Two Finiteness Theorems for Periodic Tilings of
d-Dimensional Euclidean Space

N.P. Dolbilin, A.W.M. Dress and D.H. Huson

Abstract:
Consider the d-dimensional euclidean space E^d.
Two main results are presented:
First, for any N, there exist only finitely
many different types of periodic tilings of E^d,
which are (2,4,6,...)-N-transitive.
Second, for any N, there exists only a
finite number of types of convex, periodic, and
tile-N-transitive tilings of of $E^d.
The former result (and some generalizations)
is proved combinatorially, using Delaney symbols,
whereas the proof of the latter result is based on both
geometric arguments and Delaney symbols.

To appear in: J. Discrete and Computational Geometry