![]() This in itself makes a lovely investigation for children.Īnother remarkable man who contributed enormously to the study of tessellation was the Dutch artist M.C.Escher. The two shapes are both parallelograms and the tessellation is often referred to as "Kites and Darts" :Īlthough there are small repeated sections, there is no single unit which can be copied to fill the plane. Amazingly, he managed to reduce this to only six, then just two. Using only pencil and paper, Penrose found such an arrangement but it contained many different shapes. This kind of tessellation became known as quasi-periodic, in other words at first glance there appears to be a repeating pattern, but in fact He began by investigating combinations of shapes which would produce a repeating unit, but this led on to a search for a pattern with no repetition. While studying for his PhD at Cambridge, Penrose became fascinated by the geometry of covering a plane. Octagons and squares can be arranged to form a semi-regular pattern: The image that we are likely to think of is known as a regular tessellation, where all the shapes are regular and of the same type, for example:Ī semi-regular tessellation is made up of two different regular shapes and each vertex (i.e. Traditionally, the pattern formed by a tessellation is repetitive. Two people have principally been responsible for investigating and developing tessellations: Roger Penrose, an eminent mathematician, and the artist, M.C.Escher. Tessellations are a common feature of decorative art and occur in the Presumably this is an indication of the fact that tiles of this shape are the easiest to interlock. The word tessellation itself derives from the Greek tessera, which is associated with four, square and tile. Tessellation is a system of shapes which are fitted together to cover a plane, without any gaps or overlapping. And of course, there is so much maths involved! It seems a golden opportunity to link art with maths, allowing the creative side of your children to take over. There is so much scope for practical exploration of tessellations both ![]() For many, this is their preferred method of learning and, in general, it engages pupils more effectively. So often in the classroom we try to make activities more enjoyable for the children by varying our teaching to include a more tactile or "hands on" approach. Our explorations in class of subdividing polyhedra suggest that this may be the case, as the cube was the only one which could be divided into its own likeness.'Why tessellation?' you may well be asking. What about the other regular polyhedra? As far as I can tell, none of them will work, though this assumption comes not from mathematics but from my own attempts to visualize these shapes stacking together. What about tessellating space? Cubes work well, the Rubik's Cube is an example. ![]() ![]() A hexagon would, I imagine, work just as well, though a triangle would not be as suited, because tessellations require the triangle to have alternating orientation. It's easy to make a tessellating pattern by reflecting portions of an image onto opposite sides of a square- the background of my wacky fun page is an example. Unless I'm mistaken, only three regular polygons can tessellate the plane: the triangle, square, and hexagon. Tessellations can also be done with certain regular polygons. You can use tessellations to make nifty "Magic Eye"-type stereograms here's an example. Here's a nifty Mac or PC program which lets you create and animate tesselating figures. What's more interesting is shapes which tessellate and are regular, or repeating, and among the most interesting tessellations are those which use only one figure MC Escher is famous for his work with such planar tessellations. In a sense, any painting is a tessellation, albeit with very irregular figures. To tessellate is to fill a plane with figures, such that those figures fill all available space. The discussion on folding up polygons, and the significance of the number which can fit around a single point, brings up the subject of tessellations. What other eccentricities does the fourth dimension hold? I'm prompted to wonder again if it is not an accident that our universe has a four dimensional space-time. Indeed, as this chapter tells us, fundamental analogies may not hold true across all dimensions the progression of the number of regular n-figures, from an infinite number in the plane, to five in 3-space, to three in 5-space and beyond, would seem to imply that there would be between 3 and 5 regular figures in four-space. The first thing that this chapter did was to fuel my suspicion that the fourth dimension is not an average dimension. Comments on Beyond the Third Dimension, Chapter 5 Tesselations
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