Indag. Mathem., N.S., 10 (4), 581 599 December 20, 1999 On the table and the chair by E. Arthur Robinson, Jr. Department of Mathematics, The George Washington University, Washington DC 20052, USA Communicated by Prof. M.S. Keane at the meeting of February 22, 1999 ABSTRACT We find topological models for the tiling dynamical systems corresponding to the chair and table rep-tiles. 1. INTRODUCTION A rep-tile is a polygon that can be tiled by a finite number of smaller, congruent copies of itself. Two well known examples are shown in Figure 1. Figure 1. The chair and table rep-tiles. We call these the chair and the table. Both of these rep-tiles are alsopolyominoes (cf. [2]), meaning they are edge to edge unions of squares. Given a rep-tile ~-, there is a corresponding set X of self-similar tilings of the plane. To get this, we decompose ~- into its small copies, obtaining a ~--shaped patch. Then we expand the small tiles in this patch back to their original size. Iterating this process, we obtain a sequence Xl, x2, x3... of larger and larger patches. 581