Triangulations of lattice polygons - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:42:28Z http://mathoverflow.net/feeds/question/89180 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89180/triangulations-of-lattice-polygons Triangulations of lattice polygons Daniil Rudenko 2012-02-22T12:32:17Z 2012-04-19T16:28:29Z <p>Let L be a 2-dimensional lattice and P- a lattice polygon. Suppose, it is triangulated into lattice tiangles. What are restrictions on their areas? For instance, can a lattice triangle of even area always be divided into lattice triangles of area 1? Is there any general approach to such questions? </p> http://mathoverflow.net/questions/89180/triangulations-of-lattice-polygons/89183#89183 Answer by Shaun Ault for Triangulations of lattice polygons Shaun Ault 2012-02-22T13:09:13Z 2012-02-22T13:29:24Z <p>By Pick's Theorem, the area of a lattice polytope is related to the number $i$ of lattice points in the interior and the number $b$ of lattice points on the boundary, according to the formula:</p> <p>$$A = i + \frac{b}{2} - 1$$</p> <p>Note that if $b$ is odd, then the area must be equivalent to $1/2$ mod $1$. This implies that any lattice triangle with integral area has at least one non-vertex point located on an edge. That extra point can be used to subdivide the triangle into two smaller lattice triangles.</p> <p>Concerning the splitting of even area triangles into triangles of area 1, I don't know any general results.</p> <p>Hope this helps!</p> http://mathoverflow.net/questions/89180/triangulations-of-lattice-polygons/94518#94518 Answer by Rick Walcott for Triangulations of lattice polygons Rick Walcott 2012-04-19T12:07:51Z 2012-04-19T12:23:49Z <p>This is not my own formula: A = 2i + b - 1 when the area of one triangle is 1. I haven't tested this, but I think it works on any triangular grid (not just equilateral). I did test A=sqrt(3)*(i+b/2-1)/2 on an equilateral triangle grid. This works because each square can be mapped to a parallelogram comprised of 2 equilateral triangles, and the area of the parallelogram is sqrt(3)/2.</p> http://mathoverflow.net/questions/89180/triangulations-of-lattice-polygons/94544#94544 Answer by Rick Walcott for Triangulations of lattice polygons Rick Walcott 2012-04-19T16:28:29Z 2012-04-19T16:28:29Z <p>I tested A = 2i + b - 2 for any triangular lattice where A = 1 for each triangle, and it works. Geometer's Sketchpad implements it very nicely, but it would work in GeoGebra too.</p>