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Correct identification of the Bravais lattice of a crystal is an importantstep in structure solution. Niggli reduction is a commonly used technique. Weinvestigate the boundary polytopes of the Niggli-reduced cone in thesix-dimensional space G6 by algebraic analysis and organized random probing ofregions near 1- through 8-fold boundary polytope intersections. We limitconsideration of boundary polytopes to those avoiding the mathematicallyinteresting but crystallographically impossible cases of 0 length cell edges.Combinations of boundary polytopes without a valid intersection in the closureof the Niggli cone or with an intersection that would force a cell edge to 0 orwithout neighboring probe points are eliminated. 216 boundary polytopes arefound: 15 5-D boundary polytopes of the full G6 Niggli cone, 53 4-D boundarypolytopes resulting from intersections of pairs of the 15 5-D boundarypolytopes, 79 3-D boundary polytopes resulting from 2-fold, 3-fold and 4-foldintersections of the 15 5-D boundary polytopes, 55 2-D boundary polytopesresulting from 2-fold, 3-fold, 4-fold and higher intersections of the 15 5-Dboundary polytopes, 14 1-D boundary polytopes resulting from 3-fold and higherintersections of the 15 5-D boundary polytopes. All primitive lattice types canbe represented as combinations of the 15 5-D boundary polytopes. Allnon-primitive lattice types can be represented as combinations of the 15 5-Dboundary polytopes and of the 7 special-position subspaces of the 5-D boundarypolytopes. This study provides a new, simpler and arguably more intuitive basisset for the classification of lattice characters and helps to illuminate someof the complexities in Bravais lattice identification. The classification isintended to help in organizing database searches and in understanding whichlattice symmetries are "close" to a given experimentally determined cell.

The Geometry of Niggli Reduction I: The Boundary Polytopes of the Niggli Cone