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Question: 1 below are seven shapes which extend infinitely in each...

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1. Below are seven shapes which extend infinitely in each direction. Describe these shapes symmetry groups by naming the fundamental ways to map these shapes onto each other by translations, 180- degree rotations, glide reflections (translations followed by a top-to-bottom reflection) and horizontal or vertical reflections, and explain what the multiplication rules are for these fundamental maps. For example (to get you started) the topmost shape has a fundamental translation / mapping each pair of quills onto the next pair, and a fundamental 180 rotation r around the point midway between any two arbitrarily selected quills. Rules governing the composition of these transformations are that r = e (since a 360-degree rotation changes nothing, and that Ir rt-, since a rotation after a translation would reverse the direction of the translation. The symmetry group of this shape is thus the infinite group {r.rt·IE Z}, subject to ordinary exponent-collecting rules as well as the special multiplication rules (rand (rf)n Do a similar analysis for the other six shapes
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