From teaching experience. X. Regular, semi-regular and irregular plane partitions

Abstract

The article deals with regular, semi-regular and irregular plane partitions, among the latter are partitions into equal polygons and arbitrary partitions with simple knots. It continues the author's supplementary lectures to the university textbook «Crystallography» by G. M. Popov and I. I. Shafranovsky, which was recommended to the students of geology of the St. Petersburg Mining University. The new topic is interesting because the first problem is solved elementary in the form of a Diophantine equation, the second is solved by algorithmic enumeration of variants, the third is related to the 18th Gilbert problem and is not exhausted yet, and the fourth shows the transition to the geometry of quasi-crystalline and amorphous structures. It gives examples of easily formulated but difficult to solve problems and encourages students to be scientifically creative in geometric crystallography and related disciplines (petrography, materials science). The presentation of the theory is illustrated with examples of plane partitions in urban interiors of St. Petersburg.