Many unrelated proteins and peptides have been found spontaneously to form amyloid fibers above a critical concentration. Even for a single sequence, however, the amyloid fold is not a single well-defined structure. Although the cross-β hydrogen bonding pattern is common to all amyloids, all other aspects of amyloid fiber structures are sensitive to both the sequence of the aggregating peptides and the solvent conditions under which the aggregation occurs. Amyloid fibers are easy to identify and grossly characterize using microscopy, but their insolubility and aperiodicity along the dimensions transverse to the fiber axis have complicated detailed experimental structural characterization. In this paper, we explore the landscape of possibilities for amyloid protofilament structures that are made up of a single stack of peptides associated in a parallel in-register manner. We view this landscape as a two-dimensional version of the usual three-dimensional protein folding problem: the survey of the two-dimensional folds of protein ribbons. Adopting this view leads to a practical method of predicting stable protofilament structures of arbitrary sequences. We apply this scheme to variants of Aβ, the amyloid forming peptide that is characteristically associated with Alzheimer's disease. Consistent with what is known from experiment, we find that Aβ protofibrils are polymorphic. To our surprise, however, the ribbon-folding landscape of Aβ turned out to be strikingly simple. We confirm that, at the level of the monomeric protofilament, the landscape for the Aβ sequence is reasonably well funneled toward structures that are similar to those that have been determined by experiment. The landscape has more distinct minima than does a typical globular protein landscape but fewer and deeper minima than the landscape of a randomly shuffled sequence having the same overall composition. It is tempting to consider the possibility that the significant degree of funneling of Aβ's ribbon-folding landscape has arisen as a result of natural selection. More likely, however, the intermediate complexity of Aβ's ribbon-folding landscape has come from the post facto selection of the Aβ sequence as an object of study by researchers because only by having a landscape with some degree of funneling can ordered aggregation of such a peptide occur at in vivo concentrations. In addition to predicting polymorph structures, we show that predicted solubilities of polymorphs correlate with experiment and with their elongation free energies computed by coarse-grained molecular dynamics.