We investigated the role of particle potential symmetry on self-assembly by Monte Carlo simulation with a particular view toward synthetically creating structures of prescribed form and function. First, we established a general tendency for the rotational potential symmetries of the particles to be locally preserved upon self-assembly. Specifically, we found that a dipolar particle potential, having a continuous rotational symmetry about the dipolar axis, gives rise to chain formation, while particles with multipolar potentials (e.g., square quadrupole) having discrete rotational symmetries lead to the self-assembly of “random surface” polymers preserving the rotational symmetries of the particles within these sheet structures. Surprisingly, these changes in self-assembly geometry with the particle potential symmetry are also accompanied by significant changes in the thermodynamic character and in the kinetics of the self-assembly process. Linear chain growth involves a continuous chain growth process in which the chains break and reform readily, while the growth of the two-dimensional polymers only occurs after an “initiation” or “nucleation” time that fluctuates from run to run. We show that the introduction of artificial seeds provides an effective method for controlling the structure and growth kinetics of sheet-like polymers. The significance of these distinct modes of polymerization on the functional character of self-assembly growth is illustrated by constructing an artificial centrosome structure derived from particles having continuous and discrete rotational potential symmetries.