There are at least three tilings of irregular hexagons,
illustrated above.
They are given by the following types:
(1)
(Gardner 1988). Note that the periodic hexagonal tessellation
is a degenerate case of all three tilings with
(2)
and
(3)
Amazingly, the number of plane partitions contained in an box also gives the number of hexagon tilings
by rhombi for a hexagon of side lengths , ,
, , ,
(David and Tomei 1989, Fulmek and Krattenthaler
2000). The asymptotic distribution of rhombi in a random hexagon tiling by rhombi
was given by Cohn et al. (1998). A variety of enumerations for various explicit
positions of rhombi are given by Fulmek and Krattenthaler (1998, 2000).
Cohn, H.; Larsen, M.; and Propp, J. "The Shape of a Typical Boxed Plane Partition." New York J. Math.4, 137-166,
1998.David, G. and Tomei, C. "The Problem of the Calissons."
Amer. Math. Monthly96, 429-431, 1989.Gardner, M. "Tilings
with Convex Polygons." Ch. 13 in Time
Travel and Other Mathematical Bewilderments. New York: W. H. Freeman,
pp. 162-176, 1988.Fulmek, M. and Krattenthaler, C. "The Number
of Rhombus Tilings of a Symmetric Hexagon which Contains a Fixed Rhombus on the Symmetry
Axis, I." Ann. Combin.2, 19-40, 1998.Fulmek, M.
and Krattenthaler, C. "The Number of Rhombus Tilings of a Symmetric Hexagon
which Contains a Fixed Rhombus on the Symmetry Axes, II." Europ. J. Combin.21,
601-640, 2000.
contained in an 
box also gives the number of hexagon tilings
by rhombi for a hexagon of side lengths 
, 
,
, 
, 
,
(David and Tomei 1989, Fulmek and Krattenthaler
2000). The asymptotic distribution of rhombi in a random hexagon tiling by rhombi
was given by Cohn et al. (1998). A variety of enumerations for various explicit
positions of rhombi are given by Fulmek and Krattenthaler (1998, 2000).
