We define a group structure on the set of compact “minimal” paths in R2. We classify all finitely generated subgroups of this group G: they are free products of free abelian groups and surface groups. Moreover, each such group occurs in G. The subgroups of G isomorphic to surface groups arise from certain topological 1-forms on the corresponding surfaces. We construct examples of such 1-forms f...

We establish a relationship between the word complexity and the number of generalized diagonals for a polygonal billiard. We conclude that in the rational case the complexity function has cubic upper and lower bounds. In the tiling case the complexity has cubic asymptotic growth.

We study the classical motion in bidimensional polygonal billiards on the sphere. In particular we investigate the dynamics in tiling and generic rational and irrational equilateral triangles. Unlike the plane or the negative curvature cases we obtain a complex but regular dynamics.

We say that a tiling separates discs of packing in the Euclidean plane, if each tile contains exactly one member packing. It is known elementary geometric problem to show for locally finite circular discs, there exists separating with convex polygons. In this paper we property remains true circle packings on sphere and hyperbolic plane. Moreover, plane circles are only whose similar copies can ...

We have investigated the possibility of decorating, using a bottom-up strategy, patchy particles in such a way that they self-assemble in (two-dimensional) Archimedean tilings. Except for the trihexagonal tiling, we have identified conditions under which this is indeed possible. The more compact tilings, i.e., the elongated triangular and the snub square tilings (which are built up by triangles...

A tiling of the plane with polygonal tiles is said to be strict if any point common tcl two tiles is a vertex of both or a vertex of neither. A triangle is said tc> be rational if its sides have rational length. Recently R.B. Eggleton asked if it is pos+le to strictly tile the plane <with rational triangles us@ preci.sely one triangle from each congruence clas\. In this paper we cDnstructivelv ...

This paper provides a theoretical foundation of a thinning method due to Kovalevsky for 2D digital binary images modelled by cell complexes or, equivalently, by Alexandroff T0 topological spaces, whenever these are constructed from polygonal tilings. We analyze the relation between local and global simplicity of cells, and prove their equivalence under certain conditions. For the proof we apply...

due to avoiding utilization of human figures and abstaining from idolatry, decorations have specific geometry in islamic art and architecture. one of the striking characteristics of the knots (girih) that has caused to dynamicity throughout its thousand-year history is its regeneration and diversification of the diverse geometric properties. girih tiling decoration is part of geometric arts in ...

A polyomino is a polygonal region with axis-parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling packing problems pieces container P . We give polynomial-time algorithms for deciding if can be tiled k × squares any fixed part the input (that is, union set non-overlapping squares) maximum number 2 1 dominos, allowing rotations by 90°....