The notched cube tiles Rn

Self-Affine Tiles in Rn

A self-affine tile in R is a set T of positive measure with A(T) = d ∈ $ < (T + d), where A is an expanding n × n real matrix with det (A) = m on integer, and $ = {d 1 ,d 2 , . . . , d m } ⊆ R is a set of m digits. It is known that self-affine tiles always give tilings of R by translation. This paper extends the known characterization of digit sets $ yielding self-affine tiles. It proves seve...

Cube-Tilings of Rn and Nonlinear Codes

Families of non-lattice tilings of R n by unit cubes are constructed. These tilings are specializations of certain families of nonlinear codes over GF( 2 ). These cube-tilings provide building blocks for the construction of cube-tilings such that no two cubes have a high-dimensional face in common. We construct cube-tilings of R n such that no two cubes have a common face of dimension exceeding...

Hausdorff Dimension of Boundaries of Self-affine Tiles in Rn

INTEGRAL SELF-AFFINE TILES IN Rn II. LATTICE TILINGS

Let A be an expanding n n integer matrix with j det(A)j = m. A standard digit set D for A is any complete set of coset representatives for Z n =A(Z n). Associated to a given D is a set T(A; D), which is the attractor of an aane iterated function system, satisfying T = d2D (T + d). It is known that T(A; D) tiles R n by some subset of Z n. This paper proves that every standard digit set D gives a...

DNA Tiles, Wang Tiles and Combinators

In this paper we explore the relation between Wang Tiles and Schonfinkel Combinators in order to investigate Functional Combinators as an programming language for Self-assembly and DNA computing. We show: How any combinatorial program can be expressed in terms of Wang Tiles, and again, how any computation of the program fits into a grid of tiles of a suitable finite, tile set, and finally, how ...