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Math Help - Elements of Prism

  1. #1
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    Elements of Prism

    In the attachment I have given the picture that goes with the question.


    The positions of the square faces are 1 and 6 and the positions of the rectangular faces are 2,3,4 and 5 for top, right, bottom and left respectivley. Let G denote the group of rotational symmetries of P

    a) Using the given numbering, write down all the elements of G as permutations of the face positions 1,2,3,,5 and 6.


    So far I have

    (1)(2)(3)(4)(5)(6) =e

    (1)(2345)(6) (rotation through horizontal axis by pi/2)

    (1)(24)(35)(6) (rotation throught horiz. axis by pi)

    (1)(2543)(6) (rotation through horiz. axis by 3pi/2)

    (16)(2)(35)(4) (rotation through vertical axis by pi)

    My question here is have I got them all. I wasn't sure if there should be a rotation through a diagonal axis say, starting from the bottom left of face 1 and ending at top right corner of face 6 and then rotating, or would I just end up with the same permutaions I have now.

    Hopefully you can just let me know if I have the right amount and if not a clue as to where the rest come from

    Thanx
    Attached Thumbnails Attached Thumbnails Elements of Prism-file0001.jpg  
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  2. #2
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    Quote Originally Posted by bex23 View Post
    In the attachment I have given the picture that goes with the question.


    The positions of the square faces are 1 and 6 and the positions of the rectangular faces are 2,3,4 and 5 for top, right, bottom and left respectivley. Let G denote the group of rotational symmetries of P

    a) Using the given numbering, write down all the elements of G as permutations of the face positions 1,2,3,,5 and 6.


    So far I have

    (1)(2)(3)(4)(5)(6) =e

    (1)(2345)(6) (rotation through horizontal axis by pi/2)

    (1)(24)(35)(6) (rotation throught horiz. axis by pi)

    (1)(2543)(6) (rotation through horiz. axis by 3pi/2)

    (16)(2)(35)(4) (rotation through vertical axis by pi)

    My question here is have I got them all. ...
    I've modified your drawing a little bit: I've added the axes.

    The green axis yield 4 rotations.
    The red axis yield 2 rotations.
    The blue axis yield 2 rotations.

    That means you must come out with 8 rotations in total.
    Attached Thumbnails Attached Thumbnails Elements of Prism-rotgroup_prism.gif  
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  3. #3
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    Quote Originally Posted by earboth View Post
    I've modified your drawing a little bit: I've added the axes.

    The green axis yield 4 rotations.
    The red axis yield 2 rotations.
    The blue axis yield 2 rotations.


    That means you must come out with 8 rotations in total.
    For the green axis I get the identity plus 3 rotations before coming back to where I started totaling 4 as required.

    For the red axis I get (16)(2)(35)(4) but when I rotate again I end up with the identity.

    For the blue axis I get (16)(24)(3)(5). If I rotate again I end up with the identity.

    So I am a little confused as to how the red and blue axes can have two rotations each whereas I get only one. So at the moment I have 6 rotations in total including the identity.

    Hopefully you can clarify the axes problem I seem to be having.

    Thanks again

    Bex
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