On a minimum tetrahedron in a three-dimensional lattice. Part I. Lattices with a shortest basis fulfilling b · c ≥ 0, a · c ≥ 0, a · b ≥ 0

The problem of a representative body of a three-dimensional lattice is considered. The cell fulfilling a + b + c = min is clearly not unique: even five mutually non-congruent such cells can exist in some lattices [Gruber (1973). Acta Cryst. A29, 433–440]. The idea that this number could be reduced by replacing the cell (i.e. a parallelepiped) by another, possibly more suitable, geometrical object is considered. For this object a lattice tetrahedron fulfilling the condition a + b + c + d + e + f = min is chosen, a to f being the lengths of its edges. It is called the minitetrahedron of the lattice. In this article, the problem is solved in detail for lattices that can be generated by a basis a, b, c fulfilling |a| + |b| + |c| = min, b · c ≥ 0, a · c ≥ 0, a · b ≥ 0. It turns out that in this case not more than two mutually non-congruent minitetrahedra can exist. Necessary and sufficient conditions for the uniqueness are found. They have the form of inequalities between the lengths of the edges and diagonals of the parallelepiped formed by the vectors a, b, c. A procedure for determining all minitetrahedra of a given lattice is shown. Some results are illustrated graphically and all assertions are proved mathematically.