# If we compose a glide symmetry to itself what is the result? Two different glide symmetries?

If we compose a glide symmetry to itself what is the result? Two different glide symmetries?
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Let the glide reflection be T. Without loss of generality the reflection part of T' is reflection in the y-axis. Assume that the translation part is by the vector (a, b).
The reflection takes (x,y) to (-x,y). The translation part now takes us to (-x+a, y+b). So T takes (x,y) to (-x+a, y+b).
Do it again. The reflection part takes (-x+a, y+b) to (x-a,y+b), and the translation part takes this to (x,y+2b). So T^2 is translation parallel to the y-axis by 2b.