  In this paper will be introduced large, probably complete family of complex
base systems, which are 'proper' - for each point of the space there is a
representation which is unique for all but some zero measure set. The condition
defining this family is the periodicity - we get periodic covering of the plane
by fractals in hexagonal-type structure, what can be used for example in image
compression. There will be introduced full methodology of analyzing and using
this approach - both for the integer part: periodic lattice and the fractional:
attractor of some IFS, for which the convex hull or properties like dimension
of the boundary can be found analytically. There will be also shown how to
generalize this approach to higher dimensions and found some proper systems in
dimension 3.
