The paper discusses homeomorphic types of (periodic) filings of the plane in terms of their associated Delaney symbol. Such a symbol consists of a (finite) set ~ on which three involutions ao,al and tr2 act from the right such that troa ~ = a2ao and there are two maps mol ,m12 :~ ~ ~ satisfying certain compatibility conditions. It is shown how the barycentric subdivision of a tiling can be used to define its Delaney symbol and that the symbol characterizes the tiling up to (equivariant) homeomorphisms. Furthermore, it is shown how properties of the tiling can be recognized from corresponding properties of the symbol and how this technique can be used to enumerate various types of tilings with specific properties. If necessary, this enumeration can be done by appropriate computer programs. Among other results, we have been able to vindicate the results by Griinbaum et al., announced in [8]. Finally, some recursive enumeration formulas, based on the Delaney symbol technique,, are stated.
0. Introduction
In [8] the authors carefully exhibit 508 different types of '2-homeohedral, normal' tilings of the Euclidean plane ~ = E 2 and state: 'It would be nice to be able to assert that the enumeration presented here is complete.
Unfortunately we cannot do that, though it seems reasonable to suppose that the number of omissions (if any) must be quite small .... Practically speaking, it seems that there is no way of doing this in a completely systematic manner.' We shall see below that in fact such a systematic method does exist, and show how it can be used to supply the missing proofs. More precisely, we want to explain how the theory of Delaney symbols of tilings (cf. [2]-[4], [6], [7]) can be used efficiently to reduce enumeration problems like the one mentioned above to a finite, purely combinatorial problem which can be solved easily by appropriate computer programs. In addition, we want to announce some of our findings using such programs (among them a confirmation of the result quoted above) as well as a rather general recursive enumeration formula which can be used to check such programs, but may be also of some interest of its own.
1. THE DEFINITION OF TILINGS
Let us recall that a tiling of E can be defined as a Closed connected subset 7 of IE such that all connected components of E \Tare bounded and for any point x~ T there exists a neighbourhood U of x and a natural number r(x) = rr(x) >~ 1 such that the triple (U, U ~ T, {x}) is homeomorphic to the