patterns of a 2d binary matrix

5 events

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Oct 26, 2011 at 18:45	comment	added	Piotr Migdal		Sorry for overcomplicating. It suffices to compare the original patterns with $7$ its symmetries different from the identity. Then in the normalization factor there is $7$ instead of $7*8$.
Oct 26, 2011 at 17:33	comment	added	Piotr Migdal		I like your approach. However, to get a more general answer it may be worth to take a bit larger symmetry group - identity, 3 rotations and 4 reflections (i.e. $D_4$, en.wikipedia.org/wiki/Dihedral_group). Then count differences ($d$) between all pairs (i.e. $8* 7$) and as a measure of randomness $r=k\frac{1}{8*7}\frac{25}{2n(25-n)})$, where $n$ is the number of black stones. For purely random shapes one should get $r\approx 1$, while for very symmetric $r \approx 0$. The good thing is that the formula for $r$ holds for different number of stones on the board and has the B-W symmetry.
Oct 18, 2011 at 7:13	comment	added	ttnphns		Thank you for appreciation. But the approach is just an initial stub, a general idea, and you are right saying it is expandable.
Oct 18, 2011 at 7:06	comment	added	Felix S		Thanks for your suggestion! Although I could think of several "easy" displays which are not invariant to a rotation transformation, this is a nice and easy (and expandable!) approach. I have to think about which sorts of transformation I would like to have. And I like your approach of counting points in each transformation.
Oct 17, 2011 at 20:26	history	answered	ttnphns	CC BY-SA 3.0