  When two-dimensional pattern-forming problems are posed on a periodic domain,
classical techniques (Lyapunov-Schmidt, equivariant bifurcation theory) give
considerable information about what periodic patterns are formed in the
transition where the featureless state loses stability. When the problem is
posed on the whole plane, these periodic patterns are still present. Recent
work on the Swift-Hohenberg equation (an archetypal pattern-forming partial
differential equation) has proved the existence of quasipatterns, which are not
spatially periodic and yet still have long-range order. Quasipatterns may have
8-fold, 10-fold, 12-fold and higher rotational symmetry, which preclude
periodicity. There are also quasipatterns with 6-fold rotational symmetry made
up from the superposition of two equal-amplitude hexagonal patterns rotated by
almost any angle $\alpha$ with respect to each other. Here, we revisit the
Swift-Hohenberg equation (with quadratic as well as cubic nonlinearities) and
prove existence of several new quasipatterns. The most surprising are
hexa-rolls: periodic and quasiperiodic patterns made from the superposition of
hexagons and rolls (stripes) oriented in almost any direction with respect to
each other and with any relative translation; these bifurcate directly from the
featureless solution. In addition, we find quasipatterns made from the
superposition of hexagons with unequal amplitude (provided the coefficient of
the quadratic nonlinearity is small). We consider the periodic case as well,
and extend the class of known solutions, including the superposition of
hexagons and rolls. While we have focused on the Swift-Hohenberg equation, our
work contributes to the general question of what periodic or quasiperiodic
patterns should be found generically in pattern-forming problems on the plane.
