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

Consider the d-dimensional euclidean space E d. Two main results are presented: First, for any N 2 N, the number of types of periodic equivariant tilings (T ; ?) that have precisely N orbits of (2; 4; 6 : : :)-ags with respect to the symmetry group ?, is nite. Second, for any N 2 N, the number of types of convex, periodic equivariant tilings (T ; ?) that have precisely N orbits of tiles with respect to the symmetry group ?, is nite. 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.