Earlier today, I was asked why a motion of the plane that fixes a line of points is called a reflection and I was stumped for an answer.

The best explanation I can think of is that the image of a shape under a reflection will be that shape's "mirror image" (this is mentioned on the Wikipedia page). If you look in a mirror, the image you see is a reflection of the real world. That said, in two dimensions terms like "flip" seem more intuitive.

This leads to my question:

  • Where, historically, does the analogy of "reflection" arise in mathematics?
  • $\begingroup$ It was suggested on math SE this question be asked here $\endgroup$ Commented May 9 at 20:43
  • $\begingroup$ I am afraid, your phrasing is too vague for an answer. Geometric optics goes back to antiquity, and reflections in a mirror are studied in pseudo-Euclid's Catoptrica, for example. But reflections as transformations of the plane only came to be used systematically at the end of 19th century, in Klein's Erlangen program that started transformation geometry. Optical and perspectival constructions of specific reflected figures appeared throughout in between. $\endgroup$
    – Conifold
    Commented May 10 at 7:15
  • $\begingroup$ I am not asking about optics - my question concerns transformations of the plane. $\endgroup$
    – Numeral
    Commented May 10 at 9:58
  • $\begingroup$ Again, there is no clear separating line historically between constructions in geometric optics and in plane geometry, Catoptrica is written in the same style as Elements. Projective geometers of 17-19th century can be said to deal with "transformations of the plane" (transferring points of the picture plane) or with optics (producing perspectival images). If you mean "transformations of the plane" in the modern set-theoretic sense then that starts with Klein, but his was continuous with prior work on reflecting, rotating and projecting plane figures. $\endgroup$
    – Conifold
    Commented May 11 at 6:27


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