![]() ![]() The point where two shapes meet, also known as the corner point, is referred to as the vertex. It follows that there are only three distinct types of regular tessellations: those constructed from squares, equilateral triangles, and hexagons. ![]() They are used rather frequently in works of art, patterns for clothing, designs for pottery, and blown glass windows. These days, tessellations are employed for the floors, walls, and ceilings that are found inside buildings. In addition, the tessellations that are used in architecture may be seen at Fatehpur Sikri. The Alhambra Palace in Granada, which is in the southern region of Spain, is an example of a Muslim edifice that hints at the presence of tessellations. Tessellations, which are miniature quadrilaterals used in computer games and in the construction of mosaics, were exploited by the ancient Greeks. Tessellations had been tracked all the way back to the ancient civilizations, where they were first discovered (around 4000 BC). They frequently exhibit certain qualities that are tied to their place of origin in some way. There is evidence that tessellations were used in a variety of ancient cultures across the world. The word “tessellation” originates from the Latin verb tessellate, which translates to “to pave,” or the word “ tessella,” which refers to a little, rectangular stone. The only rule is that all of the sides must fit together perfectly, with no empty spaces or overlap. But you can also make them by mixing different geometric shapes (e.g., hexagons and squares), to make tessellating patterns. Other quadrants have to be split further.As shown in the figure above, triangles can be used to make a tessellated pattern. 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. ![]()
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