TESSELLATIONS: CHARACTERISTIC, TYPES (REGULAR, IRREGULAR), EXAMPLES - MATHS - 2025

The tilings are coated surfaces one or more figures called tesserae. They are everywhere: in streets and buildings of all kinds. The tessera or tiles are flat pieces, generally polygons with congruent or isometric copies, which are placed following a regular pattern. In this way there are no spaces left uncovered 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 there is a regular tessellation, but if two or more types of regular polygons are used, then it is a semi-regular tessellation.

Figure 1. Tile floor with irregular tessellation, because the rectangles are non-regular polygons, even though the squares are. Source: Pixabay.

Finally, when the polygons that the tessellation forms are not regular, then it is an irregular tessellation.

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

History of tessellations

Tessellation has been used for thousands of years to cover floors and walls of palaces and temples of different cultures and religions.

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

Figure 2. Sumerian tessellations at the Istar gate. Source: Wikimedia Commons.

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

Penrose created a non-periodic tessellation known as the Penrose tessellation. These are just a few names of scientists who contributed much about tessellation.

Regular tessellations

Regular tessellations are made with only one 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 above 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 listing in a clockwise direction 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 sides.

This nomenclature applies to regular and semi-regular tessellations.

Example 1: Triangular tessellation

Figure 3 shows a regular triangular tessellation. 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 3 ⁶.

Figure 3. Regular triangular tessellation 3.3.3.3.3.3. Source: wikimedia commons

Example 2: Square tessellation

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

Figure 4. Square tessellation 4.4.4.4. Source: wikimedia commons.

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 6 ³.

Figure 5. Hexagonal tessellation 6.6.6. Source: wikimedia commons.

Semi-regular tessellations

Semi-regular 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 completely shared with the neighbor.

There are eight semi-regular tessellations:

1. 3.6.3.6 (tri-hexagonal tessellation)
2. 3.3.3.3.6 (blunt hexagonal tessellation)
3. 3.3.3.4.4 (elongated triangular tessellation)
4. 3.3.4.3.4 (blunt square tessellation)
5. 3.4.6.4 (rhombi-tri-hexagonal tessellation)
6. 4.8.8 (truncated square tessellation)
7. 3.12.12 (truncated hexagonal tessellation)
8. 4.6.12 (truncated tri-hexagonal tessellation)

Some examples of semi-regular tessellations are shown below.

Example 4: Tri-hexagonal tessellation

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

Figure 6. The tri-hexagonal tessellation (3.6.3.6) is an example of semi-regular tessellation. Source: Wikimedia Commons.

Example 5: Blunt hexagonal tessellation

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

Figure 7. The blunt hexagonal tessellation consists of a hexagon surrounded by 16 triangles in the 3.3.3.3.6 configuration. Source: Wikimedia Commons.

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.

Figure 8. Semi-regular tessellation composed of a triangle, a square and a hexagon in the 3.4.6.4 configuration. Source: Wikimedia Commons.

Irregular tessellations

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

Example 7

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

Figure 9. Even though all the tiles are congruent squares, this is a clear example of irregular tessellation. Source: F. Zapata.

Example 8

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

Figure 10. A tessellation formed by parallelograms is irregular, since its mosaics are non-regular polygons. Source: F. Zapata.

Example 9

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

Figure 11. Hexagons with central symmetry even when they are not regular tessellate the plane. Source: F. Zapata.

Example 10: tessellation of Cairo

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

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

Figure 12. Cairo Tessellation. Source: Wikimedia Commons.

Example 11: Al-Andalus tessellation

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

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

Figure 13. Tessellation of the Alhambra Palace. Tartaglia / Public domain

Example 12: tessellation in video games

Also known as tesellation, it is one of the most popular novelties in video games. It is about creating textures to simulate the tessellation of the different scenarios that appear in the simulator.

This is a clear reflection that these coatings continue to evolve, crossing the borders of reality.

References

1. Enjoy math. Tessellations. Recovered from: enjoymatematicas.com
2. Rubiños. Tessellations resolved examples. Recovered from: matematicasn.blogspot.com
3. Weisstein, Eric W. "Demiregular tessellation." Weisstein, Eric W, ed. MathWorld. Wolfram Research.
4. Wikipedia. Tessellation. Recovered from: es.wikipedia.com
5. Wikipedia. Regular tessellation. Recovered from: es.wikipedia.com