The formation of amyloid fibrils is often encountered in Alzheimer's disease, type II diabetes, and transmissible spongiform encephalopathies. In the last few years, however, mounting evidence has suggested that the soluble oligomers of amyloid-forming peptides are also cytotoxic agents. Understanding the early pathway steps of amyloid self-assembly at atomic detail might therefore be crucial for the development of specific inhibitors to prevent amyloidosis in humans. Using the activation-relaxation technique and a generic energy model, we study in detail the aggregation of a hexamer of KFFE peptide. Our simulations show that a monomer remains disordered, but that six monomers placed randomly in an open box self-associate to adopt, with various orientations, three possible distant low-energy structures. Two of these structures show a double-layer beta-sheet organization, in agreement with the structure of amyloid fibrils as observed by x-ray diffraction, whereas the third one consists of a barrel-like curved single-layer hexamer. Based on these results, we propose a bidirectional growth mode of amyloid fibril, involving alternate lateral and longitudinal growths.