![]() Of the 47 peer-reviewed articles listed in a recent systematic review of 3D printing in chemistry education (Pernaa & Wiedmer, 2019 ), many deal with the production of physical models of molecular and crystal structures. As many of these concepts require capacity for abstraction and spatial vision, many educators are taking advantage of the rise of 3D-printing technologies to develop interactive haptic environments for education. Some undergraduate institutions have adapted the pedagogy of hands-on research to crystallography with the use of single-crystal desktop instruments (Crundwell et al., 1999 ), and many other affordable virtual (Arribas et al., 2014 ) and tangible resources are available as educational supporting material for a number of important concepts in crystallography (Gražulis et al., 2015, and references therein). Moreover, many physics, chemistry (Fanwick, 2007 Pett, 2010 ) and materials science graduate programmes also include introductory notes on crystallography (Borchardt-Ott, 2012 ). Crystallography and X-ray techniques are present in geology graduate programmes (Hluchy, 1999 ) as an independent topic or as a significant part of mineralogy courses. The aesthetic qualities of crystals, as well as the symmetry of their idealized representations and that of lattices, can grab the attention of students and motivate them to pursue further studies of the subject. Crystallography is sometimes present in innovative teaching tasks, including project-based learning, through experiments and competitions of crystal growth. Effective application of X-ray diffraction techniques in geology, solid-state chemistry and materials science requires a basic understanding of crystallography.įrom the academic point of view, crystallography is often present in secondary school chemistry courses through the study of crystal growth and sometimes as an accessory part within earth sciences courses (as part of mineralogy). Diffraction techniques are widely used to identify and ascertain the crystal structures of all kinds of solid substances, from organic to inorganic solids, pharmaceuticals, biological substances such as proteins and viruses etc. This led to the development of modern X-ray diffraction techniques and X-ray crystallography. In the early 20th century, soon after the discovery of X-rays (1895), diffraction of an X-ray beam by a crystal contributed simultaneously to revealing the nature of X-rays (electromagnetic waves) and to confirming the space-lattice hypothesis. Systematic study of crystal shapes led to enunciation of the law of the constancy of interfacial angles and soon it was argued that crystals must consist of ordered arrangements of atoms or molecules in a lattice (space-lattice hypothesis). The most famous pair of such tiles are the dart and the kite.Ĭlick here for the lesson plan of non-periodic Tessellations.Crystallography started in the 17th century as the science for the study of the external shapes of crystals. The pattern of shapes still goes infinitely in all directions, but the design never looks exactly the same. In the 1970s, the British mathematician and physicist Roger Penrose discovered non-periodic tessellations. Whatever direction you go, they will look the same everywhere. They consist of one pattern that is repeated again and again. It may be better to show a counter-example here to explain the monohedral tessellations.Īll the tessellations mentioned up to this point are Periodic tessellations. ![]() All regular tessellations are also monohedral. If you use only congruent shapes to make a tessellation, then it is called Monohedral Tessellation no matter the shape is. You can use Polypad to have a closer look to these 15 irregular pentagons and create tessellations with them. Among the irregular pentagons, it is proven that only 15 of them can tesselate. We can use any polygon, any shape, or any figure like the famous artist and mathematician Escher to create Irregular tessellationsĪmong the irregular polygons, we know that all triangle and quadrilateral types can tessellate. The good news is, we do not need to use regular polygons all the time. If one is allowed to use more than one type of regular polygons to create a tiling, then it is called semi-regular tessellation.Ĭlick here for the lesson plan of Semi - Regular Tessellations. If you try regular polygons, you ll see that only equilateral triangles, squares, and regular hexagons can create regular tessellations.Ĭlick here for the lesson plan of Regular Tessellations. the most well-known ones are regular tessellations which made up of only one regular polygon. ![]() There are several types of tessellations. ![]()
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