On fillings of 2N-gons with rhombi

da Silva, I.P.F, On fillings of ZN-gons with rhombi, Discrete Mathematics 111 (1993) 137-144. In the first part of this paper we give a generalization of a result of Ringel [ll] on simple arrangements of pseudolines. In terms of fillings with rhombi of an N-zonogon, we obtain a way of generating every filling from a given one by successively performing the same local transformation. In the second part we interpret, via oriented matroids, fillings of N-zonogons with rhombi as families of vectors in ZN. While for results in Section 1 the finiteness of the filling is essential, the aim of Section 2 is to strengthen the possibility already pointed out by Dress [7] of obtaining de Bruijn’s results [4] on Penrose tilings from a purely combinatorial point of view. Recalling some results A$lling with rhombi of a convex polygon P is a family 9 of rhombi such that (i) if R, R’E~ then either RnR’=@ or RnR’ is a face (vertex or edge) of both and R’, (4 URE.~R=P, (iii) the length of the edge of any rhombus FEN is 1. Given a filling 9 of a convex polygon P, for every rhombus REP;, with one edge contained in an edge e of P, the zone of R, ZR, is the ‘strip’ of rhombi constructed from R in the following way. Put R, = R, e, = enR and e, the edge of R parallel to eO, Z1 = RI. For i> 1 let Ri be the rhombus of 9 which verifies RinRi_ 1 =ei_ 1, define ei as the edge of Ri opposite to ei_1 and put Zi=Zi_luRi. Since P is a convex polygon, this procedure terminates at some stage n, R, being a rhombus with one edge contained in an edge e’ of P parallel to P. From the concept of zone (see [S]) we easily obtain a characterization of what convex polygons can be filled with rhombi: a convex polygon P can be jlled with Correspondence to: Ilda P.F. da Silva, CMAF/INIC, Av. Prof. Gama Pinto, No. 2, 1699 Lisboa Codex, Portugal. 0012-365X/93/$06.00