We introduce the concept of many-to-one mapping of form to function and suggest that this emergent property of complex systems promotes the evolution of physiological diversity. Our work has focused on a 4-bar linkage found in labrid fish jaws that transmits muscular force and motion from the lower jaw to skeletal elements in the upper jaws. Many different 4-bar shapes produce the same amount of output rotation in the upper jaw per degree of lower jaw rotation, a mechanical property termed Maxillary KT. We illustrate three consequences of many-to-one mapping of 4-bar shape to Maxillary KT. First, many-to-one mapping can partially decouple morphological and mechanical diversity within clades. We found with simulations of 4-bars evolving on phylogenies of 500 taxa that morphological and mechanical diversity were only loosely correlated (R(2) = 0.25). Second, redundant mapping permits the simultaneous optimization of more than one mechanical property of the 4-bar. Labrid fishes have capitalized on this flexibility, as illustrated by several species that have Maxillary KT = 0.8 but have different values of a second property, Nasal KT. Finally, many-to-one mapping may increase the influence of historical factors in determining the evolution of morphology. Using a genetic model of 4-bar evolution we exerted convergent selection on three different starting 4-bar shapes and found that mechanical convergence only created morphological convergence in simulations where the starting forms were similar. Many-to-one mapping is widespread in physiological systems and operates at levels ranging from the redundant mapping of genotypes to phenotypes, up to the morphological basis of whole-organism performance. This phenomenon may be involved in the uneven distribution of functional diversity seen among animal lineages.