The geometry of Niggli reduction: BGAOL –embedding Niggli reduction and analysis of boundaries

Niggli reduction can be viewed as a series of operations in a six-dimensional space derived from the metric tensor. An implicit embedding of the space of Niggli-reduced cells in a higher-dimensional space to facilitate calculation of distances between cells is described. This distance metric is used to create a program, BGAOL, for Bravais lattice determination. Results from BGAOL are compared with results from other metric based Bravais lattice determination algorithms. This embedding depends on understanding the boundary polytopes of the Niggli-reduced cone N in the six-dimensional space G₆. This article describes an investigation of the boundary polytopes of the Niggli-reduced cone N in the six-dimensional space G⁶ by algebraic analysis and organized random probing of regions near one-, two-, three-, four-, five-, six-, seven- and eightfold boundary polytope intersections. The discussion of valid boundary polytopes is limited to those avoiding the mathematically interesting but crystallographically impossible cases of zero-length cell edges. Combinations of boundary polytopes without a valid intersection in the closure of the Niggli cone or with an intersection that would force a cell edge to zero or without neighboring probe points are eliminated. In all, 216 boundary polytopes are found. There are 15 five-dimensional boundary polytopes of the full G⁶ Niggli cone N.