In this paper, we introduce a new class of polyominoes, called closed paths, and study the primality their associated ideal. Inspired by an existing conjecture that characterizes polyomino ideal nonexistence zig-zag walks, classify all paths which do not contain give opportune toric representations ideals. To support conjecture, prove having no walks is necessary sufficient condition for path. ...

In a polyomino set (1,2)-achievement game the maker and the breaker alternately mark one and two previously unmarked cells respectively. The maker's goal is to mark a set of cells congruent to one of a given set of polyominoes. The breaker tries to prevent the maker from achieving his goal. The teams of polyominoes for which the maker has a winning strategy is determined up to size 4. In set ac...

A VDB topological index is defined as

In 1966, Golomb showed that any polyomino which tiles a rectangle also tiles a larger copy of itself. Although there is no compelling reason to expect the converse to be true, no counterexamples are known. In 3 dimensions, the analogous result is that any polycube that tiles a box also tiles a larger copy of itself. In this note, we exhibit a polycube (a ‘notched cube’) that tiles a larger copy...

In this paper we explore the power of geometry to overcome the limitations of non-cooperative selfassembly. We define a generalization of the abstract Tile Assembly Model (aTAM), such that a tile system consists of a collection of polyomino tiles, the Polyomino Tile Assembly Model (polyTAM), and investigate the computational powers of polyTAM systems at temperature 1, where attachment among til...

Beauquier and Nivat introduced and gave a characterization of the class of pseudo-square polyominoes that tile the plane by translation: a polyomino tiles the plane by translation if and only if its boundary word W may be factorized as W = XY X Y . In this paper we consider the subclass PSP of pseudo-square polyominoes which are also parallelogram. By using the Beauquier-Nivat characterization ...

We study the combinatorial properties and the problem of generating exhaustively double square tiles, i.e. polyominoes yielding two distinct periodic tilings by translated copies such that every polyomino in the tiling is surrounded by exactly four copies. We show in particular that every prime double square tile may be obtained from the unit square by applying successively some invertible oper...

Polyominoes, shapes made up of squares connected edge-to-edge, provide a rich source of prototiles for edge-toedge fractal tilings. We give examples of fractal tilings with 2-fold and 4-fold rotational symmetry based on prototiles derived by dissecting polyominoes with 2-fold and 4-fold rotational symmetry, respectively. A systematic analysis is made of candidate prototiles based on lower-order...