Some Helical Structures Generated by Reflexions

Abstract

There has recently been a revival of interest in the helical structure built up as a column of face-sharing tetrahedra, because of possible applications in structural crystallography (Nelson 1983). This structure and its analogues in spaces of different dimensions are investigated here. It is shown that the only crystallographic cases are the structures in one- and two-dimensional space. For three and higher dimensional space the structures are all non-crystallographic. For the physically important case of three dimensions, this result is implicit in an early discussion by Coxeter (1969). Results obtained here include explicit formulae for the positions of all vertices of the simplexes for dimensions n = 1-4 and a demonstration that, for arbitrary n, the ratio of the translation component of the screw to the edge of the simplex is {6/ n(n+ I)(n+ 2)}1/2