ABSTRACT
In theory, physical crystals can be represented by idealized mathematical lattices. Under appropriate conditions, these representations can be used for a variety of purposes, such as identifying, classifying, and understanding the physical properties of materials. Critical to these applications is the ability to construct a unique representation of the lattice. The vital link that enabled this theory to be realized in practice was provided by the 1970 paper of A. Santoro and A. D. Mighell, Determination of Reduced Cells. In the early 1960s, the Crystal Data Center at National Bureau of Standards (NBS) started to build a database with chemical and crystallographic information on all classes of materials, including inorganics, organics, minerals, and metals. The ability to determine a unique reduced cell and the subsequent achievements in lattice analysis, especially the matrix approach, have been critical milestones in crystallography. They established an important mathematical rigor in crystallography and in the materials sciences and have stimulated many practical applications.
