Tessellation

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.


Types of Tessellation

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

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

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

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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 centre point. Rotations always have a centre 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, 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 Zome System and then list the types of symmetry present in the tessellation.


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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:


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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 demi-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 (Frequently Asked Questions)

Question 1. What Shapes Cannot Tessellate?

Answer: 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.

Question 2. Is Tessellation Math or Art?

Answer: 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.

Question 3. What does Tessellate Mean?

Answer: 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°.

Question 4. What are the 3 Types of Tessellations?

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