  This work presents a construction method for interlocking assemblies based on
planar crystallographic symmetries. Planar crystallographic groups, also known
as wallpaper groups, correspond to tessellations of the plane with a tile,
called a fundamental domain, such that the action of the group can be used to
tessellate the plane with the given tile. The main idea of this method is to
extend the action of a wallpaper group so that it acts on three-dimensional
space and places two fundamental domains into parallel planes. Next, we
interpolate between these domains to obtain a block that serves as a candidate
for interlocking assemblies. We show that the resulting blocks can be
triangulated, and we can also approximate blocks with smooth surfaces using
this approach. Finally, we show that there exists a family of blocks derived
from this construction that can be tiled in multiple ways, characterised by
generalised Truchet tiles. The assemblies of one block in this family, which we
call RhomBlock, correspond to tessellations with lozenges.
