Hello friends, in today's session we are going to learn a very interesting topic which is Tessellations.
Tessellations means featuring different images which can fit together in repeated patterns just like a jigsaw puzzle, where no overlapping and spacing comes. To speak in an easy language a tessellation is something in which we have few geometric shapes and we attach them in such a way that they form a regular plane without any gaps and overlaps. We can see tessellations throughout our history, from ancient architecture to modern art.
The topic that we are studying today corresponds with the everyday term tiling which refers to applications of tessellations, generally made of clay. It also exist in the nature if you would have seen the Honeycomb its structure is also known as a tessellation. Where regular hexagons are joined together to form a tessellation.
So now we are going to learn about the different sub topics that come under it.
When discussing tessellation, we can even find multicolors ones, so it becomes very essential for us to specify whether the color is a part of the tiles or it is just a part of an illustration.
Let's now learn about the only theorem in our topic known as the four color theorem.
The theorem states that every tessellation in, with a set of four available colors if each tiled when colored with one different color in such a manner that no tiles of same color meet at a curve of positive length. We should always keep note that when the coloring guaranteed by the four-color theorem will not generally respect the symmetry of the tessellation. For this coloring to be done, we would need at least seven colors for filling.
When we take Quadrilateral into consideration, we can easily form a tessellation by using 2-fold rotational center at the mid point of all the four sides of the Quadrilateral. In the same way as we have done for the Quadrilateral we can construct a parallelogram which is subtended by a minimal set of translation vectors, it is going to start from the rotational center. We can easily divide it into two with the help of its diagonal, and consider one half of the triangle as our fundamental domain. A triangle formed using the similar process will have the same area as of that of the quadrilateral and can be easily constructed by this method, which is cutting and pasting.
Now we will look that how many types of tessellations exist. We generally classify them on two bases which are – the regular tessellation and irregular tessellation.
A regular tessellation is one which is highly symmetric and is generally made using the congruent geometric figures.
In our mathematics syllabus there are only three regular tessellations which exist, which are formed by Equilateral triangles, squares and regular hexagons. The more accurately formed Tessellation without any defects in it, is made by the edge-to-edge joining of the shapes. The tessellation formed by this method shares its one full side with the full side of the another polygon. No types of partial sharing is done in this method.
Like if you would have heard about the penrose tiles which are generally used to add some beauty to the walls in our houses, hotels etc. is also a tessellation which is periodic in pattern. There also exist a self dual tessellation which is not as important but the example will make you understand of what it is.
The Honeycomb structure which we have discussed above is known a self dual tessellation. It is also a natural Tessellation.
The tessellations are even used in our hardwares to store data by breaking them into parts like in the computer models. In the computer graphics systems, the technique is used to rearrange the datasets of different polygons and divide them into polygon structure so that it becomes easy for them to get stored. This is generally used for rendering process. The data is generally tessellated into geometric figures like triangles. The process by which it is done is also known as triangulation.
The Computer Aided Designing in our computers is generally represented in analytical 3D curves and surfaces, limited to the faces and the edges to constitute a continuous boundary of a 3D body. This is because the 3D bodies are generally difficult to analyze directly. So for them we can do the approximation with it, which then becomes really easy to analyze.
The mesh of a surface is usually generated per individual faces and edges so that the original limit vertices are included into mesh. For further processing and approximation, there are only basic parameters that needs to be defined for the surface mesh generator, these are as follows-
When basaltic lava flows, it slowly cools down due to the contraction forces between its surfaces. When the lava cools down, it generally have cracks on its surface. These cracks are often broken into hexagonal shapes forming a tessellation like figure.
So what we have read till now is all about what a tessellation is, and we just have to mug it up all. Now we come to the mathematical part, which is the number of sides of a polygon and number of sides of vertex.
If we assume an infinite long tiling, then let A be the number of sides of a polygon and B be the average number of sides which meet at the vertex. Then (A – 2 ) ( B – 2 ) = 4. For example, we have the combinations (3, 6), (31/3, 5), (33/4, 42/7), (4, 4) and (6, 3) for the tilings.
For the types of tiling that repeats itself, one can take the averages over the repeating part. For general consideration we take the averages as the limits for a region occupying the whole plane. In some cases where we have infinite row of tiles, or tiles getting smaller and smaller outwardly, the outside is not negligible and should also be taken into consideration while taking the limit. In extreme cases the limits may not exist, or depend on how the region is expanded to infinity.
in this expression f represents the number of faces of a polygon and v represents the number of vertex and X is the euler characteristic.
The above formula tells us that the number of sides of a face, summed over all faces, gives twice the total number of sides in the entire tessellation, it can be easily expressed in terms of number of faces and number of vertices. Similarly the number of sides at a vertex, summed over all vertices, also gives twice the total number of sides.
In most cases the number of sides of a face is same as the number of vertices of a face, and the number of sides meeting at a vertex is the same as the number of faces meeting at a vertex. However, in a case like two square faces touching at a corner, the number of sides of the outer face is 8, so if the number of vertices is counted the common corner needs to be counted twice. Similarly the number of sides meeting at that corner is 4, so if the number of faces at that corner is counted the face meeting the corner twice has to be counted twice.
A tile with a hole, filled with one or more other tiles, is not permissible, because the network of all sides inside and outside is disconnected. However it is allowed with a cut so that the tile with the hole touches itself. For counting the number of sides of this tile, the cut should be counted twice.
The another topic that we are going to study is the Tessellation of other spaces. As well as tessellating the 2-dimensional Euclidean plane, it is also possible to tessellate other n-dimensional spaces by filling them with n-dimensional polytopes. Tessellations of other spaces are often referred to as honeycombs.
So now I can expect that you will be able to solve the simple problems that comes under this topic, and would be well able to explained to anyone else what a tessellation is.
So thats all what we have learn in today's session and I will come back again with some other topic for you.
Tessellations means featuring different images which can fit together in repeated patterns just like a jigsaw puzzle, where no overlapping and spacing comes. To speak in an easy language a tessellation is something in which we have few geometric shapes and we attach them in such a way that they form a regular plane without any gaps and overlaps. We can see tessellations throughout our history, from ancient architecture to modern art.
The topic that we are studying today corresponds with the everyday term tiling which refers to applications of tessellations, generally made of clay. It also exist in the nature if you would have seen the Honeycomb its structure is also known as a tessellation. Where regular hexagons are joined together to form a tessellation.
So now we are going to learn about the different sub topics that come under it.
When discussing tessellation, we can even find multicolors ones, so it becomes very essential for us to specify whether the color is a part of the tiles or it is just a part of an illustration.
Let's now learn about the only theorem in our topic known as the four color theorem.
The theorem states that every tessellation in, with a set of four available colors if each tiled when colored with one different color in such a manner that no tiles of same color meet at a curve of positive length. We should always keep note that when the coloring guaranteed by the four-color theorem will not generally respect the symmetry of the tessellation. For this coloring to be done, we would need at least seven colors for filling.
When we take Quadrilateral into consideration, we can easily form a tessellation by using 2-fold rotational center at the mid point of all the four sides of the Quadrilateral. In the same way as we have done for the Quadrilateral we can construct a parallelogram which is subtended by a minimal set of translation vectors, it is going to start from the rotational center. We can easily divide it into two with the help of its diagonal, and consider one half of the triangle as our fundamental domain. A triangle formed using the similar process will have the same area as of that of the quadrilateral and can be easily constructed by this method, which is cutting and pasting.
Now we will look that how many types of tessellations exist. We generally classify them on two bases which are – the regular tessellation and irregular tessellation.
A regular tessellation is one which is highly symmetric and is generally made using the congruent geometric figures.
In our mathematics syllabus there are only three regular tessellations which exist, which are formed by Equilateral triangles, squares and regular hexagons. The more accurately formed Tessellation without any defects in it, is made by the edge-to-edge joining of the shapes. The tessellation formed by this method shares its one full side with the full side of the another polygon. No types of partial sharing is done in this method.
Like if you would have heard about the penrose tiles which are generally used to add some beauty to the walls in our houses, hotels etc. is also a tessellation which is periodic in pattern. There also exist a self dual tessellation which is not as important but the example will make you understand of what it is.
The Honeycomb structure which we have discussed above is known a self dual tessellation. It is also a natural Tessellation.
The tessellations are even used in our hardwares to store data by breaking them into parts like in the computer models. In the computer graphics systems, the technique is used to rearrange the datasets of different polygons and divide them into polygon structure so that it becomes easy for them to get stored. This is generally used for rendering process. The data is generally tessellated into geometric figures like triangles. The process by which it is done is also known as triangulation.
The Computer Aided Designing in our computers is generally represented in analytical 3D curves and surfaces, limited to the faces and the edges to constitute a continuous boundary of a 3D body. This is because the 3D bodies are generally difficult to analyze directly. So for them we can do the approximation with it, which then becomes really easy to analyze.
The mesh of a surface is usually generated per individual faces and edges so that the original limit vertices are included into mesh. For further processing and approximation, there are only basic parameters that needs to be defined for the surface mesh generator, these are as follows-
-
The maximum distance that can be allowed between the planar approximation polygon and the surface. This parameter would ensure that mesh generated is similar to the original analytical surface.
-
The maximum size that can be allowed for the approximation of polygon. This parameter ensures enough detail for further analysis.
-
The maximum angle that is allowed between two adjacent approximation polygons. This parameter would ensure that even very small humps or hollows can have significant effect to analyze.
When basaltic lava flows, it slowly cools down due to the contraction forces between its surfaces. When the lava cools down, it generally have cracks on its surface. These cracks are often broken into hexagonal shapes forming a tessellation like figure.
So what we have read till now is all about what a tessellation is, and we just have to mug it up all. Now we come to the mathematical part, which is the number of sides of a polygon and number of sides of vertex.
If we assume an infinite long tiling, then let A be the number of sides of a polygon and B be the average number of sides which meet at the vertex. Then (A – 2 ) ( B – 2 ) = 4. For example, we have the combinations (3, 6), (31/3, 5), (33/4, 42/7), (4, 4) and (6, 3) for the tilings.
For the types of tiling that repeats itself, one can take the averages over the repeating part. For general consideration we take the averages as the limits for a region occupying the whole plane. In some cases where we have infinite row of tiles, or tiles getting smaller and smaller outwardly, the outside is not negligible and should also be taken into consideration while taking the limit. In extreme cases the limits may not exist, or depend on how the region is expanded to infinity.
For a finite tessellation and a polyhedra we have
( A – 2 ) ( B – 2 ) = 4 (1 – X/f ) ( 1 – X/v ). in this expression f represents the number of faces of a polygon and v represents the number of vertex and X is the euler characteristic.
The above formula tells us that the number of sides of a face, summed over all faces, gives twice the total number of sides in the entire tessellation, it can be easily expressed in terms of number of faces and number of vertices. Similarly the number of sides at a vertex, summed over all vertices, also gives twice the total number of sides.
In most cases the number of sides of a face is same as the number of vertices of a face, and the number of sides meeting at a vertex is the same as the number of faces meeting at a vertex. However, in a case like two square faces touching at a corner, the number of sides of the outer face is 8, so if the number of vertices is counted the common corner needs to be counted twice. Similarly the number of sides meeting at that corner is 4, so if the number of faces at that corner is counted the face meeting the corner twice has to be counted twice.
A tile with a hole, filled with one or more other tiles, is not permissible, because the network of all sides inside and outside is disconnected. However it is allowed with a cut so that the tile with the hole touches itself. For counting the number of sides of this tile, the cut should be counted twice.
The another topic that we are going to study is the Tessellation of other spaces. As well as tessellating the 2-dimensional Euclidean plane, it is also possible to tessellate other n-dimensional spaces by filling them with n-dimensional polytopes. Tessellations of other spaces are often referred to as honeycombs.
So now I can expect that you will be able to solve the simple problems that comes under this topic, and would be well able to explained to anyone else what a tessellation is.
So thats all what we have learn in today's session and I will come back again with some other topic for you.
No comments:
Post a Comment