A large number of synthetic and natural compounds selforganize
into bulk phases exhibiting periodicities on the 10²⁸–10²⁶ metre scale as a consequence of their molecular shape, degree of amphiphilic character and, often, the presence of additional non-covalent interactions. Such phases are found in lyotropic systems (for example, lipid–water, soap–water), in a range of block copolymers and in thermotropic (solvent-free) liquid crystals. The resulting periodicity can be one-dimensional (lamellar phases), two-dimensional (columnar phases) or three dimensional (‘micellar’ or ‘bicontinuous’ phases). All such two and three-dimensional structures identified to date obey the rules of crystallography and their symmetry can be described, respectively, by one of the 17 plane groups or 230 space groups. The ‘micellar’ phases have crystallographic counterparts in transition-metal alloys, where just one metal atom is equivalent to a 10³ - 10⁴-atom micelle. However, some metal alloys are known to defy the rules of crystallography and form so-called quasicrystals, which have rotational symmetry other than the allowed two-, three-, four- or six-fold symmetry. Here we show that such quasiperiodic structures can also exist in the scaled-up micellar phases,representing a new mode of organization in soft
matter.