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Wikipedia: https://en.wikipedia.org/wiki/Pentagonal_tiling#Stein_.281985.29_and_Mann.2FMcLoud.2FVon_Derau_.282015.29

Media coverage: http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile

It seems that the detail is not yet published.

For those who do not see any mathematics in the question, here are two possibilities for an answer:

  • Based on the published information, is there any property that is not present in the previously known tilings?
  • Does this tiling bring any new insight on the pentagonal tilings? How does it help the full classification?
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    $\begingroup$ Maybe someone will come up with a good answer, but at this point I am wondering how this is a mathematics question? $\endgroup$
    – kantelope
    Aug 24, 2015 at 22:46
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    $\begingroup$ @JosephO'Rourke We already know that there are uncountably many tiling pentagons (any pentagon with two parallel sides, for starters; in fact the "15th pentagon" might be the first example of a tiling without continuous parameters); and some pentagons have uncountably many tilings (most easily, the pentagons that tile an infinite strip). Neither of these is what you meant, but it takes a bit of care to formulate the intended question. $\endgroup$ Aug 24, 2015 at 22:47
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    $\begingroup$ As you said, details haven't been published yet, but I suspect the most important thing to come out of the paper will not be the tiling itself, but rather the algorithm used to enumerate tilings - while nobody but the authors know for certain yet, I would presume that their work can actually be extended to enumerate all possible topologies of tilings (of pentagons) and so settle this question once and for all. $\endgroup$ Aug 24, 2015 at 23:14
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    $\begingroup$ @kantelope : I don't see how this is not a mathematics question. It is about a new mathematical discovery, which contributes to the solution of a long standing mathematical problem. The question ask about the things we can learn from it, which could be new insights, new techniques, etc. For those who are interested in the problem, this should be the first thing: look at the results and get inspired. This makes sense even before the detail is published. $\endgroup$
    – Hao Chen
    Aug 25, 2015 at 6:33
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    $\begingroup$ @StevenStadnicki : I personally would presume the opposite, because if their work could be extended straightforwardly to settle the question completely, then that would be a truly spectacular result, far more significant than the "mere" discovery of a new tiling, and I would expect that fact to be emphasized in the reporting. $\endgroup$ Aug 25, 2015 at 20:37

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What we can learn from (or be reminded of by) the new tiling, is just how hard it is to know whether we have all the tilings. According to Wikipedia, Kershner claimed to have found all pentagonal tilings in 1968, but then James found one in 1975, and Rice found three in 1977, and Stein found one in 1985, and now this new one.

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    $\begingroup$ In 1997, I had an email exchange with Doris Schattschneider, who said, "Several prominent mathematicians have privately expressed to me they think the list is now complete, but no proof is in sight." $\endgroup$ Aug 25, 2015 at 14:03

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