This final chapter methodically derives the seventeen types of wallpaper
patterns starting from the 'simpler' (rotationless) ones and then using
them to build the more involved ones, ruling out all non-existing
possibilities. The first task is based on the interplay between pattern
translations and the translations naturally associated with the pattern's
(glide) reflection, and culminates in a characterization of the
rotationless patterns in terms of their translations. The second task is
based on the said characterization and also on simple observations (often
related to isometry compositions and 'images') concerning the ways that
(glide) reflection axes may be positioned with respect to the pattern's
rotation centers.