# Talk:Isometry Groups

## Contents

## Major pass through this page

### Intro work

Got rid of the optional tag.. I suppose this section is a little daunting, and I admit I don't have my students read it, but it's not really any more or less optional than many of the other sections.

I'm trying to work this section into the flow a little better. I think the key point is that we're moving from studying the symmetries of a static figure to studying the group of motions. I tried to make that point clear in the intro section. I thought the ties to classification in bio were out of place here, they show up in the Fundamental Concepts section. Though I think it would make sense to recap that analogy somewhere, possibly the end of the Wallpaper Groups section. Bryan 13:07, 3 February 2009 (CST)

### Isometries section

Trying to stay consistent that isometries move the plane and avoid talking about objects in the plane. Illustration in Visions of Symmetry page 34 is not Escher's work, so I took that reference out. Clearly, we need an illustration here, though.

Added more detail on helical symmetry.

Did some rewriting of the isometry vs. symmetry section. Added some pix.

Bryan 13:53, 3 February 2009 (CST)

### Abstract Groups

I suggest we're either going to do a solid formal definition of a group or we're not. The current state has the four properties of a group, but since it avoids notation and lacks motivating examples, to me it's a long way from an introduction to real group theory. I'm going to attempt to re-write this section as a less formal introduction to group theory. In particular, I'm discarding the 'closure' law, because it's just getting in the way of the ideas.

Also, I'm moving the 'definition' up front. The examples-first approach wasn't well motivated. I don't think examples-first was the wrong plan - but I think we'd need to do a lot more work on it. For example, we could do the examples and then argue that they have a lot in common, then motivate the def of group. But as it was, it went: Here's Z, and it's a group because it satisfies these four properties. Now here's Q and it's a group because it satisfies these four properties. Now heres... I didn't want to rewrite those parts too much, so I put the definition first. Hope that's ok, though probably not the best solution.

The rationals needed to be re-done as the positive rationals (or it's not actually a group, natch!)

### Symmetry groups

I did quite a bit here. First, it's a section of examples, and it always was, so the headings reflect this. The pictures and discussion of D2 and D4 were always the best part of this whole chapter. They didn't change much. I didn't work much on the frieze pattern example, but it's got problems - mostly it's that there's no clear distinction between the group itself and its subgroups. It's easy to get the impression that the group is 'cyclic', when in fact it's just the translation subgroup that's cyclic. I think it's important to do an example using *p111* to get the translation part figured out, then do this fancy one.
Bryan 16:54, 3 February 2009 (CST)

## restructuring

I moved the isometries section out of Frieze patterns and into this page. The distinction between isometry and symmetry is subtle, but let's make it clear - symmetry of a figure, isometry of the plane. Since the Rosette, Frieze, and Wallpaper sections are written with the view "here's a figure, let's find it's symmetries", it seems weird to introduce isometry at that point. On the other hand, we're applying isometries all the time, and figures are preserved under isometry. So that's kind of a problem. I never use the word isometry when I teach this.. Anneke, I assume you work it in from the start? It's going to take some work to work out a form that works well.

For the immediate moment, the isometry section I moved here has a major problem: it's talking about the four types of isometry before they've been introduced/discovered by the students. In that sense, it works better here. It would be ok to stick at at the end of the Rosette page with the section on coloring/perfect coloring and the one on symmetry in science. We'd just have to focus on rotations and reflections. All three of those sections (color, science, isometry) are sort of optional. I wonder if we can organize better so the main flow is easy to follow and these extras are accessible but easy to include or skip. Bryan 21:55, 29 January 2008 (CST)

This page needs some work I think, but I put up the materials I had written up in the past. I moved some of the questions into explorations. I expect that these are not like the other explorations. This is the one section where I would want to lecture and have some class discussions before jumping into the explorations. --Barta 14:25, 6 September 2007 (CDT)

## math notation

We cannot use R2 for <math>R^2</math> because we've already established the convention of M1 and M2 meaning unrelated group elements, not powers of a group element M. I first tried R^2, but that's ugly and doesn't convey the correct notion that this is really a form of multiplication; for that reason I changed to mtath mode, <math>R^2</math>, using the LaTeX button (<math>\sqrt n</math>) on each group element with a power. This has the additional effect of putting the letter-name into italic font. That may perhaps be a good notational idea, but it means that *every* group element, with or without a power, must be math-moded. That's a lot of button-pushing, unfortunately; but maybe there's not much more to do than this one section. Furthermore, we can't just math-mode entire lines (such as several expressions strung together with equal signs), because (1) that removes the spacing around commas, and (2) we have the unsatisfying choice between having the mulitplication symbol come out as <math>x</math> or (replacing x by \times) having the entire line come out in a font many times larger than the rest of text: <math>M1 \times M2 = R</math>.
So it's a pain to LaTeX-button each group element individually, and it makes for hideously ugly input; but I don't see an alternative at this time.

Steve 22:03, 23 September 2007 (CDT)

I talked about this with Bryan and his feeling is that there are likely software developments "around the corner" that will straighten out the math formatting problem. I agree with you that some of the larger math notation in the text looks rather ugly. It may not be the best use of our time to be doing these esthetics though? I will leave that up to you.

- I expect that, "in time", software developments will improve the situation, but I hope I didn't say "around the corner", unless that corner is in a different state :-) It could be years before MathML is on Joe User's computer.Bryan 15:16, 29 September 2007 (CDT)

The comments about R^2 vs R2 you made are valid. I like the changes made. I think it may prevent some confusion. --Barta 13:17, 26 September 2007 (CDT)