One of the most famous examples of Islamic tessellation art is in the Alhambra, a huge palace located in Granada, Spain. Therefore, they embraced the abstract characteristics of tessellation and used colorful geometric tiles to create non-representational patterns. This is because many Muslims believe that the creation of living forms is solely God’s doing. Religious Islamic art is typically characterized by the absence of figures and other living beings. Perhaps the most celebrated style of tessellations can be found in Islamic art and architecture. While the Sumerians of 5th and 6th BCE used tiles to decorate their homes and temples, other civilizations around the world adapted tessellations to fit their culture and traditions the Egyptians, Persians, Romans, Greeks, Arabs, Japanese, Chinese, and the Moors all embraced repeating patterns in their decorative arts. Tessellations in Ancient Islamic Art and ArchitectureĬeramic tile tessellations in Marrakech, Morocco (Photo: Wikimedia Commons, (CC BY-SA 3.0)) Now that we’ve covered the basic math of tessellations, read on to learn about how they were used throughout history. Each vertex is surrounded by the same polygons arranged in the same recurring order. ![]() Semi-regular tessellations occur when two or more types of regular polygons are arranged in a way that every vertex point is identical. A checkerboard is the simplest example of this: It comprises square tiles in two contrasting colors (usually black and white) that could repeat forever. Regular periodic tiling involves creating a repeating pattern from polygonal shapes, each one meeting vertex to vertex (the point of intersection of three or more bordering tiles). The most common configurations are regular tessellations and semi-regular tessellations. There are many types of tessellations, all of which can be classified as those that repeat, are non-periodic, quasi-periodic, and those that are fractals. We would, for instance, still use the m × n cells for the raster, even though variations in elevation are irrelevant.An example of semi-regular tessellation (Photo: Wikimedia Commons, (CC BY-SA 3.0)) If we use any of the above regular tessellations to represent an area with minor elevation differences, then, clearly we would need just as many cells as in a strongly undulating terrain: the data structure does not adapt to the lack of relief. The cell boundaries are both artificial and fixed: they may or may not coincide with the boundaries of the phenomena of interest. An obvious disadvantage is that they are not adaptive to the spatial phenomenon we want to represent. An important advantage of regular tessellations is that we know how they partition space, and that we can make our computations specific to this partitioning. This allows us to represent continuous, even differentiable, functions. Values for other positions are computed using an interpolation function applied to one or more nearby field values. The location associated with a raster cell is fixed by convention: it may be the cell centroid (mid-point) or, for instance, its left lower corner. If one wants to use rasters for continuous field representation, one usually uses the first approach but not the second, as the second technique is usually considered computationally too intensive for large rasters. There are two approaches to refining the solution of this continuity issue: make the cell size smaller, so as to make the “continuity gaps” between the cells smaller and/or assume that a cell value only represents elevation for one specific location in the cell, and to provide a good interpolation function for all other locations that have the continuity characteristic. With square cells, this convention states that lower and left boundaries belong to the cell. ![]() Some convention is needed to state which value prevails on cell boundaries. The field value of a cell can be interpreted as one for the complete tessellation cell, in which case the field is discrete, not continuous or even differentiable. ![]() There are some issues related to cell-based partitioning of the study space. This type of tessellation is known under various names in different GIS packages: e.g. “ raster” or “raster map”. Square cell tessellation is commonly used, mainly because georeferencing of such a cell is straightforward. All regular tessellations have in common that the cells have the same shape and size, and that the field attribute value assigned to a cell is associated with the entire area occupied by the cell.
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