-all pleasant design? I ran to my growing collection of Rubik's cubes and started twiddling. At first I created 2-dimensional designs (pictures) but I soon got bored. This is essentially solving one side of the cube, and therefore below standard. Three-dimensional designs (sculptures) were more promising. And so I became a "sculptress."

Of course, a man does not live by Rubik's cubes alone. Life goes on around us.I finished my doctorate and left the university. It was necessary to find work. I chose research. At that time, by pure coincidence I stumbled across parity pars. Those are pairs of cubes with mirror reflection. For this stroke of luck I ought to thank the cube manufacturers, who scrupulously adhere to the color scheme, but don't care about reversing opposite sides of one member of this so-called parity pair. This is my, probably incorrect, terminology. If a better word is found, I will gladly use that word. Existence of those pairs forms a cornerstone of the theory of three-dimensional designs. We can construct a small 8-cube "design" having four parity pairs, which has only three colors on its six sides. The reason is that the cubes can be arranged so that internal faces that touch are colored the same. Since internal faces of a design are not seen (they are, after all, hidden inside :-) the design) only three colors are left for the six external faces, because no cube has more than six colors. This color suppression enables us to create designs that have fewer than six colors. The minimum number of colors a design has is three, but of course one has six-color designs with parity pairs. Such pairs are usually corners of a design, but designs exist with special parity-pair requirements. I will not further discuss such requirements here, but will ask the reader to figure out these concepts. It isn't that hard.

My profession took me to Texas, to one of its universities. One of the mathematics professors, a cousin of a physics professor, wanted to see my designs. I had shown him a four-color design which I really call ctyrrad. The professor examined this design, and said that he has never seen anything like this before. I cannot reproduce his next remark, because I would turn red. To make a long story short, he liked my cubes. If I would just write a book about this work, and gave it some mathematical foundation, it might be a very interesting subject. I laughed heartily. I and an author of a book, that would be truly a great joke! And about something that no one has done before, I, such an ordinary person, that would be even greater joke! Later, when I thought about it in my apartment, I came to the conclusion, that it might be a good idea. But the work involved! I let it pass. I had other things to worry about.

I started working on my book much later, after I realized that it might be of some academic value. Several factors contributed to my decision. First and foremost, I was still unable to find anyone who could do this. It would not be correct to say that no one thought of using the Rubik's cube as an art medium, but those people are creating mostly 2-dimensioal, picture-like structures, a genre which I abandoned long ago, because I found it dull and easy. I don't think those artists fully comprehend the role of parity pairs, and its tremendous impact on 3-dimensional design theory. But maybe I am too harsh on them. When I started working on my book, I did not yet have my own PC and was unaware of the Internet and its great potential for disseminating information. Another factor was a positive reception of my "cubes," particularly from Dr. David Singmaster and Sir Roger Penrose. Dr Singmaster is a mathematician, an expert on the Rubik's cube and author of several books on the subject. And Sir Roger Penrose is a famous name that needs no introduction. Dammit, I said to myself, if such outstanding mathematicians actually like my "cubes," they cannot be completely without merit. And why I am still unable to find anyone who can do it? I wrote my book in parallel with my work, and because I do not have the ability to work simultaneously on the cubes and computational accelerator physics, the cubes had to step aside. Evenings, weekends, holidays and paid vacation time was devoted to the book, for seven years. I either worked on accelerator physics or the book. Finally I finished my book and was thinking about its publication. I was advised to try to get an author's copyright. They investigate the originality of the submited work, I was told. I recognized an excellent opportunity to verify the originality of the design theory. If someone else did this, then I would probably get a letter advising me of the fact and denying me the author's cpyright. No such letter came, instead, I was granted the copyright. Does that verify the originality of the "cubes?" I would like to think so. Later I received a letter from Dorrance Publishing Co. I never heard of them before and did not communicate with them. It seems that manuscripts with author's copyrights are somewhere on the shelves of the Library of Congress. Publishing houses, always on the lookout for new ideas, send their people to look around. One of those "seekers" spotted my manuscript and told Dorrance about it. The publisher contacted me. One thing led to another and my book was published in 1997.

There was one more coincidence. I read a rather easy book on chaos and fractals. The book described a so-called box fractal and illustrated its iterations. Oh my God, I stared at the picture, this is my "checkerboard" design, only its edge cubes are solved! And the edge cubes themselves form a fractal, but then the corners and centers are solved. But then the checkerboard itself is a combination of two simple fractals... but this line of thought is no longer twiddling the cube for fun and relaxation, but a possibility of applying the cube in the theory of dynamical systems, which is very important part of modern scientific research, including stability of dynamical systems. No art, no designs, but a cold reality of science... STOP. I am already over the edge.

What do I want for these cubes? I would like a cube mania reminiscent of the 80s. But a different, more intelligent form of cube mania. We should renew the competition in speed cubing, for those who are interested. We have Olympic Games to accommodate other sports. But parallel with this activity we should also develop applications of the cube in other areas of human pursuit, in the manner I sketch here. Maybe this development will generate completely unpredictable ideas. But we need to mobilize our resources. Above all, we need people, who not only know how to solve the cube, but who also understand mathematics, physics and computers, with appropriate university education and training. Ladies and gentlemen, computers will play a crucial role in this investigation, that I guarrantee. I will gladly participate in this research effort.

I wish all a lot of enjoyment with your Rubik's cubes. HANA :-)))))