Convex tilings of the plane - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:20:57Z http://mathoverflow.net/feeds/question/75593 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75593/convex-tilings-of-the-plane Convex tilings of the plane nvcleemp 2011-09-16T12:17:04Z 2011-09-18T06:02:07Z <p>For convex polyhedra you have Steinitz's theorem characterizing them as the 3-connected planar graphs. My question is not about spheric tilings, but about periodic tilings of the euclidean plane. Is it here also the case that 3-connectivity corresponds with convexity?</p> <p>It is easy to construct an example of a 2-connected periodic tiling that is not convex. My guess is that the symmetry group prohibits some tilings from being convex and thus there are also 3-connected periodic tilings that are not convex, but I can't seem to be able to construct a counter example to support this guess.</p> <p>Is there a known counterexample for this? Or is it actually true? Any reference or hint would help.</p> <p>edit: I was looking for a periodic tiling which has no equivariantly equivalent (i.e. topologically equivalent and the same symmetry group) tiling that is convex and that is 3-connected when you look at the segments as graph edges. My feeling is that in case of the euclidean plane there might be a tiling in which you can't convexify one face, without needing to make another one nonconvex to keep the same symmetry group.</p> http://mathoverflow.net/questions/75593/convex-tilings-of-the-plane/75636#75636 Answer by Igor Pak for Convex tilings of the plane Igor Pak 2011-09-16T20:54:15Z 2011-09-16T20:54:15Z <p>You are perhaps thinking of <a href="http://en.wikipedia.org/wiki/Fary_theorem#Related_results" rel="nofollow">Tutte's "spring" theorem</a> that every 3-connected graph has an embedding with convex faces. It indeed a corollary of the (earlier) <a href="http://en.wikipedia.org/wiki/Steinitz_theorem" rel="nofollow">Steinitz theorem</a>, but Tutte's proof is of independent interest. Now, for periodic networks in the plane, you need to look at the "torus version" of the Tutte theorem. Start with <a href="http://www.cs.harvard.edu/~sjg/papers/tutte.pdf" rel="nofollow">this important paper</a> which gives a version of the result and check some further refs therein. </p>