Point reflections, take 2

I was so distracted by never having heard the term “point reflections” before that I forgot to explain in my last post why I think it’s a bad term, and why in particular I think it’s a bad term for middle schoolers.

It’s very common in math that we start out with a very simple concept (like a line in 2-dimensions) and generalize the hell out of it (line in n-dimensions, hyperbolic line, geodesic, affine line over a finite field, projective line, complex line, etc.) in ways that middle schoolers would find confusing. We then end up using the word “line” differently in different subfields of math, and that’s ok. This generalize and rename thing happens again and again at higher levels of math (say integer/algebraic integer vs. rational integer/integer), and this is also fine.

What’s unusual and pernicious is to first introduce a concept in weird generality. Yes it’s great to think about complex reflections, “reflecting about a circle”, affine reflections, etc. when the time is right. But it’s also important to have an intuition for what you’re generalizing, and in middle school that should be reflections about hyper-planes in 1, 2, and 3 dimensions, with almost all attention paid to 2-dimensions. Why? Because reflections in lines in 2-space as a concept is much richer than all order 2 affine transformations. By richer here I mean it makes it easier to think about interesting things.  How do you tell if a transformation is a reflection?  Thinking about this suggests handedness (intuitive because of mirrors) which is a deep and important property which is accessible to middle schoolers.  Thinking about this also subtly primes you for thinking about cosets and their relation to group actions (not so accessible to middle schoolers, but the priming is still there).  The fact that every tranformation can be written as a translation and then a rotation or a reflection and then a rotation is a really cool result that introduces key ideas which you could easily think about in early highschool (and again suggests more advanced topics like groupoids).  Etc.

Conversely there doesn’t seem to be nearly as much interesting math that’s going to come from thinking of point reflections as a reflection.  I could see some of the other order 2 elements leading to interesting questions (in particular “reflecting about a circle” is potentially interesting), but point reflections seem to be a waste.

14 thoughts on “Point reflections, take 2

  1. Well, okay, if you meant “it still seems incredibly counterproductive to me to use this terminology” in the context of middle school mathematics, then I agree completely.

  2. Yeah, middle school is what I was thinking about when I made that comment, I’m certainly in no position to judge which generalizations of reflections are appropriate in fields where I don’t have a lot of knowledge.

  3. Sorry about getting carried away with the last post – I agree with what you’re saying. When my math teacher introduced point reflections in school, I don’t think my classmates were particularly confused, but I also don’t think they found the concept well-motivated. Naturally, teaching these things carries nontrivial opportunity cost, and in our case, we never had the teacher take the symmetry concepts and run with them the way they do in certain programs like math camps.

  4. OK, two posts about this is taking it a little too far (and so is me commenting on each of the two posts…)

    1. Point reflections in three-dimensional are not orientation-preserving, so point reflections (albeit in space) can also make you think about handedness.

    2. “every tranformation can be written as a translation and then a rotation or a reflection and then a rotation”

    A translation followed by a (non-identity) rotation is just a rotation (with a different centre). A reflection followed by a rotation is just a glide reflection. You probably meant every plane isometry is either a translation, a rotation or a reflection followed by a translation.

    3. I’m not sure I get your point now. You say “I could see some of the other order 2 elements leading to interesting questions (in particular ‘reflecting about a circle’ is potentially interesting), but point reflections seem to be a waste.”

    Do you mean you think it’s a waste of time to teach point reflections in middle school because they don’t seem to you to lead to interesting questions? If that’s what you mean, that seems to be about the concept, not the naming of it.

    4. Some results about point reflections might be fun to teach in middle or high school with help of a dynamic geometry program (like Geometer’s Sketchpad or Cabri). For example, the composition of four point reflections is the identity if and only if the points are the vertices (in cyclic order) of a (possibly degenerate) parallelogram.

  5. My, my, my, an opportunity to get on a soap box. I will try to keep this short.
    Using a term without a corresponding concept in English (or Latin) for a middle schooler is probably dangerous. So even if some people use the word “reflection” for an order n operation, I don’t think I want my kids to learn that word just yet. Unless it is a professional using the word and explaining that the term is just used metaphorically, and as a weak metaphor at that.

    Introducing group theory to the same children is probably less dangerous, AS LONG AS THE TEACHER UNDERSTANDS THE CONCEPT. So rotations, translations, and reflections as plane isometries, and THEIR COMPOSITIONS sounds to me to be a good idea.

    Every middle schooler should be familiar with the cyclic groups and the dihedral groups (which they are) and have names attached to them.

    On the other hand, I have a little trouble with using the word “group” in this context though. The word means collection, and there is no common usage to distinguish among group, set, collection, or category. The term “collection of symmetries” while more syllabic seems better. To avoid the counter-rant, it is OK to think of a group as a set of symmetries on a given object as a first pass. That is what we mean when we say, “a group is a category with one object and all morphisms invertible.”

    Rather than sketchpad, I endorse using physical models. Even though you can’t control Z things, you get a tactile sense. As I write these lines, I imagine the feel of the pair of dice in my hand.

  6. There was a huge flap 40 years ago when grade-school mathematics introduced “negative” as the unary operation and “minus” as the description of a class of numbers. So that “negative 2 is minus” was correct. If all of you were brought up that way, you evidently survived.

    Carl

  7. I spend a lot more of my mathematical time contemplating point reflections than other kinds, because once they are generalized out of the Euclidean realm, it is the most natural way (to me) of thinking about symmetric spaces. Indeed, that is what led Loos (following in others’ footsteps) to the aesthetically pleasing definition of a symmetric space as a manifold with multiplication satisfying certain axioms.
    In some fields, for better or worse, “point reflection” is a standard term for this concept, where “standard” means that if one tried to introduce an alternative, a referee would probably balk. Should it be used in middle school? Perhaps not, but it all seems like a “sound and fury” issue to me.

  8. It is OK to think of a group as a set of symmetries on a given object as a first pass. That is what we mean when we say, “a group is a category with one object and all morphisms invertible.”

    When is it not OK to think of a group as a set of symmetries of a given object, pray tell?

  9. When is it not OK to think of a group as a set of symmetries of a given object, pray tell?

    I think at least some of us like to think of groups as having a life of their own, independent of the things they act on. Is A_5 really symmetries of the icosahedron or the alternating group on 5 points? Is one of these better than the other?

  10. I’m just amazed by Scott’s dismissive implication that thinking of a group as a set of symmetries is merely “OK as a first pass”.

    Of course a group has meaning as an abstract algebraic object (if abstract algebra’s really your thing), but as often as not, elegant solutions to problems in group theory come from getting your group to act on the right object.

    A_5 is both a collection of symmetries of the icosahedron and the even permutations of five points, and both of these points of view can be fruitful. I’d be amazed if you can tell me anything useful about A_5 by thinking of it as “a category with one object and all morphisms invertible”.

  11. Henry, my meaning was/is that good middle school students can handle groups. particularly, they can begin to think of a set of symmetries as having a composition rule. They can begin to experiment with groups and subgroups.

    I am of an age in which a group was thought of as an independent entity, not as a set of symmetries. However, they were discovered as sets of symmetries (of roots of polynomials).
    So *I* was taught to think of a group as the thing not the collection of things that act on a set. I was being dismissive of the attitude of full abstraction without a conceptual grounding based upon examples.

    The categorical comment was to point out: if you want to think of a group as a category with all morphisms invertible and one object, then you may let the object be a set upon the group acts. Most readers here know that.

    I think the thread was about using dangerous words for school children. The word “group” may be dangerous, but the idea of a group is not dangerous.

    The *idea* of point reflections is not dangerous for school kids, but the term might be. The *idea* of complex numbers for school children is certainly not dangerous, but the word *complex* might scare them.

    I am inclined to agree with Michael (7). It is a tale of sound and fury. People tend to get over the ambiguities of language. However, in teaching college students, I have found that I have to continually (not continuously) coach them on the proper use of language in the context of their courses.

  12. Scott,

    I apologise if I was overly forceful. And I completely agree with you that the word “group” may be dangerous. Indeed, many’s the time I’ve had to explain to, say, a sociologist or an immigration official that the free groups I study may not be the ones they’re thinking of!

    I suppose it’s just fundamental to my outlook as a mathematician that groups are symmetries, rather than objects that happen to satisfy a certain set of axioms and of which symmetries are an example. De gustibus non est disputandum, eh?

  13. Noah writes, inflammatorily:

    … point reflections seem to be a waste.

    I agree that the term ‘point reflection’ is confusing, especially for people just getting started on math. Physicists use parity, and crystallographers seem to use space inversion. I’d be happy with just “inversion”.

    But regardless of the term, the concept is not a waste. As Michael Kinyon pointed out, we can define a symmetric space to be a connected Riemannian manifold such for each point we have an isometry that fixes that point and acts as -1 on the tangent space. In other words, an ‘inversion’. From this we can derive a quite remarkable theory. Cartan classified symmetric spaces shortly after doing simple Lie groups. I also like how the operations of inversion about a point make a symmetric space into an algebraic structure called a “quandle”.

  14. I tend to agree with Michael Kinyon. Let me give one example of the usefulness of point-reflections. Let P stand for the isometry rot(P, 180) and let * be the composition of isometries. Then for a line m,
    (reflm)*P = P*(reflm) if and only if the point point P lies on the line m. In other words, one can capture the incidence geometry axiomatically using only pure group theory! This was at the heart of the famous axiom scheme of Bachmann characterizing absolute geometry using only involutions (= line reflections or point reflectiions).

    Still, I agree that line reflections are richer than point reflections. The above example of Bachmann and Kinyon’s reference to symmetric spaces are instances where point-reflections do play an important role.

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