a tiling of a surface is a decomposition of the surface into pieces, i.e. tiles, which cover it without gaps or overlaps. in this paper some special polygonal tiling of sphere, ellipsoid, cylinder, and torus as the most abundant shapes of fullerenes are investigated.

A tiling of a surface is a decomposition of the surface into pieces, i.e. tiles, which cover it without gaps or overlaps. In this paper some special polygonal tiling of sphere, ellipsoid, cylinder, and torus as the most abundant shapes of fullerenes are investigated.

The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting, and to provide a novel method for the construction of such polygonal tilings.

In this paper we extend general grid graphs to the grid graphs consist of polygons tiling on a plane, named polygonal grid graphs. With a cycle basis satisfied polygons tiling, we study the cyclic structure of Hamilton graphs. A Hamilton cycle can be expressed as a symmetric difference of a subset of cycles in the basis. From the combinatorial relations of vertices in the subset of cycles in th...

We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like ti...

We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like ti...

We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like ti...

Wederive ahomeomorphism invariant for those tiling spaceswhich aremadeby rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in in¢nitely manyorientations.The invariant is a quotient of C4 ech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like tiling s...

Given a finite rhombus tiling of polygonal region in the plane, associated critical Z-invariant Ising model is invariant under star-triangle transformations. We give simple matrix formula describing spin correlations between boundary vertices terms shape region. When regular polygon, our becomes an explicit trigonometric sum.

This is the second in a series of papers on conformal tilings. The overriding themes of this paper are local isomorphisms, hierarchical structures, and the type problem in the context of conformally regular tilings, a class of tilings introduced first by the authors in 1997 with an example of a conformally regular pentagonal tiling of the plane [2]. We prove that when a conformal tiling has a c...