The reconstruction of discrete two-dimensional pictures from their projections is one of the central problems in the areas of medical diagnostics, computer-aided tomography, pattern recognition, image processing and data compression. In this note, we determine the computational complexity of two open problems in this field; one of our results settles a long-standing open question. We will prove...

The problem of finding polyominoes that tile rectangles has attracted a lot of attention; see [1] for an overview, and [2, 3] for more recent results. Several general families of such polyominoes are known, but sporadic examples seem to be scarce. Marshall [2, Fig. 9] gives a polyomino of rectangular order 8, and asks if it can be generalized to a family of rectifiable polyominoes. Here we show...

We explore the art gallery problem for the special case that the domain (gallery) P is an mpolyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P . In particular, we show that ⌊ 3 ⌋ point guards are always sufficient and sometimes necessary to ...

Polyominoes are edge-connected sets of squares on the square lattice. The symbol λ usually denotes the growth constant of A(n), the sequence that enumerates polyominoes. In this paper we prove that λ ≤ 4.5685 by analyzing the growth constant of a sequence B(n), for which B(n) ≥ A(n) for any value of n ∈ N. The recursive formula for B(n) is based on the representation of a polyomino as the assem...

This paper describes a system to produce space layout topologies for architectural plans using an evolutionary approach. The layout specification is defined as a set of topological and directional constraints, which are used as a fitness function in the evolutionary system. The halfplane representation is used to represent the genotypes in the evolutionary system, for both arrangements of halfp...

This paper provides a bridge between the classical tiling theory and cellular automata on one side, and the complex neighborhood self-assembling situations that exist in practice, on the other side. A neighborhood N is a finite set of pairs (i, j) ∈ Z, indicating that the neighbors of a position (x, y) are the positions (x + i, y + j) for (i, j) ∈ N . This includes classical neighborhoods of si...

The idea of using sets of polyominoes as practical assembly puzzles can be traced back more than two and a half centuries. Les Amusmens, published in 1749, included a cross puzzle made of three Z-pentominoes and two L-polyominoes. The earliest rectangular puzzle formed from polyominoes that we found, named The Jags and Hooks Puzzle, was included in Catel’s Catalogue of 1785 [1]. It consisted of...