How Tessellation Patterns Appear in Nature and Art
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
Translation - A Tessellation in which the shape repeats by moving or sliding.
Rotation - A Tessellation in which the shape repeats by rotating or turning.
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
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 in Maths: Definition, Types & Real-World Uses
1. What is a tessellation in mathematics?
A tessellation, also known as a tiling, is a pattern created by repeating one or more geometric shapes to cover a flat surface, called a plane, without any gaps or overlaps. A key feature is that the corners, or vertices, of the shapes must fit together perfectly at each point.
2. What are some real-world examples of tessellation?
Tessellations are found all around us in both natural and man-made structures. Common examples include:
- A beehive honeycomb, which is a natural tessellation of hexagons.
- Tiled floors and bathroom walls, often using squares or rectangles.
- Brick patterns on walls and pavements.
- The intricate and artistic patterns in the work of M.C. Escher, which often use animal shapes.
- The scales on a fish or a pineapple's skin.
3. What are the main types of tessellations?
Tessellations are primarily classified into three types based on the shapes used:
- Regular Tessellation: This pattern is made using only one type of regular polygon (a shape with all sides and angles equal). Only equilateral triangles, squares, and regular hexagons can form regular tessellations.
- Semi-regular Tessellation: This is formed by using two or more types of regular polygons, with the same arrangement of polygons at every vertex.
- Non-regular Tessellation: This pattern uses shapes that are not regular polygons, such as irregular triangles or custom-designed interlocking shapes.
4. Why can only equilateral triangles, squares, and regular hexagons form regular tessellations?
The ability of a regular polygon to tessellate by itself depends entirely on its interior angle. For shapes to fit together perfectly at a vertex without gaps or overlaps, the sum of the angles around that point must be exactly 360 degrees.
- Equilateral Triangles: Each angle is 60°, and 6 × 60° = 360°.
- Squares: Each angle is 90°, and 4 × 90° = 360°.
- Regular Hexagons: Each angle is 120°, and 3 × 120° = 360°.
5. How does tessellation connect the concepts of mathematics and art?
Tessellation serves as a perfect bridge between mathematics and art. Mathematics provides the fundamental rules: the geometry of shapes, the properties of angles, and the principles of symmetry and transformations (like rotation and reflection). Art applies these rules to create visually stunning and complex patterns. Artists like M.C. Escher famously used the mathematical principles of tessellation to create repeating patterns of interlocking figures like birds and lizards, turning a geometric concept into a creative masterpiece.
6. What do vertex configurations like '4.8.8' or '3.3.4.3.4' mean in tessellations?
This numbering system, known as a vertex configuration, is used to describe semi-regular tessellations. The numbers indicate the type and sequence of regular polygons that meet at any single vertex. Each number represents the number of sides of a polygon. For example:
- 4.8.8: This means one square (4 sides) and two octagons (8 sides) meet at every vertex.
- 3.3.4.3.4: This means two equilateral triangles (3 sides), a square (4 sides), another triangle, and another square meet in that specific order around each vertex.
7. Can irregular shapes be used to create a tessellation?
Yes, absolutely. While regular tessellations are limited to three shapes, many irregular polygons can tile a plane perfectly. A common misconception is that only regular shapes tessellate. In fact, any triangle (whether scalene, isosceles, or right-angled) and any quadrilateral (including concave ones) can be used to form a tessellation. This flexibility allows for an infinite variety of non-regular tiling patterns.
8. What role do geometric transformations play in creating tessellation patterns?
Geometric transformations are the actions used to repeat a shape or a set of shapes across a plane to build the tessellation. The primary transformations involved are:
- Translation: Simply sliding the shape to a new position without rotating or flipping it.
- Rotation: Turning the shape around a fixed point (often a vertex or the centre of an edge).
- Reflection: Flipping the shape across a line to create a mirror image.
