Reduced cells based on extremal principles

It is known that the Buerger cell, a + b + c = abs min, is ambiguous. Uniqueness is usually achieved by an additional system of inequalities which leads to the generally accepted Niggli cell. However, this system is rather unusual and does not suggest any geometrical meaning for the Niggli cell. In this paper four types of unique cells originating from the Buerger cell are introduced by means of simple conditions which have an extremal character. Any of these cells may stand for a reduced cell and has an express geometrical property. One of the four types coincides with the Niggli cell, which is thus given a geometrical interpretation. Systems of inequalities are shown that allow recognition of the cell of any type and algorithms are presented for achieving it. An algorithm for obtaining all Buerger cells of a lattice is included. The use of the reciprocal lattice enables the definition of four further unique cells which, however, need not be Buerger cells and are not discussed in detail. The mathematics must deal with a number of inequalities which often contain square roots, and sometimes rather intricate technical tricks are required.