Avperiodic

Avperiodic refers to a property of a set of points or objects in geometry and mathematics, particularly in the context of Voronoi diagrams and Delaunay triangulations. An avperiodic set is a collection of points that, when used to construct a Voronoi diagram, results in a Delaunay triangulation that contains an infinite number of disjoint triangles of the same area. This property was first identified by Ronald Graham and Gerhard Rote in the early 1990s.

The term "avperiodic" is derived from the concept of aperiodicity, which is a characteristic of certain mathematical structures, such as quasicrystals. In the context of Voronoi diagrams and Delaunay triangulations, aperiodicity refers to the fact that the structure of the triangles in the triangulation does not repeat in a regular or periodic manner. This property is particularly interesting in the context of quasicrystals, which are materials that lack long-range order but still exhibit periodic diffraction patterns.

An avperiodic set of points has several important properties. First, it is dense in the sense that the points are evenly distributed throughout the space. Second, the Voronoi diagram of an avperiodic set is entirely contained within a ball of a fixed radius around the origin. Finally, the Delaunay triangulation of an avperiodic set is composed of a single type of triangle, all of which have the same area.

The study of avperiodic sets has far-reaching implications for fields such as geometry, topology, and materials science. Researchers have been actively investigating avperiodic sets and their properties, exploring new avenues for theoretical and practical applications. The discovery of avperiodic sets has contributed significantly to the development of our understanding of geometric structures and their mathematical properties.