Quartz aggregates revisited

Quartz aggregates that formed by coalescence of quartz crystals in the magma or in hydrothermal solution are considered. If the individuals have rhombohedral faces in contact, there will be two special cases: parallel intergrowths and intergrowths that agree in orientation and contact plane with Esterel twins grown from a twinned nucleus. For all other known cases, i.e. when the relative orientation satisfies the Japan, Sardinian, Tiflis, Zyndel-A or Samshvildo law, both individuals have exactly coinciding short symmetry translations in only one direction in the contact plane (monoperiodic twins). If a rhombohedral face is in contact with a prism face, monoperiodic twins will occur if the relative orientation satisfies the Zinnwald, Disentis or a proposed hypothetic law. The orientation of the two lattices can be expressed by a 180° rotation about an axis with low indices independent of c/a in the case of the Esterel, Japan and Sardinian laws. The same is true for the Tiflis and Zyndel-A laws only if they are redefined, and not at all in the case of the Samshvildo, Zinnwald, Disentis and hypothetic laws. When the two individuals have rhombohedral faces in contact, there will even be exact two-dimensional coincidence (multiplicity σ) in the contact plane and exact three-dimensional coincidence (multiplicity Σ) in space if the square of the axial ratio c/a is rational. Indications are found that (despite their high values) these multiplicities may be related to the frequency of occurrence of intergrowths in low- and high-quartz.