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Irregular tessellation patterns12/28/2023 ![]() ![]() Other types of tessellations exist, depending on types of figures and types of pattern. no tile shares a partial side with any other tile. 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. Here are a variety of basic geometric shapes that can tessellate from this same pattern, including a hexagon, triangle, square, trapezoid, parallelogram, pentagon (irregular), rhombus (diamond), and rectangle:Ĭopyright © 2014 Chris McMullen, author of the Improve Your Math Fluency series of math workbooksĬlick to view my Goodreads author page.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. The same pattern can make a tessellation with stars and hexagons: ![]() The lattice structure below can be shaded in several different ways to create simple geometric patterns that tessellate:įor example, here is a tessellation composed of hexagons: Some of the more extreme examples of this can be seen in M.C. Even arrangements of curved objects can tessellate. There are many other shapes that tessellate, such as stars combined with other shapes. (Quadrilaterals are polygons with four sides.) Although regular pentagons don’t tessellate, some irregular polygons can (such as the pentagon made by placing an isosceles triangles on a square, as children often do to draw a simple picture of a house). (A regular polygon is one with equal sides and angles.) All quadrilaterals can form tessellations. Tessellations can also be made from irregular polygons. For example, it won’t work with pentagons. ![]() Not any regular polygon will work, however. Simple tessellations can be made by creating a two-dimensional lattice out of regular geometric shapes, like equilateral triangles, squares, and hexagons. A tessellation is a repeated two-dimensional geometric pattern, with tiles arranged together without any space or overlap. ![]()
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