This is a job for the Frobenius/Burnside/Polya/Messerschmitt/Merill/Lynch/Pierce/Fenner/Smith counting formula. On Tue, Dec 6, 2011 at 11:58 AM, Dan Asimov <dasimov@earthlink.net> wrote:
What are all the ways that 30-, 60-, and 90-degree angles can be arranged about the origin in the plane?
Let two ways be equivalent if one can be carried into the other by either a rotation or a flip of the plane -- i.e., by an element of the group O(2).
This appears to be a messy counting job but certainly doable by hand because of its small scale.
As a first step, I found there are 19 partitions of 12 using only 1's, 2's, and 3's (appended).
Can anyone suggest a smart way to go about this, or is there no good way to shorten the task?
--Dan
----------------------- 3+3+3+3
2+2+2+3+3
2+2+2+2+2+2
1+2+3+3+3
1+2+2+2+2+3
1+1+2+2+3+3
1+1+2+2+2+2+2
1+1+1+3+3+3
1+1+1+2+2+2+3
1+1+1+1+2+3+3
1+1+1+1+2+2+2+2
1+1+1+1+1+2+2+3
1+1+1+1+1+1+3+3
1+1+1+1+1+1+2+2+2
1+1+1+1+1+1+1+2+3
1+1+1+1+1+1+1+1+2+2
1+1+1+1+1+1+1+1+1+3
1+1+1+1+1+1+1+1+1+1+2
1+1+1+1+1+1+1+1+1+1+1+1 -----------------------
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