  The rapidly developing field of optomechanics aims at the combined control of
optical and mechanical (solid-state or atomic) modes. In particular, laser
cooled atoms have been used to exploit optomechanical coupling for
self-organization in a variety of schemes where the accessible length scales
are constrained by a combination of pump modes and those associated to a second
imposed axis, typically a cavity axis. Here, we consider a system with many
spatial degrees of freedom around a single distinguished axis, in which two
symmetries - rotations and translations in the plane orthogonal to the pump
axis - are spontaneously broken. We observe the simultaneous spatial
structuring of the density of a cold atomic cloud and an optical pump beam. The
resulting patterns have hexagonal symmetry. The experiment demonstrates the
manipulation of matter by opto-mechanical self-assembly with adjustable length
scales and can be potentially extended to quantum degenerate gases.
