From cube-lovers-errors@mc.lcs.mit.edu Fri Sep 12 17:53:00 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA20354; Fri, 12 Sep 1997 17:52:59 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Fri Sep 12 17:08:04 1997 Date: Fri, 12 Sep 1997 17:07:47 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: isoglyphs In-Reply-To: <199708182216.SAA00604@sun30.aic.nrl.navy.mil> To: cube-lovers@ai.mit.edu Reply-To: Jerry Bryan Message-Id: On Mon, 18 Aug 1997, Dan Hoey wrote: > A "chiral isoglyph" is one in which the handedness of the glyph is > taken into account in testing for isoglyphy,* so that the glyph > appears only in one variety. > > Mike used "achiral" for an isoglyph that fails to be a chiral > isoglyph, though I would tend to use "non-chiral". I would rather use > "achiral" for a situation that lacked chirality, as in an isoglyph of > a mirror-symmetric glyph. Let me see what I can do to muddy these waters. It seems to me that we might ought to consider the chirality of an isoglyph as being a different issue than the chirality of a glyph. I think the two are clearly related, but I am not sure that the one necessarily derives from the other. As to a glyph, it seems to me that a glyph is chiral only if conjugating the position by each of the four reflections of the square yields a different set of positions than does conjugating the position by each of the four rotations of the square. Hence, you can have a glyph which occurs in right-handed or left-handed forms, or one that doesn't. This is the simple part. I think the situation with isoglyphs is a little more complicated. For example, form an isoglyph using both the right-handed and the left-handed forms of a chiral glyph. You might have 6 right-handed glyphs and 0 left-handed glyphs, 5 right-handed glyphs and 1 left-handed glyph, etc. If there are unequal numbers of right-handed and left-handed glyphs, then it seems natural to define the handedness of the isoglyph as being that of the dominate glyph. But what if there are three right-handed glyphs and three left-handed glyphs? Up to symmetry, there are only two ways to partition the six faces of a cube into two sets of three faces. For example, the F, U, and B faces can be of the same chirality, or the F, U, and R faces can be of the same chirality (or any conjugates of these choice of faces). In the first case, the cube is partitioned like a universal joint, or maybe like a cubic baseball. Such a position seems to me to lack chirality. In the second case, three faces with the same chirality cluster around a common corner. Again, such a position seems to me to lack chirality. So an isoglyph which lacks chirality can contain chiral glyphs. On the other hand, even on an isoglyph consisting of three right-handed and three left-handed glyphs, you still might be able to find a distinguishing characteristic of the right hand part that was different from the left-handed part. For example, the glyph boundaries which were internal to the right-handed part of the isoglyph might be continuous whereas the glyph boundaries which were internal to the left-handed part of the isoglyph might not be continuous. Or for another example, the rotations of the three right-handed faces relative to each other might be different than the rotations of the left-handed faces relative to each other. (By the way, I have not verified that any of these positions I have described are actually in G. I guess I am thinking in terms of the constructible group of the facelets -- conceptually, peeling all the facelets off and reattaching them.) On the other hand, two glyphs which lack chirality when placed side by side can be chiral. For example, XOOXXX (the base glyph is XXX XXXOXO OXO XOOOXO OXO ) I really haven't thought through the implications of using six glyphs instead of two, but it seems to me quite likely that an isoglyph could be constructed using six glyphs which lack chirality and which are the same pattern, and where the we could attribute chirality to the isoglyph as a whole. I have thought about this in terms of Herbert's Cube Explorer 1.5 program. The pattern editor has a check box for continuous. If you don't check the box, the program finds both continuous and non-continuous isoglyphs. If you do check the box, it finds only continuous ones. So I have considered what would happen if the program had a check box for chiral. What should it do? The obvious thing would be that in normal operation, it would consider conjugates of both rotations and reflections of the square when building an isoglyph from a glyph, but that if the chiral box were checked it would consider only conjugates of rotations of the square. But is that sufficient to satisfy our various definitions of chiral, achiral, and/or non-chiral? I'm not sure. Maybe Dan or Mike would be kind enough to clarify further their thoughts on this issue. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990