5 and 10 order transforms in four dimensional crystal families

摘要

The aim of this conference paper is to present some algebraic methods, algorithms for the determination of the fixed point sets and the orbits of the four dimensional crystallographic groups. Our main topics are the following: - transformations of n-dimensional symmetry groups by (n + 1) * (n + 1) matrices; - examples for determining the Wyckoff positions and fixed point sets in the case of some space groups of three dimensional monoclinic crystal family; - fixed point sets and orbits of some groups of four dimensional decagonal and icosahedral crystal family; - four dimensional symmetry transformations with five and ten orders.