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The following are exercises from The Four Pillars of Geometry; I'm not sure what they are stating, for example I don't know what the addition of prime (an apostrophe) to a line means. There are no examples or references.

3.6.5 Show that the reflections in lines $L$, $M$, and $N$ (in that order) have the same outcome as reflections in lines $L'$, $M'$, and $N$, where $M'$ is perpendicular to $N$.

3.6.6 Next show that reflections in lines $L'$, $M'$, and $N$ have the same outcome as reflections in lines $L'$, $M''$, and $N'$ where $M''$ is parallel to $L'$ and $N'$ is perpendicular to $M''$.

3.6.7 Deduce from 3.6.6 that the combination of any three reflections is a glide reflection

I don't need to know how to do the problem, just what it's asking.

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    $\begingroup$ It sounds like there should be a diagram to go along with the problem. As far as I know, the prime mark in the context does not have any standard meaning. Double check to make sure you haven't missed a diagram somehwere. $\endgroup$
    – Jonathan Gleason
    Jul 16, 2011 at 1:21
  • $\begingroup$ I don't know for sure (I don't have this particular book and there isn't enough context to tell), but it is somewhat common to use notation like $A'$ for the image of $A$ under some transformation, $A''$ for the image of $A'$, etc. $\endgroup$
    – Isaac
    Jul 16, 2011 at 1:44
  • $\begingroup$ The bothersome thing is there is no diagram! Somehow the meaning is obvious! It's especially awkward that the the notation is applied to lines. This question languished at artofproblemsolving as well. $\endgroup$
    – ttt
    Jul 16, 2011 at 2:00
  • $\begingroup$ My original supposition was incorrect and I am now sure that there isn't any missing information, diagram, or context. (See my answer, below.) $\endgroup$
    – Isaac
    Jul 16, 2011 at 4:34
  • $\begingroup$ Here is the scribd link for the relevant section of the book. The author claims that a picture will be helpful. If you assume whatever and proceed to draw your own, you would get what he means to get the intended result. $\endgroup$
    – kuch nahi
    Jul 16, 2011 at 4:52

1 Answer 1

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$L'$, $M'$, $M''$, and $N'$ are just more lines, with no specific known relationship to the original $L$, $M$, and $N$, except that they happen to take their places in the three-reflection composition.

Essentially, these are three steps to showing that the composite of any three reflections is a glide reflection. Let me rephrase the steps:

  1. Given three arbitrary lines $\ell_1=L$, $\ell_2=M$, and $\ell_3=N$, let $T=R_{\ell_3}\circ R_{\ell_2}\circ R_{\ell_1}$ (reflect over $\ell_1$, then over $\ell_2$, then over $\ell_3$). Show that there exist some lines $\ell_4=L'$ and $\ell_5=M'$ with $\ell_3\perp\ell_5$ such that $R_{\ell_3}\circ R_{\ell_5}\circ R_{\ell_4}=T$.

  2. Given the above, show that there exist some lines $\ell_6=M''$ and $\ell_7=N'$ with $\ell_6\parallel\ell_4$ and $\ell_7\perp\ell_6$ such that $R_{\ell_7}\circ R_{\ell_6}\circ R_{\ell_4}=T$ (probably through $R_{\ell_3}\circ R_{\ell_5}\circ R_{\ell_4}=T$ from part 1).

  3. Conclude that the composite of any three reflections (our original $T$) is a glide reflection—that is, we know from the previous two parts that $T=R_{\ell_7}\circ R_{\ell_6}\circ R_{\ell_4}$, so explain why $R_{\ell_7}\circ R_{\ell_6}\circ R_{\ell_4}$ a glide reflection.

(Please let me know if this is still confusing. I am now sure that there is not a missing diagram or some other context that you have left out, but the original notation is definitely hard to follow and I'm not sure that my notation is any better.)

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  • $\begingroup$ Just to check myself on (1). I'm going to work backwards. Because $M'$ can be varied so long as it is perpendicular to $N$ the point $R_{L'}\circ R_{M'}$ lies on a line parallel to $N$. To get back to the original point $P$ all we have to do is give a line that can reflect so that $R_{L'}$ is on the line parallel to $N$ that intersects $N\circ T$. - I can see why people don't answer much in comments, I hope that can be parsed. $\endgroup$
    – ttt
    Jul 18, 2011 at 16:38

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