An approach is presented to the tripartite problem of modeling physical solids mathematically, representing the models in a computer, and using representations in geometric algorithms. Examples are primarily from the domain of manufacturing and design of discrete goods, but the results reported here have wider significance. Mathematical definitions can formalize many of our intuitions about three-dimensional (3-D) objects and operations on them. Representation-free (mathematical) models and functions allow formal properties to be defined for characterizing geometric representations. Three common representation schemes for 3-D objects are described briefly, along with some of their formal and informal properties. A rigorous, as opposed to ad hoc, approach to modeling has several advantages. Broadly, the conceptual complications and ambiguities which are endemic to ad hoc problem statements and solutions may be avoided by appealing to a precise mathematical semantics. Mathematical rigor is mandatory in applications such as automatic manufacturing in which correctness must be guaranteed and consistency and validity maintained.