Throughout history it has been seen that mathematics is a fundamental and very important foundation in the life of human beings. These have been very useful in the daily interaction of people, as a process of maintaining order and balance in the social structure. Mathematics is present in the places we visit, and in the things we see and do every day. They have also helped in the creation of artifacts and resources that have facilitated and been a breakthrough in the life of mankind. There are many benefits and contributions that have made mathematics in our lives, and because of them is that we have much of the things that have had a great impact on our society.
An example of this is the importance of mathematics in art. There are several themes or concepts that influence one way or another in the realization of works of art and objects that we use or see day by day. These include the golden section, tessellations, fractals and digital art. With these concepts, artists and designers have been able to answer questions such as, what is the most harmonious and perfect way of dividing an object into two parts? And what is the relationship between the measures of the parts that constitute an object so that it is beautiful?
In classical antiquity, Greek Plato noticed a way of dividing a segment in a harmonic and pleasing way, which he called The Section. In the year 300 BC, Another Greek called Euclides discovered geometrically how to divide a segment into two parts so that it was pleasing to the eye, which he called the Golden Section, also known as the Golden Ratio or Divine Proportion. In order to divide a segment into the Golden Section, the smaller part divided by the larger part must be equal to the larger part divided by the total length. Here is an example of a segment divided into Golden Section:
An example of this is the importance of mathematics in art. There are several themes or concepts that influence one way or another in the realization of works of art and objects that we use or see day by day. These include the golden section, tessellations, fractals and digital art. With these concepts, artists and designers have been able to answer questions such as, what is the most harmonious and perfect way of dividing an object into two parts? And what is the relationship between the measures of the parts that constitute an object so that it is beautiful?
In classical antiquity, Greek Plato noticed a way of dividing a segment in a harmonic and pleasing way, which he called The Section. In the year 300 BC, Another Greek called Euclides discovered geometrically how to divide a segment into two parts so that it was pleasing to the eye, which he called the Golden Section, also known as the Golden Ratio or Divine Proportion. In order to divide a segment into the Golden Section, the smaller part divided by the larger part must be equal to the larger part divided by the total length. Here is an example of a segment divided into Golden Section:
where 1-x divided by x is equal to x divided by 1. This way of dividing a segment constituted the basis on which the art and architecture of the Greeks were founded.
The relation between the two parts in which the segment is divided is called φ (we write Phi and pronounce fi), which is equal to 1.618033988..., an irrational number (has infinite decimal numbers and is not periodic), better known as Golden Number. It was named in honor of Fidias, the Greek architect who made the sculptural decoration of the Parthenon, and who used the Golden Ratio in the creation of many of his works.
The relation between the two parts in which the segment is divided is called φ (we write Phi and pronounce fi), which is equal to 1.618033988..., an irrational number (has infinite decimal numbers and is not periodic), better known as Golden Number. It was named in honor of Fidias, the Greek architect who made the sculptural decoration of the Parthenon, and who used the Golden Ratio in the creation of many of his works.
Where do we find the Golden Ratio?
We can find the Golden Ratio basically in everything around us, but we almost never realize that it is present. An example of where the Golden Ratio is found is in the body and people's faces. Within art it can be found in several drawings, paintings and works done by famous artists. Among these is The Man of Vitruvius, by Leonardo Da Vinci. This is because the ratio between the distance from the navel to the feet and the distance from the head to the navel is φ, as well as the ratio between the height of the man and the distance from the navel to the feet. In addition, the ratio between the height of the man (side of the square) and the distance from the navel to the tip of the hand (radius of the circumference) is the golden number.
Tessellations
A tessellation is created by covering a plane with a certain pattern of figures without empty spaces between them or overlapping. To make a tessellation, you must involve technique, geometry, art and decoration. It can be a tiled floor or a plane with ceramics or tiles. They may be squares, triangles, rectangles, but there are also other shapes with which the desired surface can be coated. They have also been used as decorative motifs for furniture, clothing, carpets, tapestries, etc.
To tile a plane, you can use geometric figures called polygons. A polygon is a flat geometric figure established or limited by three or more straight lines, and having three or more angles and vertices. To these can be applied some types of transformations in the plane, called isometries, which are those transformations that do not alter the shape or size of the figure, and that only involves a change of position of it (in the orientation or in the sense). This results in the initial and final figures being similar, and geometrically congruent.
The polygons with which you can tessellate a plane can be regular, semiregular or irregular. The tessellation is regular if all the polygons that conform it are regular. If it is semi-regular it means that it is made up of two or more regular polygons. On the other hand, if it is irregular means that the tessellation is formed by non-regular polygons.
To tile a plane, you can use geometric figures called polygons. A polygon is a flat geometric figure established or limited by three or more straight lines, and having three or more angles and vertices. To these can be applied some types of transformations in the plane, called isometries, which are those transformations that do not alter the shape or size of the figure, and that only involves a change of position of it (in the orientation or in the sense). This results in the initial and final figures being similar, and geometrically congruent.
The polygons with which you can tessellate a plane can be regular, semiregular or irregular. The tessellation is regular if all the polygons that conform it are regular. If it is semi-regular it means that it is made up of two or more regular polygons. On the other hand, if it is irregular means that the tessellation is formed by non-regular polygons.
Regular tessellations
Semi-regular tessellations
Irregular tessellations
Fractals
A fractal is a geometric object whose basic structure, fragmented or irregular, is repeated at different scales. The term was proposed by the mathematician Benoît Mandelbrot in 1975, and comes from the Latin "fractus", which means broken or fractured. Fractals are generated on computers with mathematical formulas or algorithms, and with a very small set of data. Its algorithm is defined by a key feature: iteration. An iteration is the repetition of "something" an "infinite" amount of times. For this reason, fractals are generated through iterations of a fixed geometric pattern.
Within the computational field, the process of fractal transformation is done with images that contain many pixels. Each of these is "increasing infinitely", without ceasing to be the same. In addition to being beautiful and interesting, they are used to generate effects in the programming, representation and analysis of a great variety of complex processes in various fields, such as: Biology, Mathematics, Physics, Chemistry, Geology, Art, etc.
A fractal geometric object is attributed the following characteristics: it is too irregular to be described in traditional geometric terms. It is also autosimilar, which means that its shape is made of smaller copies of the same figure (the copies are similar to the whole: same shape but different size). Another is that, when expanding a fractal, it reveals more details without having a limit. In addition, its area or surface is finite (has limits), but its perimeter or length are infinite (no limits).
Within the computational field, the process of fractal transformation is done with images that contain many pixels. Each of these is "increasing infinitely", without ceasing to be the same. In addition to being beautiful and interesting, they are used to generate effects in the programming, representation and analysis of a great variety of complex processes in various fields, such as: Biology, Mathematics, Physics, Chemistry, Geology, Art, etc.
A fractal geometric object is attributed the following characteristics: it is too irregular to be described in traditional geometric terms. It is also autosimilar, which means that its shape is made of smaller copies of the same figure (the copies are similar to the whole: same shape but different size). Another is that, when expanding a fractal, it reveals more details without having a limit. In addition, its area or surface is finite (has limits), but its perimeter or length are infinite (no limits).
The aforementioned concepts: the Golden Ratio, Phi (golden number), the tessellations and fractals, are widely used in art and are present in our daily lives. In addition, they are closely related to mathematics. Mathematics has a great importance in the creation of works of art, by influencing them in different ways. For this reason, it can be said that the link between these two subjects helps human beings to grow intellectually and artistically, according to the way in which they apply them in their lives.
Author: Arlyn V. Padín López
Undergraduate student at the University of Puerto Rico, Aguadilla Campus
Undergraduate student at the University of Puerto Rico, Aguadilla Campus