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As you can read here: http://www.npr.org/sections/thetwo-way/2015/08/14/432015615/with-discovery-3-scientists-chip-away-at-an-unsolvable-math-problem there are now 15 known convex pentagons, or nonregular pentagons with the angles pointing outward, that can "tile the plane."

Recently I have stumbled upon this site: http://domesticat.net/quilts/sunshine (see the 2nd image from the top:)

Hirschhorn tiling

This tiling uses 60-160-80-100-140 pentagons, all boarders are equal in size. It has a Hirschhorn circle in the middle with 18 "arms" that are stretching outwards. Those arms get thicker by adding 2 pentagons with each level. The picture above shows the 2 and a half levels.

Here you can see a bathroom floor tiled with such pattern. It shows more levels, but not a full circle: http://burtleburtle.net/bob/other/bathroom.html

bathroom floor

Question: How can it be proved that such a tiling covers the plane?

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    $\begingroup$ The tile itself here is not new -- it's a case of "type 1" in the Wikipedia article on pentagonal tilings because two of the sides are parallel. Therefore it does not count as a 16th type. $\endgroup$ – Henning Makholm Mar 24 '16 at 12:34
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    $\begingroup$ Specifically, it's the non-periodic tiling here: en.wikipedia.org/wiki/… $\endgroup$ – David K Mar 24 '16 at 13:49

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