Symmetries

May 2010
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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.

 

Grandad

MHF Hall of Honor
Dec 2008
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Hello gomes
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 \(\displaystyle 72^o\), 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
 

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May 2010
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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! :)



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
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?
 

Grandad

MHF Hall of Honor
Dec 2008
2,570
1,416
South Coast of England
Hello gomes
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 \(\displaystyle 0^o, 72^o, 144^o, 216^o, 288^o\).

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


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:
\(\displaystyle \begin{pmatrix}\cos144^o&\sin144^o\\ \sin144^o&-\cos144^o\end{pmatrix}\)
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:
\(\displaystyle ABCDE \to AEDCB\)
and the \(\displaystyle 72^o\) rotation is:
\(\displaystyle ABCDE \to BCDEA\)
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 \(\displaystyle 108^o\) are the angle at each vertex. Each angle at the centre of the pentagon is \(\displaystyle \frac{360}{5}=72^o\).

Grandad
 
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May 2010
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Thanks for your help, I understand it now!

For example, reflection in the purple line produces the following permutation:
\(\displaystyle ABCDE \to AEDCB\)

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/Coordinate_rotations_and_reflections


but on this page, why is both the rotation matrice and reflection matrice only theta?
http://en.wikipedia.org/wiki/Orthogonal_matrix#Lower_dimensions
 

Grandad

MHF Hall of Honor
Dec 2008
2,570
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South Coast of England
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 \(\displaystyle ABCDE \to AEDCB\) 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 \(\displaystyle \theta/2\).

Grandad
 
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May 2010
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That's what \(\displaystyle ABCDE \to AEDCB\) means.

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

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

cheers again!!
 
May 2010
39
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See, for example, here.For example, reflection in the purple line produces the following permutation:
\(\displaystyle ABCDE \to AEDCB\)

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?
 

Grandad

MHF Hall of Honor
Dec 2008
2,570
1,416
South Coast of England
Hello gomes
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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