Tessellations in Grade VIII

Hii guys, so today we are going to study about a very new and different topic of Maths which is Tessellations. How many of you know what a Tessellations is? I believe not many but still a few might know. So let's start without wasting any time.

Tessellation is the process in which we create a two dimensional plane by repeatedly joining the geometric shapes without any overlaps and gaps. In simple language a tessellation is something in which we have some shapes and we join them together to form a regular sheet without any overlaps and gaps. We can see tessellations throughout our history, from ancient architecture to modern art.

For Example the Jigsaw puzzle we used to play when we were little. Its the best example we can get for our topic. It corresponds with the everyday term tiling which refers to applications of tessellations, often made of glazed clay. It also exist in the nature like the Honeycomb which also has a Tessellation structure.

For Example -

a tessellation of triangles

a tessellation of squares

a tessellation of hexagons

The above given figures are some examples of Tesselations, for more understanding.

Now let's study the different topics that comes under it.

When discussing about the tilings a form of tessellation, we see that they are multicolored, so we need to specify whether the colors are the part of tiling or just the part of an illustration.

Let's come to the theorem in our topic, it is known as the four color theorem.

The theorem states that every tessellation in every Euclidean plane, with a set of four available colors, each tiled when colored in one color such that no tiles of equal color meet at a curve of positive length. Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. To produce a coloring which does, as many as seven colors may be needed.

Any arbitrary Quadrilateral when taken can also form a tessellation with 2-fold rotational centre at the mid point of all the four sides of the Quadrilateral. In a similar manner, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational center. We can easily divide this by one diagonal, and take one half of the triangle as fundamental domain. Such a triangle formed will have the same area as of the quadrilateral and can be constructed from it by cutting and pasting.

Now we come over to the types of tessellations, we classify then in two types – the regular tessellation and irregular tessellation.

A regular tessellation is one which is highly symmetric and is made of congruent regular polygon. There are only three regular tessellations which exist. They are of Equilateral triangles, squares and regular hexagons. We even have a more accurate one which is edge-to-edge tessellation. In this type of tessellation the side of one polygon is fully shared with the sides of another polygon. There is no room for partial sharing of sides.

The most common example of the apperiodic pattern is the Penrose tilings which is formed using two different types of polygons and adds beauty to the walls. We also have the self dual tessellation and the example for it is the Honeycomb. Another example for it is shown in the figure below.

 

Tessellations is not only used in architecture but also to design the computer models. In the computer graphics, tessellation technique is used manage the datasets of polygons and divide them into polygon structure. This is generally used for rendering. The data is generally tessellated into triangles also known as triangulation.

The CAD application in our computers is generally represented in analytical 3D curves and surfaces.

The mesh of a surface is usually generated per individual faces and edges so that original limit vertices are included into mesh. The following three basic fundamentals for surface mesh generator are defined to ensure the approximation of the original surface that suits the need for the further processing

• The maximum allowed distance between the planar approximation polygon and the surface. This parameter ensures that mesh is similar enough to the original analytical surface (or the polyline is similar to the original curve).
• The maximum allowed size of the approximation polygon (for triangulations it can be maximum allowed length of triangle sides). This parameter ensures enough detail for further analysis.
• The maximum allowed angle between two adjacent approximation polygons (on the same face). This parameter ensures that even very small humps or hollows that can have significant effect to analysis will not disappear in mesh.

Some geodesic domes are designed by tessellating the sphere with triangles that are as close to equilateral triangles as possible.

So as we have read above that a honeycomb is a natural tessellation, there are some more tessellations in our nature like when the lava flows, it often displays columnar jointing, due to the contraction forces that are present, it causes the lava to break into cracks when it is cooled. These cracks often gets broken in the hexagonal columns. The tessellation is even present in the flower petals, leaves, tree bark, or a fruit. It is not easily visible but when we look through the leaf etc. with the help of a microscope then we can easily see the formation.

It seems like what we have learned till now seems to be theoretical, so let's now have something mathematical in it.

Now the topic that comes under it is, Number of Sides of a Polygon versus the number of Sides at a 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), (6, 3) for the tilings.

For a tiling which repeats itself, one can take the averages over the repeating part. In the general case the averages are taken as the limits for a region expanding to the whole plane. In cases like an infinite row of tiles, or tiles getting smaller and smaller outwardly, the outside is not negligible and should also be counted as a tile 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 characterstic.

The formula follows observing that the number of sides of a face, summed over all faces, gives twice the total number of sides in the entire tessellation, which can be expressed in terms of the number of faces and the number of vertices. Similarly the number of sides at a vertex, summed over all vertices, also gives twice the total number of sides. From the two results the formula readily follows.

In most cases the number of sides of a face is the 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 has 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 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 this is all we have for the topic tessellations and I hope that you would have understood what a tessellation is. This topic we have learned is totally a theoretical one, but it do have some problems to do, and I expect that you will be able to understand how to solve them by using the only formula which we have learned so far in this topic.