Then, we shifted the shape horizontally by 6 units to the right. The tessellations shown here are from Suad, Alim, Mohamed, Ruhan and Era. The example in Figure 10.112 shows a trapezoid, which is reflected over the dashed line, so it appears upside down. At the end of the inquiry, I displayed some of the tessellations under a visualiser, which elicited an intriguing question from one of the students who had noticed the angles chosen for the quadrilaterals were all less than 180 o : "Would it work if the quadrilateral has a reflex angle?" I encouraged students to write in the angles that met at a point to verify that they summed to 360 o. They had to think carefully about how to transform the shape. The quadrilaterals presented a challenge even to the students with the highest prior attainment, particularly when the size of the angles were similar. Īs our time was limited, I directed the students to cut out a triangle or quadrilateral from card and, after measuring and noting down the interior angles, tessellate their shape on paper. To tessellate is to cover a surface with a pattern of repeated shapes, especially polygons, that fit together closely without gaps or overlapping. We compared their ideas with a formal definition (below) and agreed that they were consistent. Will it work with all the types of triangles?Īfter showing them pictures of tessellations, the students began to construct an understanding of the concept: Clockwise from top left: a pineapple, a turtle, Giants Causeway, a honeycomb. They had no prior knowledge of tessellations and, unsurprisingly, that was their first question about the prompt:ĭoes it mean that triangles fit into quadrilaterals? Do they "perfectly overlap"?ĭo triangles and quadrilaterals do it in the same way? The prompt gave them an opportunity to see angle facts in a new context. Andrew Blair reports on how the inquiry progressed: Hunt using an irregular pentagon (shown on the right).A year 7 mixed attainment class at Haverstock school (Camden, UK) inquired into the prompt during a 50-minute lesson. In computer graphics, tessellation is the dividing of datasets of polygons (sometimes called vertex sets) presenting objects in a scene into suitable structures for rendering. Another spiral tiling was published 1985 by Michael D. A simple tessellation pipeline rendering a smooth sphere from a crude cubic vertex set using a subdivision method. 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. These tessellations do not have repeating patterns. Notice how each gecko is touching six others. The following "gecko" tessellation, inspired by similar Escher designs, is based on a hexagonal grid. Here, all the shapes are still regular polygons, but one may use more than one type of regular polygon. After sharing some students work, tell them this particular tessellation is called semi-regular tessellations. By their very nature, they are more interested in the way the gate is opened than in the garden that lies behind it." For example, this designs configuration is 3.4.6.4 3.4.6.4 3.4.6.4. 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. A unique art form is enabled by modifying monohedral tessellations.
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