The tessellated are surfaces covered by one or more figures called *tiles*. They are everywhere: in streets and buildings of all kinds. The tesserae or tiles are flat pieces, usually polygons with congruent or isometric copies, which are placed following a regular pattern. In this way, there are no spaces left without being covered and the tiles or mosaics do not overlap.

In the case that a single type of mosaic formed by a regular polygon is used, then we have a regular tilingbut if two or more types of regular polygons are used, then it is a semiregular tessellation.

Finally, when the polygons that form the tessellation are not regular, then it is a irregular tiling.

The most common type of tessellation is the one formed by rectangular and particularly square mosaics. In figure 1 we have a good example.

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**History of tessellations**

For thousands of years, tessellation has been used to cover floors and walls of palaces and temples of different cultures and religions.

For example, the Sumerian civilization that flourished around 3500 BC in southern Mesopotamia, between the Euphrates and Tigris rivers, used tessellations in their architecture.

Tessellations have also aroused the interest of mathematicians of all ages: starting with Archimedes in the 3rd century BC, followed by Johannes Kepler in 1619, Camille Jordan in 1880, up to contemporary times with Roger Penrose.

Penrose created a non-periodic tiling known as *Penrose tiling. *AND*these* These are just a few names of scientists who contributed a lot about tessellation.

## regular tessellations

Regular tessellations are made with a single type of regular polygon. On the other hand, for the tessellation to be considered regular, every point of the plane must:

-Belong to the interior of the polygon

-Or to the edge of two adjacent polygons

-Finally, it can belong to the common vertex of at least three polygons.

With the previous restrictions it can be shown that only equilateral triangles, squares and hexagons can form a regular tessellation.

### Nomenclature

There is a nomenclature to denote tessellations that consists of enumerating clockwise and separated by a point, the number of sides of the polygons that surround each node (or vertex) of the tessellation, always starting with the polygon with the lowest number. from sides.

This nomenclature is applied to regular and semi-regular tessellations.

### Example 1: Triangular tessellation

Figure 3 shows a regular triangular tiling. It should be noted that each node of the triangular tessellation is the common vertex of six equilateral triangles.

The way to denote this type of tessellation is 3.3.3.3.3.3, which is also denoted by 36.

### Example 2: Square tessellation

Figure 4 shows a regular tessellation composed only of squares. It should be noted that each node of the tessellation is surrounded by four congruent squares. The notation that applies to this type of square tessellation is: 4.4.4.4 or alternatively 44

### Example 3: Hexagonal tessellation

In a hexagonal tessellation each node is surrounded by three regular hexagons as shown in figure 5. The nomenclature for a regular hexagonal tessellation is 6.6.6 or alternatively 63.

## semiregular tessellations

The semiregular tessellations or Archimedean tessellations consist of two or more types of regular polygons. Each node is surrounded by the types of polygons that make up the tessellation, always in the same order and the edge condition is fully shared with the neighbor.

There are eight semiregular tilings:

**3.6.3.6 (tri-hexagonal tessellation)**

**3.3.3.3.6 (blunt hexagonal tiling)**

**3.3.3.4.4 (elongated triangular tessellation)**

**3.3.4.3.4 (blunt square tessellation)**

**3.4.6.4 (rhombi-tri-hexagonal tessellation)**

**4.8.8 (truncated square tiling)**

**3.12.12 (truncated hexagonal tiling)**

**4.6.12 (truncated tri-hexagonal tessellation)**

Some examples of semiregular tessellations are shown below.

### Example 4: Tri-hexagonal tessellation

It is the one that is composed of equilateral triangles and regular hexagons in the structure 3.6.3.6, which means that a node of the tessellation is surrounded (up to completing one turn) by a triangle, a hexagon, a triangle and a hexagon. Figure 6 shows such a tessellation.

**Example 5: Blunt hexagonal tiling**

Like the tessellation in the previous example, this one also consists of triangles and hexagons, but its distribution around a node is 3.3.3.3.6. Figure 7 clearly illustrates this type of tessellation.

### Example 6: rhombi-tri-hexagonal tessellation

It is a tessellation consisting of triangles, squares and hexagons, in the configuration 3.4.6.4, which is shown in figure 8.

## irregular tilings

Irregular tessellations are those that are formed by irregular polygons, or by regular polygons but that do not meet the criterion that a node is the vertex of at least three polygons.

### Example 7

Figure 9 shows an example of an irregular tessellation, in which all the polygons are regular and congruent. It is irregular because a node is not the common vertex of at least three squares and there are neighboring squares that do not completely share an edge.

### Example 8

The parallelogram tessellates a flat surface, but unless it is a square it cannot form a regular tessellation.

### Example 9

The non-regular hexagons with central symmetry tessellate a flat surface, as shown in the following figure:

### Example 10: Tessellation of Cairo

It is a very interesting tessellation, made up of pentagons with sides of equal length but with unequal angles, two of which are right and the other three have 120º each.

Its name comes from the fact that this tessellation is found on the pavement of some of the streets of Cairo in Egypt. Figure 12 shows the tessellation of Cairo.

### Example 11: Al-Andalus tessellation

The tessellation during in some parts of Andalusia and North Africa are characterized by geometry and epigraphy, as well as ornamental elements such as vegetation.

The tessellation of palaces such as that of the Alhambra were made up of tiles made up of ceramic pieces of many colors, with multiple (not to say infinite) shapes that unleashed geometric patterns.

### Example 12: Tessellation in video games

Also known as tessellation, it is one of the most booming innovations in video games. It is about the creation of textures to simulate the tessellation of the different scenarios that appear in the simulator.

This is the clear reflection that these coatings continue to evolve beyond the borders of reality.

## References

Enjoy math. Tessellations. Recovered from: disfrutalalasmatematicas.com

blondies Tessellations solved examples. Recovered from: matematicasn.blogspot.com

Weisstein, Eric W. «Demiregular tessellation.» Weisstein, Eric W., ed. MathWorld. Wolfram Research.

Wikipedia. tessellation. Recovered from: en.wikipedia.com

Wikipedia. Regular tessellation. Recovered from: en.wikipedia.com