triangulations of torus, general, and Euler number. (Hopefully more interesting/relevant) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T21:09:39Z http://mathoverflow.net/feeds/question/22480 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22480/triangulations-of-torus-general-and-euler-number-hopefully-more-interesting-r triangulations of torus, general, and Euler number. (Hopefully more interesting/relevant) Herb 2010-04-25T03:52:32Z 2010-04-26T17:28:48Z <p>Hi, everyone:</p> <p>I have been going over some simplicial homology recently, hoping to get<br> some geometric insight that I don't know how to get from the algebraic machinery alone.</p> <p>I have been trying to find the homology of the torus this way, i.e., by triangulating it ( i.e., finding a carrier for the torus), but the smallest triangulation I have been able to do , has 18 triangles/faces --I checked it works; there are 8 vertices and 26 edges. Still: does anyone know of a simpler triangulation, ie., one with a smaller total number of triangles (and, of course, fewer vertices and edges resp.). ?</p> <p>I had tried the long shot of solving the very simple equation:</p> <p>V-E+F =0 </p> <p>in positive integers.</p> <p>but this alone does not seem to help . Any ideas.?. Any ideas for finding minimal triangulations of surfaces, or higher-dimensional manifolds.?</p> <p>Thanks.</p> http://mathoverflow.net/questions/22480/triangulations-of-torus-general-and-euler-number-hopefully-more-interesting-r/22481#22481 Answer by David Eppstein for triangulations of torus, general, and Euler number. (Hopefully more interesting/relevant) David Eppstein 2010-04-25T04:12:08Z 2010-04-25T05:21:26Z <p>If you're just looking to glue triangles together along their edges, you can do it with two triangles, glued together to form a square, and then with opposite sides of the square glued to form a torus in the usual way. The resulting mesh has one vertex and three edges.</p> <p>But if the triangles have to form a simplicial complex (meaning that the intersection of any two triangles is empty, a single vertex, or an edge) then I think the smallest mesh for a torus has 14 triangles, connected to each other in the pattern of the <a href="http://en.wikipedia.org/wiki/Heawood_graph" rel="nofollow">Heawood graph</a>. The resulting mesh has seven vertices and 21 edges. It can be embedded into space as the <a href="http://en.wikipedia.org/wiki/Cs%C3%A1sz%C3%A1r_polyhedron" rel="nofollow">Császár polyhedron</a>.</p> http://mathoverflow.net/questions/22480/triangulations-of-torus-general-and-euler-number-hopefully-more-interesting-r/22622#22622 Answer by John Palmieri for triangulations of torus, general, and Euler number. (Hopefully more interesting/relevant) John Palmieri 2010-04-26T17:28:48Z 2010-04-26T17:28:48Z <p>For the particular case of a simplicial complex structure for a torus, David Eppstein is right: the minimal triangulation has 7 vertices, 21 edges, and 14 triangles. For a sphere, the minimal triangulation has \$(v,e,f) = (4, 6, 4)\$. For a real projective plane, the minimal triangulation has \$(v,e,f) = (6, 15, 10)\$.</p> <p>For the general situation of finding minimal triangulations of manifolds, Frank Lutz has written <a href="http://arxiv.org/abs/math/0506372" rel="nofollow">a nice preprint</a>, and he also has some information and other references on <a href="http://www.math.tu-berlin.de/diskregeom/stellar/" rel="nofollow">The Manifold Page</a>. There are plenty of unsolved problems in this area, it seems...</p>