  DNA nanoparticles with three-fold coordination have been observed to
self-assemble in experiment into a network equivalent to the hexagonal (6.6.6)
tiling, and a network equivalent to the 4.8.8 Archimedean tiling. Both networks
are built from a single type of vertex. Here we use analytic theory and
equilibrium and dynamic simulation to show that a model particle, whose
rotational properties lie between those those of the vertices of the 6.6.6 and
4.8.8 networks, can self-assemble into a network built from three types of
vertex. Important in forming this network is the ability of the particle to
rotate when bound, thereby allowing the formation of more than one type of
binding motif. The network in question is equivalent to a false tiling, a
periodic structure built from irregular polygons, and possesses 40 particles in
its unit cell. The emergence of this complex structure, whose symmetry
properties are not obviously related to those of its constituent particles,
highlights the potential for creating new structures from simple variants of
existing nanoparticles.
