Wikipedia:Reference desk/Archives/Mathematics/2015 August 17
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August 17
[edit]Testing tiling
[edit]This is a question at the intersection of mathematics and computing. I was recently reading about the discovery of a new pentagon tiling of the plane [1] (e.g. pentagonal tiling). The discoverers apparently used a computer program to search a large space of potential pentagons in order to discover one capable of tiling. I was wondering, given an arbitrary polygon, how do you test whether it tiles the plane? What does an algorithm to do that look like? In the the general case, such as the new pentagon found, you have to combine many rotations and reflections of the polygon in order to describe the required tiling. Because of that it seems like figuring out how to arrange the tiles in a repeating pattern could be fairly complicated, and you might have consider a large number of possible tile arrangements. Is there an efficient way to do that? Dragons flight (talk) 06:28, 17 August 2015 (UTC)
- The breakthrough of the scientists involved seems to have been discovering the efficient way (and persuading a uni to let them use the cluster!). As far as I can tell, they haven't yet published their algorithm, so you may to have wait to learn the details. Still, any pentagonal tiling should be periodic (as opposed to schemes like Penrose tiles that are aperiodic), and that means that the pattern will be regular and crystalline – all tilings should be equivalent to a simple translation tiling of a (perhaps rather complex) composite polygon (the new pentagon tiling is equivalent to a simple translation of a shape comprising twelve pentagons arranged in a kind of S). Finding complex polygons built of pentagons that tile (without need for arbitrary rotation) is one productive approach to finding new pentagonal tilings. Smurrayinchester 08:12, 17 August 2015 (UTC)
- In pentagonal tiling, there is a small illustration of the 15 tilings. I presume that for some (many?) of the 15 tilings that there are multiple (infinite?) variations of the scheme. For example #1 of the 15 appears to be a skewed variant of "Dual-tiling" #3 lower down the page - and obviously that 'design' can be stretched at will, and the angle of the "end-line" rotated to form an infinite set of variants. I observe that "Dual-tiling" #2 seems to be a symmetric version of tiling #5 and "dual-tiling" #1 seems to be a symmetric version of tiling #4.
- BUT the article has very little (no?) information on the shapes and limitations of the 15 patterns. Where would we find such info and how easy would it be to add it to that article? -- SGBailey (talk) 12:26, 17 August 2015 (UTC)
- This does suggest that most (all?) of the solutions are each part of an infinite family. Interestingly, it looks like the 14 previous examples all required no more than 8 pentagons to define their fundamental block that could be translated across the plane without rotation. The new solution appears to have 12 pentagons in its fundamental block. Dragons flight (talk) 14:25, 17 August 2015 (UTC)
- There's some discussion [2], it's reddit but seems to have comments from one of the people who worked on discovering the 15th (which suggests they just started so it's easily possible more could be found). Incidently I found that reddit discussion from Talk:Pentagonal tiling#Set of 15 isohedral pentagonal tilings, from their username, the editor who commented there and also added the details to our article, may be someone else well known (the author of the above linked page according to their article) although possibly not very active here. Nil Einne (talk) 19:49, 17 August 2015 (UTC)
- FWIW, I found [3] to be somewhat helpful in better understanding what's being referred to here, and about the families. One point which is mentioned in our article and some sources but not in others is that AFAIK, the 15 tilings are only referring to convex pentagons. Nil Einne (talk) 03:31, 19 August 2015 (UTC)
- There's some discussion [2], it's reddit but seems to have comments from one of the people who worked on discovering the 15th (which suggests they just started so it's easily possible more could be found). Incidently I found that reddit discussion from Talk:Pentagonal tiling#Set of 15 isohedral pentagonal tilings, from their username, the editor who commented there and also added the details to our article, may be someone else well known (the author of the above linked page according to their article) although possibly not very active here. Nil Einne (talk) 19:49, 17 August 2015 (UTC)
- This does suggest that most (all?) of the solutions are each part of an infinite family. Interestingly, it looks like the 14 previous examples all required no more than 8 pentagons to define their fundamental block that could be translated across the plane without rotation. The new solution appears to have 12 pentagons in its fundamental block. Dragons flight (talk) 14:25, 17 August 2015 (UTC)