![]() The idea is similar to dividing a number by one of its factors. He experimented with practically every geometric shape imaginable and found the ones that would produce a regular division of the plane. Escher became obsessed with the idea of the “regular division of the plane.” He sought ways to divide the plane with shapes that would fit snugly next to each other with no gaps or overlaps, represent beautiful patterns, and could be repeated infinitely to fill the plane. The Dutch graphic artist was famous for the dimensional illusions he created in his woodcuts and lithographs, and that theme is carried out in many of his tessellations as well. ![]() These movements are termed rigid motions and symmetries.Ī good place to start the study of tessellations is with the work of M. The topic of tessellations belongs to a field in mathematics called transformational geometry, which is a study of the ways objects can be moved while retaining the same shape and size. We will explore how tessellations are created and experiment with making some of our own as well. There are countless designs that may be classified as regular tessellations, and they all have one thing in common-their patterns repeat and cover the plane. ![]() These two-dimensional designs are called regular (or periodic) tessellations. It may be a simple hexagon-shaped floor tile, or a complex pattern composed of several different motifs. Repeated patterns are found in architecture, fabric, floor tiles, wall patterns, rug patterns, and many unexpected places as well. In this section, we will focus on patterns that do repeat. What's interesting about this design is that although it uses only two shapes over and over, there is no repeating pattern. Notice that there are two types of shapes used throughout the pattern: smaller green parallelograms and larger blue parallelograms. The illustration shown above (Figure 10.101) is an unusual pattern called a Penrose tiling. Apply translations, rotations, and reflections.Other quadrants have to be split further.\)Īfter completing this section, you should be able to: After the first split, the southeast quadrant is entirely green, and this is indicated by a green square at level two of the tree. To construct a quadtree, the field is successively split into four quadrants until all parts have only a single value. Figure: An 8 x8, three value raster (here, three colours) and its representation as a region quadtree. Therefore, a quadtree provides a nested tessellation: quadrants are only split if they have two or more different values. When a conglomerate of cells has the same value, they are represented together in the quadtree, provided their boundaries coincide with the predefined quadrant boundaries. Quadtrees are adaptive because they apply Tobler’s law. ![]() The links between them are pointers, i.e. a programming technique to address (or to point to) other records. In the computer’s main memory, the nodes of a quadtree (both circles and squares in the Figure) are represented as records. The procedure produces an upside-down, tree-like structure, hence the name “quadtree”. This procedure stops when all the cells in a quadrant have the same field value. The quadtree that represents this raster is constructed by repeatedly splitting up the area into four quadrants, which are called NW, NE, SE, SW for obvious reasons. It shows a small 8×8 raster with three possible field values: white, green and blue. A simple illustration is provided in the Figure above. It is based on a regular tessellation of square cells, but takes advantage of cases where neighbouring cells have the same field value, so that they can be represented together as one bigger cell. A well-known data structure in this family - upon which many more variations have been based - is the region quadtree.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |