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May 9th, 2011, 02:49 PM   #1
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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:
Each edge of a cube are to be colored. Find the cycle index polynomial for this problem, and then use it to calculate the amount of unique cubes with 2 different colors.
I understand how to find a cycle index polynomial for a 2D plane, like coloring each corner of a triangle :
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:

plane-plane 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.
corner-corner axes. 8 corners, 4 axes which are rotated 120 deg. With an order of 3.
edge-edge 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 plane-plane 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).
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May 9th, 2011, 10:43 PM   #2
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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:
Originally Posted by pineapple
The problem i'm having is:
If I for example take the plane-plane 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?
My guess is that for each axis, you can rotate 90° clockwise, or you can rotate 90° counter-clockwise, hence the 2.

-Ormkärr-
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