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Thread: 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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    Here's one way...

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

    For example, if your figure is a square, then let $\displaystyle V(F)=\{1,2,3,4\}$, 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):

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

    Reflections:

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

    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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