Tessellation

Dhristi JEE 2022-24

Introduction

Tessellation is any recurring  pattern of symmetrical and interlocking shapes . Therefore tessellations have to have no gaps or overlapping spaces. Tessellations are from time to time referred to as “tilings' '. Strictly, but, the phrase tilings refers to a pattern of polygons (shapes with straight aspects) simplest. Tessellations can be formed from ordinary and abnormal polygons, making the patterns they produce yet more interesting. Tessellations of squares, triangles and hexagons are the simplest and are frequently visible in normal existence, as an instance in chess boards and beehives. 


Tessellations and The Way They are Utilized in Structure

Tessellations are a crucial part of arithmetic because they may be manipulated to be used in artwork and structure. One artist specifically, MC Escher, a Dutch artist, integrated many complicated tessellations into his artwork. Tessellations are used appreciably in regular objects, especially in buildings and walls. They are part of an area  of mathematics that often appears easy to recognize and research indicates that Tessellations are in truth complicated.


Early Records of Tessellations 

The Latin root of the word tessellations is tessellate, which means ‘to pave’ or ‘tessella’, which means a small, rectangular stone. Tessellations have been located in many historic civilizations internationally. They often have precise characteristics depending on where they may be from. Tessellations had been traced all of the way back to the Sumerian civilizations (around 4000 BC). Tessellations were used by the Greeks, as small quadrilaterals utilized in video games and in making mosaics. Muslim structure suggests evidence of tessellations and an example of this is the Alhambra Palace at Granada, inside the south of Spain. Fatehpur Sikri additionally shows tessellations used in architecture. Nowadays tessellations are used inside the floors, partitions and ceilings of buildings. they're extensively utilized in artwork, designs for garb, ceramics and stained glass windows.


What are Tessellations?

In Latin, the word 'tessera' means a small stone cube. They were used to make up 'tessellata' - that are the mosaic pictures that form floors and tilings in Roman buildings.


The term has become more specialized and is often used to refer to pictures or tiles, mostly in the form of animals and other life forms, which cover the surface of a plane in a symmetrical way without leaving gaps or overlapping.

 

Styles of Tessellations

  • Semi-Regular Tessellations

While two or 3 varieties of polygons share a commonplace vertex, then a semi-normal tessellation is fashioned. There are nine specific varieties of semi-normal tessellations which include combining a hexagon and a rectangle that each include a one-inch aspect. some different instances of a semi-normal tessellation that is usual with the useful resource of combining  hexagons with  equilateral triangles.

  • Demi-Regular Tessellations

A demi tessellation may be formed by way of placing a row of squares, then a row of equilateral triangles (a triangle with identical aspects) which can be alternated up and down forming a line of squares when blended. Demi  tessellations usually incorporate  vertices.

  • Non-Regular Tessellations

A non-regular  tessellation may be defined as a group of shapes which have the sum of all interior angles equaling 360 stages. There are once more no overlaps or you can say there are not any gaps, and non-regular tessellations are fashioned typically using polygons that are not ordinary.


Types of Tessellation

  1. Translation - A Tessellation in which the shape repeats by moving or sliding.

  2. Rotation - A Tessellation in which the shape repeats by rotating or turning.

  1. Reflection - A Tessellation in which the shape repeats by reflecting or flipping. 


Classifying Tessellations

  • A Normal Tessellation is a tessellation that is made by repeating a regular polygon. understand that an ordinary polygon has the same angles and aspects. Regular tessellations may be made using an equilateral triangle, a rectangular, or a hexagon.


Some tessellations can be named after the use of a variety of machines. you will first select a vertex within the pattern; recall that a vertex is a nook of a polygon. It doesn't count which vertex you select. Then, pick out the polygons round it according to the number of facets each one has. As we study the examples that comply with, we will exercise naming them.


The image of tessellation shows a tessellation crafted from equilateral triangles which is probably translated horizontally. Let's practice naming it. First, we select a vertex inside the pattern. Next, we rely on how many polygons meet at that vertex. There are six. every polygon is a triangle. on the grounds that every triangle has three sides, that is a 3.3.3 tessellation. Each 3 represents a triangle that meets at the vertex.

  • Semi-Regular Tessellations are tessellations which are fabricated from  or greater everyday polygons. The photo of a semi-everyday tessellation is made of hexagons and equilateral triangles. Focusing on the hexagons, we will see the pattern is created via a way of rotating the triangles around the elements of the hexagons.Using our strategy for naming tessellations, we discover that it is a three.3.6 tessellation. The trees constitute the two triangles and the six represents the hexagon.

 

Difference Between the Four Types of Tessellation


Translation

Reflection

Rotation

Glide Reflection

A translation can be defined as a shape that is simply translated, or slid, across the paper and drawn again in another place.


The translation basically shows the geometric shape in the same alignment as the original; it does not turn or flip.

A reflection can be defined as a shape that has been flipped.  Most commonly flipped directly to the right or left (over a "y" axis) or flipped to the top or the bottom (over an "x" axis), reflections can also be done at a particular angle.


If a reflection has been done correctly, an imaginary line can be drawn right through the middle, and the two parts will be symmetrical "mirror" images. To reflect any shape across an axis is to plot a special corresponding point for every point in the original shape.

Rotation is spinning the pattern around a point that is rotating it. A rotation, or turn, occurs when an object is moved in a circular fashion around a central point that does not move.


A good example of a rotation is one "wing" of a pinwheel that turns around the center point. Rotations always have a center and they also have an angle of rotation.

In glide reflection, translation and reflection are used concurrently much like the following piece by Escher, Horseman. There is no reflectional symmetry, nor is there any rotational symmetry.

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What are Keplerʼs Tessellations?

The German astronomer named Johannes Kelper was the one who discovered the planets have elliptical orbits, and was also interested in the problem of tessellations that involve pentagons. The figures replicate some patterns he published involving regular pentagons, regular decagons, and other different polygons. Make one of these with the Zone System and then list the types of symmetry present in the tessellation.

 

What are Regular Tessellations?

A regular tessellation can be defined as a highly symmetric, edge-to-edge tiling made up of regular polygons, all of the same shape. There are only three regular tessellations: those made up of squares, equilateral triangles, or regular hexagons.

 

For Example:

Firstly you need to choose a vertex and then count the number of sides of the polygons that touch it. In the example given above of a regular tessellation of hexagons, next to the vertex there are a total of three polygons and each of them has six sides, so this tessellation is called "6.6.6".

 

Semi-Regular Tessellations

When two or three types of polygons share a common vertex, then a semi-regular tessellation is formed. There are nine different types of semi-regular tessellations including combining a hexagon and a square that both contain a one-inch side. Another example of a semi-regular tessellation that is formed by combining two hexagons with two equilateral triangles.

 

Demi-Regular Tessellations

There are twenty different types of semi-regular tessellations; these are tessellations that combine two or three polygon arrangements. A demi-regular tessellation can be formed by placing a row of squares, then a row of equilateral triangles (a triangle with equal sides) that are alternated up and down forming a line of squares when combined. Demi-regular tessellations always contain two vertices.

 

Non-Regular Tessellations

A non-regular tessellation can be defined as a group of shapes that have the sum of all interior angles equaling 360 degrees. There are again no overlaps or you can say there are no gaps, and non-regular tessellations are formed many times using polygons that are not regular.

 

Other Types

There are two other types of tessellations which are non-periodic tessellations and three-dimensional tessellations. A three-dimensional tessellation uses three-dimensional forms of various shapes, such as octahedrons. A non-periodic tessellation is known to be a tiling that does not have a repetitious pattern. Instead, the tiling evolves as it is created, yet still contains no overlapping or gaps.

 

Which Shapes are Conducive for Tessellation and Why?

In a tessellation, whenever two or more polygons meet at a point (or two or more polygons meet at a particular vertex), the internal angles must add up to 360°. Only three regular polygons (shapes with all sides and angles equal) can form a tessellation by themselves - triangles, squares, and hexagons.

FAQs on Tessellation

1. What Shapes Cannot Tessellate?

There are shapes that are unable to tessellate by themselves. Circles, for example, cannot tessellate. Not only do they not have angles, but it is important to know that it is impossible to put a series of circles next to each other without a gap.

2. Is Tessellation Math or Art?

Tessellation, or tiling, is the covering of the plane by closed shapes, called tiles, without gaps or overlaps 17, page 157. Tessellations have many real-world examples and are a physical link between art and mathematics. Simple examples of tessellations are tiled floors, brickwork, and textiles.

3. What does Tessellate Mean?

A tessellation is a pattern created with identical shapes that fit together with no gaps. Regular polygons tessellate if the interior angles of the polygons can be added together to make 360°.

4. What are the 3 Types of Tessellations?

There are three types of tessellations: Translation, Rotation, and Reflection.

5. Can tessellations overlap?

A tessellation is a sample of shapes repeated to fill a plane. The shapes of Tessellations do not overlap.

6. What are the number of varieties of tessellations present?

There are 3 types of normal tessellations: triangles, squares and hexagons.

7. What are the main features of tessellations?

The key functions of tessellations are that there should be no gaps or overlaps in shapes. The identical discern (or institution of figures) come collectively to absolutely cowl a wall or floor or a few different planes. This calls for the vertices to fit together.

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