Recent advances in the fabrication of quasicrystals in soft matter systems have increased the length scales for quasicrystals into the mesoscale range (20 to 500 ångströms). Thus far, dendritic liquid crystals, ABC-star polymers, colloids and inorganic nanoparticles have been reported to yield quasicrystals. These quasicrystals offer larger length scales than intermetallic quasicrystals (a few ångströms), thus potentially leading to optical applications through the realization of a complete photonic bandgap induced via multiple scattering of light waves in virtually all directions. However, the materials remain far from structurally ideal, in contrast to their intermetallic counterparts, and fine control over the structure through a self-organization process has yet to be attained. Here we use the well-established self-assembly of surfactant micelles to produce a new class of mesoporous silicas, which exhibit 12-fold (dodecagonal) symmetry in both electron diffraction and morphology. Each particle reveals, in the 12-fold cross-section, an analogue of dodecagonal quasicrystals in the centre surrounded by 12 fans of crystalline domains in the peripheral part. The quasicrystallinity has been verified by selected-area electron diffraction and quantitative phason strain analyses on transmission electron microscope images obtained from the central region. We argue that the structure forms through a non-equilibrium growth process, wherein the competition between different micellar configurations has a central role in tuning the structure. A simple theoretical model successfully reproduces the observed features and thus establishes a link between the formation process and the resulting structure.