As another example for 4 with the translation glide reflection and horizontal

# As another example for 4 with the translation glide

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As another example, for (4) with the translation, glide reflection, and horizontal reflection, we start with the same fundamental domain as the previous problem, and restrict it to { ( x, y ) | 0 x 1 , y 0 } since the reflection sends y 7→ - y . The vertical edges of the strip are still identified with one another, so the result is a cylinder which is “infinite” only on one end. The other examples are more intricate, and require some thought. There also may be errors in the pictures, so caveat lector (reader beware). Problem 7. Rigid Motions of the Plane. We saw that the frieze groups contained symmetries of the following types: reflections about some line, glide reflections centered on some line, rotations about some point, and translations along some line. In fact, these are all the possible rigid motions of the plane. Describe each of the following compositions of transformations as a single motion from the list above. No need for proofs here, just pictures are fine. One way to figure this out is to choose particularly simple lines to reflect about and see where particular points go, just to get a sense for what happens. 1. Rotation by α radians, counter-clockwise around the origin followed by rotation by β radians counter- clockwise around the origin. 2. Translation by a vector ( a, b ) followed by translation by a vector ( c, d ). 3. Reflection through some line 1 followed by reflection through a different line 2 . 4. Translation by (1 , 1) followed by rotation by π 2 counter-clockwise about the origin. 5. Rotation by π 2 counter-clockwise about the origin followed by translation by (1 , 1). 4
6. Rotation by π 2 counter-clockwise about the origin followed by rotation by π 2 clock-wise around the point (2 , 0). Solution: (1) Rotation by α + β (2) Translation by ( a + c, b + d ) (3) Depends on whether lines are parallel or intersecting; if parallel then translation, if intersecting then rotation (4) Translation by vector depending on input (5) Rotation about a different point (6) Rotation about (2 , 0) is the same as translating by ( - 2 , 0), rotating, then translating back. So we can think of this as the composition rotation-translation-rotation-translation. This is overall a rotation about a different point (find the point!). 5

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• Winter '16
• Ms. Lauri Crestani
• Wallpaper group, Symmetry group, Rotational symmetry, Reflection symmetry