Remind them that they can only use one kind of regular polygon for each of their designs. Which regular polygons tessellate Regular Octagons: Dont tessellate: This is called a semi-regular tessellation since more than one regular polygon is used. Let students use regular polygons to create tessellations. Invite them to create regular tessellations. In a regular tessellation, all the shapes are the same regular polygon and all the vertices are the same. Tessellations are also used in computer graphics where objects to be shown on screen are broken up like tessellations so that the computer can easily draw it on the monitor screen. The first ones are called Regular Tessellations. Each of these has many fascinating properties which mathematicians are continuing to study even today. There are many other types of tessellations, like edge-to-edge tessellation (where the only condition is that adjacent tiles should share sides fully, not partially), and Penrose tilings. There are eight such tessellations possible All the other rules are still the same.įor example, you can use a combination of triangles and hexagons as follows to create a semi-regular tessellation. If you look at the rules above, only rule 2 changes slightly for semi-regular tessellations. If you use a combination of more than one regular polygon to tile the plane, then it's called a "semi-regular" tessellation. The mathematics to explain this is a little complicated, so we won't look at it here So what's unique to those 3 shapes (triangle, square and hexagon)? As it turns out, the key here is that the internal angles of each of these three is an exact divisor of 360 (internal angle of triangle is 60, that of square is 90, and for a hexagon is 120). A honeycomb is a perfect example of a tessellated pattern in nature: tiny hexagons repeated without gap or overlap. You can see that there is a gap and that's not allowed. Demi-regular tessellations are those that use non-regular or non-geometric shapes, such as those popularized by M.C. Let's try with pentagons and see what shape we come up with. You may wonder why other shapes won't work. Let me show you examples of these two here. What are the other two? They are triangles and hexagons. Of course, you would have guessed that one is a square. Each vertex (the points where the corners of the tiles meet) should look the same.All the tiles must be the same shape and size and must be regular polygons (that means all sides are the same length).The tessellation must cover a plane (or an infinite floor) without any gaps or any overlaps.There are only three rules to be followed when doing a "regular tessellation" of a plane If you use only one kind of polygon to tile the entire plane - that's called a "Regular Tessellation"Īs it turns out, there are only three possible polygons that can be used here. There are different kinds of tessellations – the ones of most interest are tessellations created using polygons. The word “Tiling” is also commonly used to refer to "tessellations". Of course, when we are talking about floors, the shapes used to cover it are mostly rectangles or squares (in fact, the word " tessellation" comes from the Latin word tessella - which means " small square"). The one difference here is that technically a plane is infinite in length and width so it's like a floor that goes on forever. That is a good example of a "tessellation". And you'll notice that the floor is covered with some tiles or marbles of different shapes. That is a flat surface - called a "plane" in mathematical terms. To explain it in simpler terms – consider the floor of your house. In the rightmost figure, we used octagons and squares in tiling, which is considered as a semi-regular tessellation. The polygons shown in Figure 7 are some of the tiles which are not regular polygons. A tessellation is simply is a set of figures that can cover a flat surface leaving no gaps. We will not limit, of course, our creativity by using only regular polygons in tiling floors. For example, the cube has Schläfli symbol. Another related symbol is the Coxeter–Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. A Schläfli symbol describing an n-polytope equivalently describes a tessellation of an ( n − 1)-sphere. The Schläfli symbol describes every regular tessellation of an n-sphere, Euclidean and hyperbolic spaces. This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
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