From cube-lovers-errors@mc.lcs.mit.edu Tue Mar 16 15:39:51 1999 Return-Path: Received: from sun28.aic.nrl.navy.mil (sun28.aic.nrl.navy.mil [132.250.84.38]) by mc.lcs.mit.edu (8.9.1a/8.9.1-mod) with SMTP id PAA02419 for ; Tue, 16 Mar 1999 15:39:50 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Message-Id: <36EB0B28.1169@ameritech.net> Date: Sat, 13 Mar 1999 19:04:40 -0600 From: Hana Bizek Reply-To: hbizek@ameritech.net To: cube-lovers@ai.mit.edu Subject: parity pairs References: <36E7D3A7.1796@ameritech.net> > [Moderator's note: By parity pairs, I rather suspect he means mirror-image > pairs.] Let me tell you what I mean by parity pairs, why very few have probably heard about this concept and why they are crucial in 3-dimensional (3-d) cube art. Suppose one has two cubes of identical color scheme such that the color on both cubes' up, down, front and back faces are exactly the same. If the color of the left face of one cube is identical to the color of the right face of the other cube, such a pair of cubes is said to form a parity pair. The color scheme is still identical, but the ORIENTATION of the faces is reversed for one of the members of the pair. One cannot obtain parity pairs by conventional cube manipulations, but must obtain them either from the manufacturer, or switch the faces themselves manually. I would prefer to buy such pairs from the toymaker, for it pains me to tamper illegally with those stickers. I have devised a simple algorithm to do it as painlessly as possible, but it still is a pain. But will a manufacturer sell me parity pairs? The reason so few people know about parity pairs is that such pairs are moot in solution algorithms. You do not need to concern yourself at all with parity pairs, you just have one cube and painlessly solve it. Ditto for 2-dimensional (2-d) designs (unless you treat them as lxmx1) designs. However, they are essential in 3-d cube art. They are responsible for reflection-equivalent designs, designs of fewer than six colors and ultimately fractal design prototypes. They also determine special symmetries in a 3-d design. They are the cornerstone of 3-d design theory. Without their presence all of the 3-d designs I have constructed would not be possible. Why all this self-serving fuss about parity pairs and 3-d designs? The point is this: given four parity pairs, one can construct a 2x2x2 larger clean design, that has three colors only on its six faces. The internal faces that touch are colored the same. Those colors are hidden inside the design or suppressed. Such an array of cubes, when used as corners, produce, e.g., reflection-equivalence in a design. Go to your cube collection, extract four parity pairs and see for yourselves. So I think you got the idea, Mr. Moderator. Just one slight correction; I am a "she," not a "he." You will find this almost incredible, but women too, love the cube. Hana Bizek (female) physicist and 3-d Rubik's cube designer [Moderator's note: On the contrary, there are several women on cube-lovers, and Dame Kathleen Ollerenshaw is well-known as one of the earliest writers about Rubik's cube and one of the first victims of Cubist's Thumb. I just didn't know that "Hana" was a woman's name, and I had forgotten that this information was presumed by a mention of you in the archives. I apologize for the oversight. As for nomenclature, the reason no one knows about "parity pairs" is that the term is ambiguous--"parity" could refer to representatives of any even division of a set into two parts. If you wish to enable people to know what you mean without going through your somewhat confusing description, then you should use the term "enantiomorphic pairs", "chiral pairs", or "mirror-image pairs". I believe these are the standard terms used by chemists, physicists, and everyone else, respectively. There is an interesting question, though, which your hobby may give you a particular ability to answer. According to _Rubik's Cubic Compendium_, the most common color scheme has red opposite orange, blue opposite green, and white opposite yellow. This permits two mirror-image color schemes, distinguished by whether red, white, and blue go clockwise or anticlockwise around a corner. The question is whether there is a tendency for one of these schemes to predominate, and if so, which and by how much? For instance, one enantiomorph predominates extremely strongly in the manufacture of dice, though I don't know why. --Dan ]