* Andy Latto <andy.latto@pobox.com> [Oct 22. 2013 08:32]:
On Mon, Oct 21, 2013 at 3:52 AM, Joerg Arndt <arndt@jjj.de> wrote:
Pick's theorem (cf. http://en.wikipedia.org/wiki/Pick's_theorem):
Square grid, every second column shifted by a half unit: A = i/2 + b/2 - 1
where b counts only the boundary points whose local neighborhood is not concave (i.e., 90 degrees of outside, 270 degrees of inside).
What about a square of side 2? Doesn't this have i = 1 and b = 4?
I do not see which configuration you mean. Btw. the formula for the triangular lattice seems to be correct, see Ren Ding, John R.\ Reay: The Boundary Characteristic and Pick's Theorem in the Archimedean Planar Tilings, Journal of Combinatorial Theory, Series A, vol.44, no.1, pp.110-119, (January-1987). http://www.sciencedirect.com/science/article/pii/009731658790063X % "Ren Ding" == "Ding Ren" Ren Ding, Krzysztof Kolodziejczyk, John Reay: A new Pick-type theorem on the hexagonal lattice, Discrete Mathematics, vol.68, no.2-3, pp.171-177, (1988). http://www.sciencedirect.com/science/article/pii/0012365X88901100
Andy Latto andy.latto@pobox.com
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Regards, jj