2 FINITENESS THEOREMS FOR PERIODIC TILINGS OF D-DIMENSIONAL EUCLIDEAN-SPACE

Consider the d-dimensional euclidean space E-d. Two main results are p re sented: First, for any N is an element of N, the number of types of periodic equivariant tilings (T, Gamma) that have precisely N orbits of (2, 4, 6,...)-flags with respect to the symmetry group Gamma, is fi nite. Second, for any N is an element of N, the number of types of con vex, periodic equivariant tilings (T, Gamma) that have precisely N orb its of tiles with respect to the symmetry group Gamma, is finite. The former result (and some generalizations) is proved combinatorially, us ing Delaney symbols, whereas the proof of the latter result is based o n both geometric arguments and Delaney symbols.