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?
  • 1
    $\begingroup$ Is it known that there are only a finite number of convex pentagons that tile the plane monohedrally? Or might there be (for all we know) a countably infinite number? Or an uncountable number? $\endgroup$ – Joseph O'Rourke Aug 24 '15 at 22:41
  • 5
    $\begingroup$ 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 '15 at 22:46
  • 4
    $\begingroup$ @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. $\endgroup$ – Noam D. Elkies Aug 24 '15 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. $\endgroup$ – Steven Stadnicki Aug 24 '15 at 23:14
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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. $\endgroup$ – Timothy Chow Aug 25 '15 at 20:37

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.

  • 3
    $\begingroup$ 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." $\endgroup$ – Timothy Chow Aug 25 '15 at 14:03

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