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Math Help - Symmetry

  1. #1
    Junior Member
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    Symmetry

    Prove that the set of symmetries of a figure F in the plane forms a group

    how do i define each element of the set ?
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  2. #2
    Newbie lepton's Avatar
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    Sep 2009
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    Here's one way...

    Let V(F)=\{v_1,\ldots, v_n\} be the set of vertices of F. Then you can think of the group of symmetries as bijections (permutations) of V(F). Your law of composition is then function composition.

    For example, if your figure is a square, then let V(F)=\{1,2,3,4\}, where the positions of 1,2,3,4 in the plane are (1,1), (-1,1), (-1,-1), (1,-1), respectively. What are the symmetries of the square? We have rotations and reflections:

    Rotations (counter clockwise):

    \pi/2\leftrightarrow (1234),
    \pi\leftrightarrow (13)(24),
    3\pi/2\leftrightarrow (1432).

    Reflections:

    through x-axis \leftrightarrow (14)(23),
    through y-axis \leftrightarrow (12)(34),
    through (2,4)-diagonal \leftrightarrow (13),
    through (1,3)-diagonal \leftrightarrow (24),

    and of course we can leave the poor square alone=identity.

    The set of vertices can also be changed to the set of edges of the figure, and when in >2 dimensions, you can consider the set of faces too. There are other ways to define such a group, but I find this the most intuitive.
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