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May 9th, 2011, 01:49 PM  #1  
Newbie Joined: May 2011 Posts: 1 Thanks: 0  Finding the cycle index polynomial for platonic solids
First off all, I hope you can excuse my English a little as I am not learning this subject (Groups and Symmetry) in English, but rather in Norwegian, so some translations or wordings may be a little weird. Basically, what I am not able to do is problems like this: Quote:
A triangle has a group D3 = {e, r, r^2, s, rs, r^2s}. So I go through each g in D3 and find the cycle index. e has cycle index (*)(*)(*), so r,r^2 has cycle index (***), so ... The cycle polynomial being the polynomial for everything summed together. Now back to my original problem. I don't really know the group for a cube. I do have a table like the one on this page which helps. Now what I've been instructed to do is check this: Check every: Corner, edge and plane on the solid and find an axis. For the cube I have these axes: planeplane axes. There are 6 planes, so 3 of these axes. These axes can be rotated either by 90 deg, or 180 deg (with an order of 4 or 2 respectively) so the polynomial needs to be found for both cases. cornercorner axes. 8 corners, 4 axes which are rotated 120 deg. With an order of 3. edgeedge axes. 12 edges, 6 axes which are rotated 180 deg. With an order of 2. The problem i'm having is: If I for example take the planeplane axis with a rotation of 90 deg (order of 4). According to this page on wikipedia there are 6 such rotation. There are 3 axes, so it's 3*2 in this case. Where does the number 2 come from? (I've looked through a lot of problems like this, and the number of cycles is always: Amount of similar axes * ?(order). ? being Euler's Totient Function, though I'm not sure if it's just a coincidence or if it works every time).  
May 9th, 2011, 09:43 PM  #2  
Senior Member Joined: Jun 2010 Posts: 618 Thanks: 0  Re: Finding the cycle index polynomial for platonic solids
pineapple, I confess that I did not follow all of your links, but still maybe I can answer one question for now, and maybe if I have more time, the rest later. Quote:
Ormkärr  

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