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While the hexagonal lattice is ubiquitous in two dimensions, the body centered cubic lattice and the face centered cubic lattice are both commonly observed in three dimensions. A geometric variational problem motivated by the diblock copolymer theory consists of a short range interaction energy and a long range interaction energy. In three dimensions, and when the long range interaction is given by the nonlocal operator $$(-\Delta)^{-3/2}$$, it is proved that the body centered cubic lattice is the preferred structure. 

The free energy of a ternary system, such as a triblock copolymer, is a sum of two parts: an interface energy determined by the size of the interfaces separating the micro-domains of the three constituents, and a long range interaction energy that serves to prevent unlimited micro-domain growth. In two dimensions a parameter range is identified where the system admits stable stationary disk assemblies. Such an assembly consists of perturbed disks made from either type-I constituent or type-II constituent. All the type-I disks have approximately the same radius and all the type-II disks also have approximately the same radius. The locations of the disks are determined by minimization of a function. Depending on the parameters, the disks of the two types can be mixed in an organized way, or mixed in a random way. They can also be fully separated. The first scenario offers a mathematical proof of the existence of a morphological phase for triblock copolymers conjectured by polymer scientists. The last scenario shows that the ternary system is capable of producing two levels of structure. The primary structure is at the microscopic level where disks form near-perfect lattices. The secondary structure is at the macroscopic level forming two large regions, one filled with type-I disks and the other filled with type-II disks. A macroscopic, circular interface separates the two regions.