Biodiversity can be divided into two aspects: richness (the number of species or other taxa in a community or sample) and evenness (a measure of the distribution of relative abundances of different taxa in a community or sample). Sample richness is typically evaluated using rarefaction, which normalizes for sample size. Evenness is typically summarized in a single value. It is shown here that Hurlbert's probability of interspecific encounter (Δ1), a commonly used sample-size independent measure of evenness, equals the slope of the steepest part of the rising limb of a rarefaction curve. This means that rarefaction curves provide information on both aspects of diversity. In addition, regional diversity (gamma) can be broken down into the diversity within local communities (alpha) and differences in taxonomic composition among local communities (beta). Beta richness is expressed by the difference between the composite rarefaction curve of all samples in a region with the collector's curve for the same samples. The differences of the initial slopes of these two curves reflect the beta evenness thanks to the relationship between rarefaction and Δ1. This relationship can be further extended to help interpret species-area curves (SAC's). As previous authors have described, rarefaction provides the null hypothesis of passive sampling for SAC's, which can be interpreted as regional collector's curves. This allows evaluation of richness and evenness at local and regional scales using a single family of well-established, mathematically related techniques.