![]() In fact, in working with tessellating shapes and incorporating their patterns into his work, M.C. Escher have used the intriguing optical effect of tessellations to create a surreal mood. A branch of science known as x-ray crystallography studies the repeating arrangements of identical objects in nature, sort of a natural form of tessellation. Tessellating patterns cut across many different disciplines. He worked on the problem of creating a set of shapes that would tile a surface without a repeating pattern, called quasi-symmetry. In the present day, Oxford mathematician Sir Roger Penrose has devoted much time to the study of recreational mathematics and tessellations. In 1619, Johannes Kepler published the first formal study of tessellations. In fact, the nature of mosaic art naturally gives rise to some tessellating patterns. Sumerian wall decorations, an early form of mosaic dating from about 4000 B.C., contain examples of tessellations. Tessellation patterns are very old, and are found in many cultures around the world. For example, the "Fish n' Chicks" animation below shows how you can alter a square to create an irregular shape that tessellates a surface. Tessellations made from regular polygons (equilateral triangles, squares, and hexagons) are usually referred to as tilings however, tessellations can be made from many irregular shapes as well. Semi-regular tessellations, on the other hand, use a combination of different regular polygons, such as the pattern above, and you can typically see examples of these patterns in the tilework of bathroom and kitchen floors. You can find examples of these on chess- or checkerboards. Patterns using only one regular polygon to completely cover a surface are called regular tessellations. Circles, for instance, would not create a tessellation by themselves, because any arrangement of circles would leave gaps or overlaps.ĭespite the limitations on the types of shapes that can form this intriguing pattern, there are many varieties of tessellations. Not all shapes, however, can fit snugly together. There are usually no gaps or overlaps in patterns of octagons and squares they "fit" perfectly together, much like pieces of a jigsaw puzzle. Lizard tiles by Ben Lawson.Geometry formally defines a tessellation as an arrangement of repeating shapes which leaves no spaces or overlaps between its pieces. Hexagonal and rhombic tessellations from Wikimedia Commons. Triangular tessellation from pixababy.If you want to try a more complicated version, cut two different squiggles out of two different sides, and move them both.Color in your basic shape to look like something - an animal? a flower? a colorful blob? Add color and design throughout the tessellation to transform it into your own Escher-like drawing. The shape will still tessellate, so go ahead and fill up your paper.Then move it the same way you moved the squiggle (translate or rotate) so that the squiggle fits in exactly where you cut it out. On a large piece of paper, trace around your tile. Tape the squiggle into its new location.It’s important that the cut-out lines up along the new edge in the same place that it appeared on its original edge.You can either translate it straight across or rotate it. Cut out the squiggle, and move it to another side of your shape.Draw a “squiggle” on one side of your basic tile.The first time you do this, it’s easiest to start with a simple shape that you know will tessellate, like an equilateral triangle, a square, or a regular hexagon. Here’s how you can create your own Escher-like drawings. Tessellations are often called tilings, and that’s what you should think about: If I had tiles made in this shape, could I use them to tile my kitchen floor? Or would it be impossible? The first two tessellations above were made with a single geometric shape (called a tile) designed so that they can fit together without gaps or overlaps. So we’ll focus on how to make symmetric tessellations. ![]() ![]() It’s actually much harder to come up with these “aperiodic” tessellations than to come up with ones that have translational symmetry. The Penrose tiling shown below does not have any translational symmetry. Many tessellations have translational symmetry, but it’s not strictly necessary. ![]() The idea is that the design could be continued infinitely far to cover the whole plane (though of course we can only draw a small portion of it). \)Ī tessellation is a design using one ore more geometric shapes with no overlaps and no gaps. ![]()
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