  In many mathematical models for pattern formation, a regular hexagonal
pattern is stable in an infinite region. However, laboratory and numerical
experiments are carried out in finite domains, and this imposes certain
constraints on the possible patterns. In finite rectangular domains, it is
shown that a regular hexagonal pattern cannot occur if the aspect ratio is
rational. In practice, it is found experimentally that in a rectangular region,
patterns of irregular hexagons are often observed. This work analyses the
geometry and dynamics of irregular hexagonal patterns. These patterns occur in
two different symmetry types, either with a reflection symmetry, involving two
wavenumbers, or without symmetry, involving three different wavenumbers. The
relevant amplitude equations are studied to investigate the detailed
bifurcation structure in each case. It is shown that hexagonal patterns can
bifurcate subcritically either from the trivial solution or from a pattern of
rolls. Numerical simulations of a model partial differential equation are also
presented to illustrate the behaviour.
