Yesterday Malia and I were discussing some middle school math material (she’s a learning designer for a video game company) and she mentioned this thing called a “point reflection.” Apparently this is another name for what I learned as “symmetry about the origin,” which is to say it’s 180-degree rotation about some point. In other words, it is *not a reflection*. Reflections reverse orientation, right? Why would you want to confuse kids like that, how are they going to figure out what a reflection is when you go and tell them some rotation is actually a reflection?

Did any of the rest of you learn to use the word reflection for this concept? Am I wrong here? Anyone know how long this term has been around? Who came up with it?

A little googling confirms that in fact this is a Regents approved concept. And I guess I can see where they’re coming from, a point reflection is the same as applying an ordinary reflection about your point on each line through the point you’re “reflecting about.” Similarly for a “line reflection” in 3-space you could draw all planes through that line and then reflect through the line in each of those planes to get a “reflection” on 3-space. But it still seems incredibly counterproductive to me to use this terminology.

Wikipedia seems to suggest a split in this terminology with Point_reflection and Reflection_(linear_algebra) agreeing with the Regents and Reflection_(mathematics) agreeing with me. The linear algebra article strikes me as particularly nutty talking about things like “non-orthogonal reflections.”

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In high school we called it central-point reflection (in Hungarian).

If you can call projection the projection map to any subspace in an Euclidean vector space, then I do not see what is wrong calling your transformation reflection through a subspace.

I’m too lazy at the moment to follow all the links to put my comment in proper perspective, but…

First of all, in my field (high-dimensional convex geometry), the idea of reflecting in a subspace of arbitrary dimension is quite familiar. Whether such a reflection preserves orientation depends on the parity of the codimension of the fixed space. It seems you would restrict use of the word “reflection” for what I would clarify as “reflection in a hyperplane” (hyperplane being a standard-in-some-fields term for a subspace of codimension 1).

As for non-orthogonal reflections, they come up naturally in some contexts – consider for example the symmetries of a tiling of the plane by parallelograms. Another point to consider is that any such reflection is in fact orthogonal, but with respect to some other inner product on space. If the context you are working in doesn’t include a preferred inner product (or includes more than one) then you will want to be able to talk about such things.

“extremely counterproductive” is an exaggeration, but I appreciate your passion about terminology. ;)

In Spanish, even among mathematicians, symmetry about a point is called reflection in the point. Growing up with this terminology it doesn’t sound so strange to me: reflection in an affine subspace X is (as you said) traveling to the nearest point of X and then continuing an equal distance further. This definition is very clear and geometric and applies equally to a hyperplane or a point.

I agree with you that this is not standard among English speaking mathematicians; I think it’s a mismatch between middle school and university math. There are many of these, for example: in middle school a circumference is a curve and a circle is a two-dimensional set. At university you learn to call the curve a circle and the two-dimensional thing a disk. (This happened to me in Mexico, but I think the same example occurs in English speaking countries, right?)

I’m not sure that there is anything wrong with this definition, except that it is not the one mathematicians use.

The canonical example of a reflection should be actual reflection in a mirror, in other words, an orthogonal linear map that fixes a 2-plane and negates a ray.

Plausible generalizations are:

(1) A diagonalizable linear map whose eigenvalues are .

(2) A diagonalizable linear map whose eigenvalues are , where has multiplicity one.

(3) An orthogonal linear map whose eigenvalues are .

(4) A orthogonal linear map whose eigenvalues are , where has multiplicity one.

(I’m sure this is not a complete list.) Mathematicians have chosen (2), but it is not clear to me that it is intrinsically a more important concept than (1) or (3).

Oh, and yes, I am pretty sure that I learned that language in middle school.

I’d note that they’re extending the idea of “reflection” only to 180-degree rotations, not to all rotations. In particular, the symmetries in question have order 2. (Off the top of my head, I think ordinary reflections and 180-degree rotations are the only linear transformations of order 2, but the coffee hasn’t quite kicked in yet.) As a combinatorialist, that’s what I care about.

I suspect that there is somebody out there who requires reflections to be orientation-reversing, like you seem to want, but allows reflections in subspaces of any odd codimension. I haven’t thought about this too hard, but a “point reflection” in 3-space, taking (x, y, z) to (-x, -y, -z), seems more natural to me as a name than a point reflection in 2-space.

We could use a little more acknowledgement that there is no unique thing known as “standard mathematical English terminology” (even with all those qualifiers). In particular, regarding David’s comment #4, I think most people in my field would pick (3), not (2), as the definition of “reflection”, although they would probably be happy to accept (1) as a reasonable definition as well.

I learned this terminology in elementary or middle school, but since I grew up in New York, that is consistent with the terminology being Regents-approved. Even in professional mathematics, I don’t think the notion of reflection is a point of universal agreement.

People who study complex and p-adic reflection groups define reflections to be finite order linear transformations A such that (A-I) has rank 1. There are also notions of reflection in characteristic 2, but they are often called orthogonal transvections. Closer to your original point, there are papers about the monster that call the -1 involution of the Leech lattice a reflection.

The translations in comment 2 also arise in representation theory, since the translations in an affine Weyl group are really composites of reflections (in the sense of Noah) in the root space. This space has a singular inner product, because there is a line that is orthogonal to everything including itself. If you promote one more dimension, an affine reflection group can be viewed as a group of reflections in spacelike vectors in Lorentz space that fix a lightlike line, i.e., the one that was orthogonal to everything. Reflections in spacelike vectors correspond to reflections in geodesic hyperplanes in a hyperboloid, so affine reflection groups can then be viewed as special cases of hyperbolic reflection group, where a boundary point (corresponding to the lightlike line) is fixed. The translations and reflections can be seen by examining the action on a horosphere containing the fixed point, which has a flat geometry.

“… a reflection in the origin, sometimes called inversion…. In the plane [this] is not a reflection, but a turning through 180 degrees; and, generally, inversion in the origin is a reflection only in spaces of an odd number of dimensions. If the number is even, it is a rotation.” –Felix Klein, “Elementary Mathematics from an Advanced Standpoint: Geometry”

I have been rereading Weyl’s “Symmetry” recently and discovered that he calls reflections (about lines in the plane) “improper rotations”. I find that even more unnatural than calling 180 degree rotation a reflection about a point.

I learned the term “reflection through a point” from Serge Lang and Gene Murrow’s high school geometry textbook, so I presume Lang must not have objected to it, which says something.