Penrose tilings using two different polygons are the most famous example of tessellations that create aperiodic patterns. There are regular versus irregular, periodic versus aperiodic, symmetric versus asymmetric, and fractal tessellations, as well as other classifications. Other types of tessellations exist, depending on types of figures and types of pattern. no tile shares a partial side with any other tile.
Tessellation definition full#
An edge-to-edge tessellation is even less regular: the only requirement is that adjacent tiles only share full sides, i.e.
The arrangement of polygons at every vertex point is identical. Only three regular tessellations exist: those made up of equilateral triangles, squares, or hexagons.Ī semiregular tessellation uses a variety of regular polygons there are eight of these. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting.Ī regular tessellation is a highly symmetric tessellation made up of congruent regular polygons. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational center. As fundamental domain we have the quadrilateral. For an asymmetric quadrilateral this tiling belongs to wallpaper group p2. To produce a coloring which does, as many as seven colors may be needed, as in the picture at right.Ĭopies of an arbitrary quadrilateral can form a tessellation with 2-fold rotational centers at the midpoints of all sides, and translational symmetry with as minimal set of translation vectors a pair according to the diagonals of the quadrilateral, or equivalently, one of these and the sum or difference of the two. Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. The four color theorem states that for every tessellation of a normal Euclidean plane, with a set of four available colors, each tile can be colored in one color such that no tiles of equal color meet at a curve of positive length. When discussing a tiling that is displayed in colors, to avoid ambiguity one needs to specify whether the colors are part of the tiling or just part of its illustration. (This tiling can be compared to the surface of a torus.) Tiling before coloring, only four colors are needed. Escher used this concept in many of his works.If this parallelogram pattern is colored before tiling it over a plane, seven colors are required to ensure each complete parallelogram has a consistent color that is distinct from that of adjacent areas. To see some interesting patterns, check out. I know you have seen these patterns before. The word “tessellate” is derived from the Ionic version of the Greek word “tesseres,” which in English means “four.” The first tilings were made from square tiles. Here is a definition of tessellation from .Ī tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.Īnother word for a tessellation is tilting.Ī dictionary * will tell you that the word “tessellate” means to form or arrange small squares in a checkered or mosaic pattern. He explained that the class learned about tessellation to do the work. One young man stood in the hall with many colorful pieces of paper with design on them.
The students had an evening where they displayed art, science, math and engineering projects from their efforts this year. I heard it at a middle school of all things.
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