Hierarchical Tilings
- PDF / 399,240 Bytes
- 7 Pages / 420.48 x 639 pts Page_size
- 4 Downloads / 239 Views
HIERARCHICAL TILINGS Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063 ABSTRACT Tiling problems arise in many areas of science, and they raise fascinating but difficult mathematical questions. Nonperiodic hierarchical tilings are particularly interesting and important. We discuss several examples of such tilings and indicate some of the unsolved problems related to them.
Tiling theory is an ancient subject, with roots in natural philosophy and the practical arts. It is concerned with the ways in which a surface or space can be covered by copies of shapes of a few kinds, without overlapping or leaving gaps. It also seeks to answer questions about the properties of the patterns created in this way. Today, tiling theory deals not only with such questions as "in what ways can I cover this floor, or this stove, with linoleum or porcelain tiles?", but also "what are the patterns produced in a tissue by cell division?" and "what is the most fruitful way to represent the growth and structure of a crystal?" To answer these questions, we need the answers to others: "which shapes can be packed together to fill the plane, or space, or cover a bounded surface, without gaps or overlaps?" "Which sets of shapes can be grouped into larger shapes similar to themselves?" Tiling theory continues to thrive on the interplay of the particular and the abstract. In talking about tiles and tilings, we should be clear about what we mean by these words. For example, a tiled roof is not a tiling in our sense, because the roof tiles overlap. The surface or space to be tiled may bounded or unbounded, of any dimension. Definition. A tiling is a partition of a space S into nonoverlapping cells called tiles. Each tile is congruent to one element of a finite set of prototiles. The prototiles can vaiy greatly in shape but it must be possible to continuously deform them into disks. For simplicity, we will only consider tilings of the entire plane, in which the usual laws of Euclidean geometry are assumed to hold. The most interesting tilings are those for which the number of prototiles is small. If there is only one prototile, the tiling is said to be monohedral. These definitions and this terminology, though informal, agree with their more formal counterparts in the definitive Tilings and Patterns [1]. We note that the word "tile" is also used as verb, as in "the plane can be tiled by squares" or "the regular pentagon does not tile the plane". When we say that a shape tiles, or is a tile, it is understood that an infinite number of congruent copies of it are on hand for constructing the tiling. It is natural to ask, "which shapes tile, either by themselves or in conjunction with a small number of others?" Unfortunately, one of the remarkable achievements in tiling theory of the last few decades was the proof that there is no algorithm for determining whether copies of a given shape will tile the plane [2]. There are of course many particular classes of shapes that are known to tile, and many tests by which candidates
Data Loading...
