As a Materials Science major with strong background in crystallography I would love to see more questions on the subject which is why I would like to try and clarify matters here. There is indeed quite a distinction between a crystallographic point group and space group. The question mentioned two particular groups, 2/m and T sub h. Both of these are point groups ONLY. Point groups refer to the symmetry elements present at a certain point in space. Consider a simple cubic lattice structure, i.e. one atom on each corner of the cube. Each corner of the cube has the same point group as neccessitated by the translational symmetry of any crystalline lattice. There are even more point group symmetries present at other points in the unit cell (such as the middle .5,.5,.5 coordinates and at points such as .25,.25,.25). To put it simply, a crystallographic space group refers to all the point groups created when translational symmetry is introduced to a particular atomic arrangement, which itself may have its own point group. There are 230 of these space groups while only 32 point groups. For the simple cube with one atom on each corner, the space group is Primative Cubic (P m 3 bar m I think), while each corner of the cube probably has some point group which shows a 4 fold axis, 3 fold axis, mirror, etc. through that point. The distinction becomes more pronounced when you consider groups of really high symmetry with really complicated repeating motifs. I guess the easiest way to put it may be that a point group is in the reference frame of a point in space, while a space group is relative to a particular unit cell. I suppose both would have been acceptable before the specific examples were given since both can be notated as mentioned in the question. Some of what I wrote may be oversimplified as I am writing this off of the top of my head from how I justified it to myself, but I just wanted to point out there difference between the two definitions. -Peter Stone, MIT --- In quizbowl_at_yahoogroups.com, matt979 <no_reply_at_y...> wrote: > The protest... > > (Before going into detail I want (speaking for myself) to thank R. > Robert Hentzel for his thoroughness, and also to thank David Farris > for handling the protest as best a protesting player can: Remaining > calm, courteous, and deferential throughout, he was able not only to > explain very precisely what his contention was but also to suggest > useful research angles. If there were handbook on how lodge a > protest with class, this would be a prime example. Anyhow, those > kudos -- and my explaining of my part of the research -- were what > motivated me to post this myself rather than deferring to R.) > > Berkeley A and Michigan A faced each other in the first round of > seeded play, as #2 versus #3. Pending protest resolution, the clock > ran out in the second half with Michigan A ahead, 300-295. > > If you don't care about mathematics, the short version is that after > some research, we concluded that a question needed to be thrown out > and replaced off the clock. By the time we concluded this, with all > other results in, it was clear that this one question would not only > decide the match but also literally settle which team would be in the > final. Both teams strongly suggested that, under the circumstances, > playing a full game would be much more equitable than letting one > question determine their fate. They did indeed play that game; > Berkeley won. > > (For what it's worth, Berkeley A had also defeated Michigan A in a > Saturday morning match that was very high-scoring and neither close > nor a blowout.) > > The question (typos, if any, are mine from retyping it): > --- > They can be designated by Hermann-Mauguin [MAW-gan] symbols, like 2- > slash-m, or Schoenflies symbols, like [T sub h]. While mathematics > permits an infinite number of them, the three-dimensional structures > of crystals forbid (*) symmetry axes of order higher than 6, leaving > just 32. For 10 points--name these collections of operations that > leave at least one atom in a molecule, or a single point in space, > unchanged. > > answer: (crystallographic) _point group_s (accept _space group_ > before "infinite"; do not accept "crystal class") > --- > > A Michigan player took an early neg with "tesselation"; once the > question was complete, a Berkeley player rang in and > said "crystallographic groups." On being prompted, he > said "symmetry" and was ruled incorrect. > > Berkeley's claim was that the answer "crystallographic group" was > sufficiently complete and correct that it should have been accepted > rather than prompted. Michigan's counterclaim was that "tesselation" > was correct at the point of the buzz; if their protest was accepted > then Berkeley's protest would be moot. > > The protest committee decided to send teams to their next round and > do research on whether "crystallograhpic group" would be acceptable. > In practice that fell to me, using the stat room's Internet > connection, at the temporary expense of updating results/stats. > > For various (obvious) reasons I wanted to get both a conclusive > answer and as quick as possible an answer. > > If "crystallographic groups" were commonly used as a synonym > for "point groups", I supposed that there would exist a resource > listing the two as equivalent. If there is, I was unable to find it. > > On the other hand, suppose the term "crystallographic groups" > referred not only to "crystallographic point groups" but also to > groups that don't fit the clues of the question. I imagined that I > could find a counterexample -- say, a paper that referred > to "crystallographic groups" with an "axis of symmetry" higher than > 6. Having found such a paper (it seemed to treat "crystallographic > groups" as a synonym of "space groups"), I was prepared to deny > Berkeley's protest. > > R., wanting better closure than that, did a Google search that the > Berkeley team had suggested ( http://www.google.com/search? > hl=en&ie=UTF-8&oe=UTF-8&q=32+%22crystallographic+group%22 ). > Although I had run this same search and given up after finding > nothing useful in the first few links, R. delved deeper and found > reasonable, if not overwhelming, evidence that some mathematicians do > simply say "crystallographic group" rather than "crystallographic > point group." > > Discuss as motivated... :-) > > (Also feel free to send feedback to feedback_at_n... about anything > ICT-related.)
This archive was generated by hypermail 2.4.0: Sat 12 Feb 2022 12:30:47 AM EST EST