Pentagon Tiling
There are at least 15 classes of convex pentagonal tilings, as illustrated above. The first five were discovered during investigations of German mathematician Karl Reinhardt in 1918. After a gap of 50 years, R. B. Kershner found three more in 1968. Richard James subsequently discovered a ninth type of pentagonal tiling in 1975 and over the next few years, Marjorie Rice discovered another four types. Rolf Stein found a 14th tiling in 1985. The most recently discovered 15th tiling was found by Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington Bothell in 2015 using a computer to exhaustively search through a large but finite set of possibilities (Bellos 2015).
It has not been proven whether these 15 cases exhaust all possible tilings, but no others are known.
Note that the tile in the 14th tiling is essentially different from the others because it is unique (up to similarity), while all the others form families with at least
one parameter. For this special tile, all angles are determined, and all sides are
in fixed ratios. Its angles 
and 
are 
and 
, respectively. The tiling by
this tile is also unique (B. Grünbaum, pers. comm.).
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