Radix Representation and Rep-tiling

1. Rep-tiling. This paper provides a short survey of recent work by the author on self-replicating tilings, a subject that has recently come under independent investigation by several mathematicians, for example Grr ochenig and Hass 11], Grr ochenig and Madyeh 10], Kenyon 14], Kitto, Vince and Wilson 15] and Vince 21] 22]. These tilings have application to computer addressing 5] 15] 20] and to wavelets 10], but the main concern of this paper is the construction and clas-siication of rep-tilings. In particular, this involves the connection between radix representation of points in a lattice and self-replicating tilings of Euclidean space. Section 2 concerns radix representation, and Secion 3 connects radix representation with self-replicating tiling. The common wall tilings by squares or by hexagons are examples of lattice tilings. A lattice in R n is the set of all integer combinations of n linearly independent vectors. A lattice tiling is a tiling of R n by translates of a single compact tile T by a lattice L. In other words, the tiling is T = fx + T j x 2 Lg. Two distinct tiles in a tiling are allowed to intersect only on their boundary, not at an interior point. The self-replicating a property goes back at least to 1964. Golomb 7] deened a plane gure F to be rep-k if F can be tiled by k congruent similar gures. Three rep-4 gures are shown in Figure 1. The notion of tiling and the notion of self-replicating can be combined so that each tile in a tiling is a rep-k gure. A k-similarity tiling is a tiling T , by copies of a single tile, such that for some similarity , the image (T) is, in turn, tiled by k tiles of T , for each T 2 T. This is essentially the deenition of Grunbaum and Shephard in their comprehensive book \Tilings and Patterns" 12]. Three