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Math Help - Symmetries

  1. #1
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    Symmetries

    Im not sure how to find the other 2 coordinates, how would i find it?

    and how would i give the coordinates of the vertices in terms of sines and cosines, of appropriate angles?

    Thanks.

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  2. #2
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    Hello gomes
    Quote Originally Posted by gomes View Post
    Im not sure how to find the other 2 coordinates, how would i find it?

    and how would i give the coordinates of the vertices in terms of sines and cosines, of appropriate angles?

    Thanks.

    Your diagram is incorrect - see the one I have attached.

    The angles at O are all 72^o, so the coordinates of the points B, C, D and E are all in the form (\cos\theta, \sin\theta), where \theta = 72^o, 144^o, 216^o, 282^o respectively.

    Can you complete the rest of the question?

    Grandad
    Attached Thumbnails Attached Thumbnails Symmetries-untitled.jpg  
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  3. #3
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    Thanks! I tried carrying it on, this is what i've done so far.
    -------------------------------
    Draw a diagram of the pentagon and label the vertices. Done.

    Give the coordinates of the vertices in terms of sines and cosines of appropriate angles. Done.

    List the symmetries of the pentagon. How would i present my answer? There are 5 rotational symmetries, and 5 reflection symmetries. How would i list each one out?

    For each reflection in this list, add the line of the reflection to your diagram. I've added it in the diagram below, is that correct?

    Choose two of these reflections and write each symmetry in terms of the chosen reflections. I chose the purple line, and said its reflection in x-axis, with the matrice on the diagram. How would I write each symmetry in terms of the chosen reflection?

    For each symmetry, give the resulting permutation of the vertices. How would I do this?

    Thanks alot!



    Quote Originally Posted by Grandad View Post
    Hello gomesYour diagram is incorrect - see the one I have attached.

    The angles at O are all 72^o, so the coordinates of the points B, C, D and E are all in the form (\cos\theta, \sin\theta), where \theta = 72^o, 144^o, 216^o, 282^o respectively.

    Can you complete the rest of the question?

    Grandad
    Thanks, could you explain to me why the angles are 72 degrees, and hence (cos72, sin 72)? I thought the internal angles are meant to be 108?
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  4. #4
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    Hello gomes
    Quote Originally Posted by gomes View Post
    Thanks! I tried carrying it on, this is what i've done so far.
    -------------------------------
    Draw a diagram of the pentagon and label the vertices. Done.

    Give the coordinates of the vertices in terms of sines and cosines of appropriate angles. Done.

    List the symmetries of the pentagon. How would i present my answer? There are 5 rotational symmetries, and 5 reflection symmetries. How would i list each one out?
    Correct. Each rotational symmetry has centre at the origin; the five angles of rotation are 0^o, 72^o, 144^o, 216^o, 288^o.

    Each reflection line passes through the origin and one of the five vertices of the pentagon.


    For each reflection in this list, add the line of the reflection to your diagram. I've added it in the diagram below, is that correct?
    Yes.

    Choose two of these reflections and write each symmetry in terms of the chosen reflections. I chose the purple line, and said its reflection in x-axis, with the matrice on the diagram. How would I write each symmetry in terms of the chosen reflection?
    You are correct for the reflection in the purple line.

    I am not sure exactly what information is wanted for one of the other reflections. But if you choose the red line (through B), the matrix that represents it is:
    \begin{pmatrix}\cos144^o&\sin144^o\\ \sin144^o&-\cos144^o\end{pmatrix}
    See, for example, here.
    For each symmetry, give the resulting permutation of the vertices. How would I do this?
    For example, reflection in the purple line produces the following permutation:
    ABCDE \to AEDCB
    and the 72^o rotation is:
    ABCDE \to BCDEA
    Do you see how to continue?

    Thanks, could you explain to me why the angles are 72 degrees, and hence (cos72, sin 72)? I thought the internal angles are meant to be 108?
    The interior angles of 108^o are the angle at each vertex. Each angle at the centre of the pentagon is \frac{360}{5}=72^o.

    Grandad
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  5. #5
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    Thanks for your help, I understand it now!

    Quote Originally Posted by Grandad View Post
    For example, reflection in the purple line produces the following permutation:
    ABCDE \to AEDCB
    Do you see how to continue?
    Thanks, erm, could you explain this one? I thought reflecting it sends:

    B to E, E to B
    C to D, D to C
    A stays the same?

    -------------------
    on this page which you gave me, why is the rotation matrice only theta, but reflection two-theta?
    http://en.wikipedia.org/wiki/Coordin...nd_reflections


    but on this page, why is both the rotation matrice and reflection matrice only theta?
    http://en.wikipedia.org/wiki/Orthogo...wer_dimensions
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  6. #6
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    Quote Originally Posted by gomes View Post
    Thanks for your help, I understand it now!



    Thanks, erm, could you explain this one? I thought reflecting it sends:

    B to E, E to B
    C to D, D to C
    A stays the same?
    That's what ABCDE \to AEDCB means.

    on this page which you gave me, why is the rotation matrice only theta, but reflection two-theta?
    Coordinate rotations and reflections - Wikipedia, the free encyclopedia


    but on this page, why is both the rotation matrice and reflection matrice only theta?
    Orthogonal matrix - Wikipedia, the free encyclopedia
    Read it carefully. On the second page the reflection is in a line at an angle of \theta/2.

    Grandad
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  7. #7
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    Quote Originally Posted by Grandad View Post
    That's what ABCDE \to AEDCB means.

    Read it carefully. On the second page the reflection is in a line at an angle of \theta/2.

    Grandad
    thanks, sorry i got confused about the first bit.

    cheers again!!
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  8. #8
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    Quote Originally Posted by Grandad View Post
    See, for example, here.For example, reflection in the purple line produces the following permutation:
    ABCDE \to AEDCB
    Do you see how to continue?
    For each symmetry, give the resulting permutation of the vertices.

    thanks, lets say instead of writing ABCDE, to AEDCB......do you think the question wants it like:
    (cos72,sin72) --- > (cos282, sin282)?



    Is that how I would go about doing it?
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  9. #9
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    Hello gomes
    Quote Originally Posted by gomes View Post
    For each symmetry, give the resulting permutation of the vertices.

    thanks, lets say instead of writing ABCDE, to AEDCB......do you think the question wants it like:
    (cos72,sin72) --- > (cos282, sin282)?


    Is that how I would go about doing it?
    The question doesn't specify how you are to give the answer. It does tell you to label the vertices, so I should think that it would be OK to use these labels in your answers.

    Grandad
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