Pentagon tilings by congruentconvexpentagons fall into 15
classes, 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-Mann 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).
The tiles in the 14th and 15th families are unique up to similarity, while the first 13 families have at least one parameter
(Mann et al. 2018). For each of these two tiles, all angles
are determined and all sides are in fixed ratios. The angles and of the 14th tile are and , respectively. The tiling
by this tile is also unique (B. Grünbaum, pers. comm.).
and 
of the 14th tile are 
and 
, respectively. The tiling
by this tile is also unique (B. Grünbaum, pers. comm.).
-Block Transitive Tilings." Geom. Dedicata 194,
141-167, 2018.