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 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 distorted hexagons are often observed. This work analyses the geometry and dynamics of distorted hexagonal patterns. These patterns occur in two different types, either with a reflection symmetry, involving two wave numbers, or without symmetry, involving three different wave numbers.
The relevant amplitude equations are studied to determine 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. The transcritical bifurcation with D3 symmetry that occurs for regular hexagons unfolds into either two pitchfork bifurcations or two saddlenode bifurcations.
Numerical simulations of a model partial differential equation are also presented to il~trate the behaviour.