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