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.

http://img217.imageshack.us/img217/3319/4234w.jpg

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- Jun 4th 2010, 04:05 AMgomesSymmetries
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.

http://img217.imageshack.us/img217/3319/4234w.jpg - Jun 4th 2010, 05:36 AMGrandad
Hello gomesYour diagram is incorrect - see the one I have attached.

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

Can you complete the rest of the question?

Grandad - Jun 4th 2010, 06:16 AMgomes
Thanks! I tried carrying it on, this is what i've done so far.

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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! :)

http://img691.imageshack.us/img691/2190/123jki.jpg

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? - Jun 4th 2010, 07:16 AMGrandad
Hello gomesCorrect. Each rotational symmetry has centre at the origin; the five angles of rotation are $\displaystyle 0^o, 72^o, 144^o, 216^o, 288^o$.

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

Quote:

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?**

Quote:

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?**

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:$\displaystyle \begin{pmatrix}\cos144^o&\sin144^o\\ \sin144^o&-\cos144^o\end{pmatrix}$See, for example, here.

Quote:

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

$\displaystyle ABCDE \to AEDCB$and the $\displaystyle 72^o$ rotation is:

$\displaystyle ABCDE \to BCDEA$Do you see how to continue?

Quote:

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?

*vertex*. Each angle at the*centre*of the pentagon is $\displaystyle \frac{360}{5}=72^o$.

Grandad - Jun 4th 2010, 07:30 AMgomes
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?

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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 - Jun 4th 2010, 07:36 AMGrandad
That's what $\displaystyle ABCDE \to AEDCB$ means.

Quote:

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

Grandad - Jun 4th 2010, 07:44 AMgomes
- Jun 4th 2010, 08:01 AMgomes
- Jun 4th 2010, 10:14 AMGrandad