Les colonnes cylindriques transparentes unicolores et bicolores

Abstract

By unrolling the surface of a cylinder into a double-faced plane, the cylindrical, double-faced columns, with single-coloured (\cal j) or two-coloured (\cal j2) patterns and their symmetries, are studied. A symmetry is the isometric mapping of the column and its pattern set onto themselves, and is characterized by the retention or change of the patterns between the two faces, or by the colour-retention or colour-change of the patterns. All possible classes of these discontinuous columns are derived for the cases with and without translation. As extreme cases, the columns with an infinite-fold cylinder axis and those with an infinite screw pitch are given. For particular values of m, the rotational component order of the fundamental rototranslation, the 945 types of proper crystallographic, two-coloured, transparent, columns \cal j2 2c are deduced. Illustrations are given for the unrolling into a double-faced plane of some types of \cal j2 2c columns, of the two-coloured, biracial columns without translation, {\bar \cal j}2 2, and of the two-coloured, semicontinuous oblique columns.