New Compounds In3Ti2AO10, In6Ti6BO22, and Their Solid Solutions (A: Al, Cr, Mn, Fe, or Ga; B: Mg, Mn, Co, Ni, Cu, or Zn): Synthesis and Crystal Structures

Archetypes are defined as embeddings of minimal nets with maximal symmetry. A simple geometrical construction is proposed to construct the archetype associated with two-line-connected graphs with more than one cycle and without loops and a criterion is derived to check the embedding. The periodicity of the archetype is equal to the cyclomatic number of the quotient graph of the net, and the factor group of its space group with respect to the normal subgroup of all translations is isomorphic to the automorphism group of the quotient graph. Orthogonal projections are considered to ensure the generation of periodic structures with three-dimensional coordination polyhedra.