相关论文: Aperiodic sets of three types of convex polygons

In this paper, the tiling of the Euclidean plane with regular hexagons whose vertices are occupied by carbon atoms is called the graphene. We describe six different ways to generate the graphene by the means of group theory. There are two…

We briefly review the standard methods used to construct quasiperiodic tilings, such as the projection, the inflation, and the grid method. A number of sample Mathematica programs, implementing the different approaches for one- and…

We prove that every homogeneous convex polyhedron with only one unstable equilibrium (known as a mono-unstable convex polyhedron) has at least $7$ vertices. Although it has been long known that no mono-unstable tetrahedra exist, and…

Ammann bars are formed by segments (decorations) on the tiles of a tiling such that forming straight lines with them while tiling forces non-periodicity. Only a few cases are known, starting with Robert Ammann's observations on Penrose…

We present three non-equivalent procedures to obtain entwining (non-constant) tetrahedron maps. Given a tetrahedron map, the first procedure incorporates its underlying symmetry group. With the second procedure we obtain several classes of…

A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons…

This is a chapter in an incoming book on aperiodic order. We review results about the topology, the dynamics, and the combinatorics of aperiodically ordered tilings obtained with the tools of noncommutative geometry.

We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the…

We introduce the problem of partitioning 2D regions (usually convex regions) into mutually congruent pieces ('tiles').

We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information about the connectedness of interior and boundary sets, and on…

We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas…

There is only one type of tilings of the sphere by $12$ congruent pentagons. These tilings are isohedral.

We give a set of tiles that enforces the sphinx tiling substitution system; the tiles are thus aperiodic.

Pentagonal subdivision gives three families of edge-to-edge tilings of the sphere by congruent pentagons. Each family forms a two dimensional moduli. We describe the moduli in detail.

We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and…

We construct a polygonal spiral by arranging a sequence of regular $n$-gons such that each $n$-gon shares a specified side and vertex with the $(n+1)$-gon in the construction. By offering flexibility for determining the size of each $n$-gon…

This study introduces a novel approach to composite design by employing aperiodic monotiles, shapes that cover surfaces without translational symmetry. Using a combined computational and experimental approach, we study the fracture behavior…

Polygon spaces have been studied extensively, and yet missing from the literature is a simple property that every polygon has: dimension. This is distinct (possibly) from the dimension of the ambient space in which the polygon lives. A…

A tiling is a decomposition of a polygon into finitely many non-overlapping triangles. We prove that if a regular n-gon, $n \geq 5$, $n \neq 28$, can be tiled with similar right triangles, then one of the angles of these triangles is in…

A triangulation of a surface with fixed topological type is called irreducible if no edge can be contracted to a vertex while remaining in the category of simplicial complexes and preserving the topology of the surface. A complete list of…