Hunt using an irregular pentagon (shown on the right). Another spiral tiling was published 1985 by Michael D. The first such pattern was discovered by Heinz Voderberg in 1936 and used a concave 11-sided polygon (shown on the left). Lu, a physicist at Harvard, metal quasicrystals have "unusually high thermal and electrical resistivities due to the aperiodicity" of their atomic arrangements.Īnother set of interesting aperiodic tessellations is spirals. The geometries within five-fold symmetrical aperiodic tessellations have become important to the field of crystallography, which since the 1980s has given rise to the study of quasicrystals. According to ArchNet, an online architectural library, the exterior surfaces "are covered entirely with a brick pattern of interlacing pentagons." An early example is Gunbad-i Qabud, an 1197 tomb tower in Maragha, Iran. The patterns were used in works of art and architecture at least 500 years before they were discovered in the West. Medieval Islamic architecture is particularly rich in aperiodic tessellation. There are also tessellations used for spiritual or religious purposes and decorations. We have tessellations that man creates for a visual appeal in housing and other architecture. There are natural tessellations that just occur in different situations around us. These movements are termed rigid motions and symmetries. Tessellations are used in many different realms. The topic of tessellations belongs to a field in mathematics called transformational geometry, which is a study of the ways objects can be moved while retaining the same shape and size. These tessellations do not have repeating patterns. We will explore how tessellations are created and experiment with making some of our own as well. Notice how each gecko is touching six others. The following "gecko" tessellation, inspired by similar Escher designs, is based on a hexagonal grid. By their very nature, they are more interested in the way the gate is opened than in the garden that lies behind it." In doing so, they have opened the gate leading to an extensive domain, but they have not entered this domain themselves. This further inspired Escher, who began exploring deeply intricate interlocking tessellations of animals, people and plants.Īccording to Escher, "Crystallographers have … ascertained which and how many ways there are of dividing a plane in a regular manner. His brother directed him to a 1924 scientific paper by George Pólya that illustrated the 17 ways a pattern can be categorized by its various symmetries. According to James Case, a book reviewer for the Society for Industrial and Applied Mathematics (SIAM), in 1937, Escher shared with his brother sketches from his fascination with 11 th- and 12 th-century Islamic artwork of the Iberian Peninsula. The most famous practitioner of this is 20 th-century artist M.C. Escher & modified monohedral tessellationsĪ unique art form is enabled by modifying monohedral tessellations. Finally, color your design with markers, colored pencils or crayons.A dual of a regular tessellation is formed by taking the center of each shape as a vertex and joining the centers of adjacent shapes. (Remember that whatever details you add to one shape, will need to be added to EVERY shape! Keep your details simple.)ĩ. Trace over your pencil lines with a Sharpie and add details to each shape to help others recognize what you “saw” in it. Repeat this step until your whole paper is covered and there are no gaps or spaces.Ĩ. There shouldn’t be any gaps or overlapping. Now, pick up your tile and place it next to your traced design, as if it were a piece fitting into a jigsaw puzzle. (I use 12″x18″ paper when I do this with 6th graders.)Ħ. Place your tile on the center of a 9″x12″ paper and carefully trace around it. Lightly sketch your idea onto your tile…. Turn your newly created shape (we’ll call this your “tile”) in different directions and use your imagination to see if it “looks like” anything. (For older students, you can make this project more challenging by having them repeat this step on an adjacent side of their card, as in the sample project above.)Ĥ. If you include a corner in your cut, it makes it easier to line the shape up on the opposite side. Now, tape the shape so that it is exactly across from the spot you cut it from. (The lines on your index card will show you if you’ve flipped or turned it!)ģ. Next, cut a shape from one side of your 3″x3′ card, and slide it to the opposite side of the card, without flipping it over or turning it. Polygon – a shape with three or more sidesĢ. Tessellation – a pattern made with polygons that completely fills a space with no gaps, spaces or overlaps. Escher – a Dutch artist (1898-1972) who is best known for his mathematically inspired drawings and prints which displayed great realism, while at the same time showing impossible perspective, eye trickery and metamorphosis.
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