Half-turns are shown to be the main determinants of many experimental Alzheimer’s Aβ fibril structures. Fibril structures contain three half-turn types, βαRβ, βαLβ and βεβ which each result in a ∼90° bend in a β-strand. It is shown that only these half-turns enable cross-β stacking and thus the right-angle fold seen in fibrils is an intrinsic feature of cross-β. Encoding a strand as a conformational sequence in β, αR, αL and ε(βL), pairwise combination rules for consecutive half-turns are used to decode this sequence to give the backbone path. This reveals how structures would be dramatically affected by a deletion. Using a wild-type Aβ(42) fibril structure and the pairwise combination rules, the Osaka deletion is predicted to result in exposure of surfaces that are mutually shielding from the solvent. Molecular dynamics simulations on an 11-mer β-sheet of Alzheimer’s Aβ(40) of the Dutch (E22Q), Iowa (D23N), Arctic (E22G), and Osaka (E22Δ) mutants, show the crucial role glycine plays in the positioning of βαRβ half-turns. Their “in-phase” positions along the sequence in the wild-type, Dutch mutant and Iowa mutant means that the half-folds all fold to the same side creating the same closed structure. Their out-of-phase positions in Arctic and Osaka mutants creates a flatter structure in the former and an S-shape structure in the latter which, as predicted, exposes surfaces on the inside in the closed wild-type to the outside. This is consistent with the gain of interaction model and indicates how domain swapping might explain the Osaka mutant’s unique properties.