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A new and efficient algorithm based on multipole techniques is presented which calculates the electrostatic forces and potentials in macroscopic periodic assemblies of particles. The fast multipole algorithm (FMA) can be used to compute forces within then-particle unit cell inO(n) time. For the cubic lattice, forces due to a 3^k× 3^k× 3^klattice of images of the unit cell, containing 3^3knparticles, can be computed inO(nk²+k³logk) time to arbitrary precision. The algorithm was readily added onto an existing FMA implementation, and computational results are presented. Accurate electrostatic computations were done on a 3⁸× 3⁸× 3⁸region of 100000-particle unit cells, giving a volume of 28 quadrillion particles at less than a twofold cost over computing the forces and potentials in the unit cell alone. In practice, ak= 4…6 simulation approximates the true infinite lattice Ewald sum forces (including the shape-dependent dipole …