Mallard's law recast as a Diophantine system: fast and complete enumeration of possible twin laws by [reticular] [pseudo] merohedry

Mallard's law of observation states that known twins by reticular pseudomerohedry have low twin index m and low obliquity δ between a lattice row and a lattice plane. Crystals in the twin are related by exact reflection in the lattice plane, exact rotation by π about the perpendicular to the lattice plane, or by exact rotation about the lattice row by 2kπ/q, with q = 2, 3, 4 or 6, and 1 ≤ k < q. In known cases, m is up to five or six, and δ is up to five or six degrees. Mallard's converse problem is then about finding all pairs of indices for rows and planes leading to twin indices not larger than m and to obliquities that are at most δ. Mallard's law is recast as the Diophantine pair constituted by the equality h · u = n and the inequality |h × u| < n tan δ. If a primitive reference system is used, the integer n is either m, 2m or 3m. A direct general solution of this system for u given n, h, δ and lattice data is detailed. That straightforward solution involves a tiny two-dimensional grid for u, considerably reducing the number of permutations to be considered. If the primitive reference system is Buerger-reduced, then moduli for indices of solutions h cannot exceed 3m, thus establishing a simple way to produce complete solutions. A program called OBLIQUE was designed on such principles. An implementation is available for free execution at http://ylp.icpet.nrc.ca/oblique/. OBLIQUE is also an interactive tool in Toth Information Systems' Materials Toolkit (http://www.tothcanada.com/), an exploitation framework currently operating on CRYSTMET and ICSD crystal structure data. The example of quartz cell data is described with m_max = 3 and δ_max = 6°.