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Math Help - geometry- generating set

  1. #1
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    geometry- generating set

    in a pentagon, there are 10 symmetries ,5 rotational ,5 reflections, this question has baffled me for a while: from 2 symmetries (a generating set) every other symmetry can be described as a combination of these 2 symmetries but whenever i work out the other symmetries using the generating set it takes me forever using trial and error mucking about with composite functions are there laws as to generate the other symmetries,
    i know a rotation can be expressed as reflection o reflection for example. but i find it very annoying not knowing. il try be more specific if this question is bit vague,tho ive done my best.thanks
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  2. #2
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    Quote Originally Posted by skystar View Post
    in a pentagon, there are 10 symmetries ,5 rotational ,5 reflections, this question has baffled me for a while: from 2 symmetries (a generating set) every other symmetry can be described as a combination of these 2 symmetries but whenever i work out the other symmetries using the generating set it takes me forever using trial and error mucking about with composite functions are there laws as to generate the other symmetries,
    i know a rotation can be expressed as reflection o reflection for example. but i find it very annoying not knowing. il try be more specific if this question is bit vague,tho ive done my best.thanks
    Label your pentagon with vertices 1,2,3,4,5. Any symettry must preserve the order these vertices are names. For example, suppose the top point of the pentagon is 1, and to the right it is 2. Then rotating the pentagon once clockwise will make the new top point 5, and to the right it is 1, followed by 2, all the way back to 5. Thus, this is an okay symettry. However, replace 3 with 1. The resulting configuration is not a symettrical motion because it breaks apart the configuration of the pentagon.

    Using basic counting we can show there are precisely 10 symettries of a pentagon. We have five slots to place the the number 1. Haven chosen a slot we can place 2 either to the right or left of that point, hence there are just two more slots. Haven done that all the other numbers are completely determined because they need to preserve the configuration of the pentagon. Thus, there are a total of 5\cdot 2 = 10 symettries.

    Let \rho be a rotation by 72 degree clockwise. Then \rho,\rho^2,\rho^3,\rho^4 are all distinct rotations. While \rho^5 = e, where e is the identity transformation.

    Draw a line segment from 1 through the middle of the pentagon. Let \theta be the rotation through this line (so 1 stays fixed). Then \theta^2 = e. Now confirm that \rho\theta , \rho^2 \theta , \rho^3 \theta, \rho^4 \theta are all distinct. Thus, we have created 10 distinct symettries. And we see that each one is made out of \rho, a rotation, and \theta - a reflection.
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