The Cordoban number was discovered and used by Arab mathematicians in Cordoba, Spain during the Middle Ages, and was used in the architecture and art of the time to achieve a harmonious and balanced aesthetic.

To obtain the Cordoban number, these procedure is followed:

An example of the presence of the Cordoban number in art can be found in the Mosque-Cathedral of Cordoba, one of the masterpieces of Islamic architecture in Spain. In the construction of the Mosque-Cathedral, measurements based on the Cordoban number were used to achieve a harmonious distribution of arches, columns, and vaults.

Another example of the presence of the Cordoban number in art can be found in Renaissance painting. The famous Italian painter Leonardo da Vinci used the Cordoban number in his artwork "La Gioconda" (also known as "Mona Lisa") to achieve a sense of balance and harmony in the composition.

In summary, the Cordoban number is an irrational number used in the architecture and art of the Middle Ages to achieve a harmonious and balanced aesthetic. Examples of its presence in art include the Mosque-Cathedral of Cordoba and the Renaissance painting of Leonardo da Vinci.

Botticelli's "The Birth of Venus" is a masterpiece of the Italian Renaissance and aestheticism and is an example of the presence of the golden ratio. The painting depicts the goddess Venus emerging from the sea, standing on a seashell, and being blown towards the shore by the winds. The painting's vibrant colors, beautiful composition, and idealized figures are a celebration of the beauty and grace of the human form. 

Botticelli's "The Birth of Venus" has several relationships with the golden ratio. For example, Venus' position in the shell fits a logarithmic spiral, which is a geometric shape commonly found in nature and can also be constructed using the golden ratio. Additionally, the edges of the shell and Venus' body fit the proportions of the golden ratio, adding a sense of balance and harmony to the painting. Therefore, we can say that the use of the golden ratio in "The Birth of Venus" contributes to the sense of beauty and aesthetic perfection associated with the work.

Drawing a Reuleaux triangle is quite simple. Starting from an equilateral triangle and centering on one of the vertices, an arc of circumference is drawn that connects two other vertices. The operation is repeated for each vertex and thus, eliminating the initial triangle, the Reuleaux triangle is obtained.

One example of this is the Former St Mark’s, Low Moor.

Former St Mark’s is a building on Huddersfield Road, Low Moor, Bradford, located in England. It was designed by Mallinson and Healy, and built by Catherine Mawer, between 1855-1857. Catherine was an architectural sculptor who worked with her husband. She has a lot of works, for example, Mawer Memorial or Susannah Blesard monument. But we are going to focus on Former St Mark’s. 

It is listed as a Second Grade Building in England, which means it has a particular historical and architectural meaning, and there are some rules and regulations protecting it. There is a particular item that was the striking element for us, and this is the font. In this sculpture we can see a special element, the triangle of Reuleaux. It is situated at the top of the columns, around half of the height of this piece, where there is a kind of arch but with a slight curvature. It is a triangle with constant width, which is obtained by the intersection of three equal circles who have their centers in the vertex of a equilateral triangle. Franz Reuleaux is the responsible of this mechanism, he was a german mechanical engineer and a lecturer of the Berlin Royal Technical Academy, he was often called “father of kinematics”. Nowadays we can see this triangle in many places, such as pencils, guitar picks and also in some signs and corporate logos. 

Topology is the branch of mathematics that deals with the properties of geometric objects that are preserved under continuous transformations. In topology, we can define a Möbius band as a surface that can be obtained by taking a rectangular strip of paper, giving it a half-twist, and then gluing the ends together.

Mathematically, we can represent the Möbius band as a subset of three-dimensional Euclidean space, given by the following parametric equations:

x(u, v) = (1 + (v/2)cos(u/2))cos(u)

y(u, v) = (1 + (v/2)cos(u/2))sin(u)

z(u, v) = (v/2)sin(u/2)

where 0 ≤ u ≤ 2π and -1 ≤ v ≤ 1.

These equations describe a twisted surface that has only one side and one edge. The parameter u controls the twist of the surface, while the parameter v controls the width of the band.

The Möbius band has many interesting properties that make it a fascinating object of study in mathematics and physics. For example, it has only one side, which means that if you start at any point on the surface and move in a straight line, you will eventually end up back where you started, but on the other side of the band. This property is related to the concept of orientability in topology.

The Möbius band also has a non-trivial topology, which means that it cannot be deformed into a flat surface without tearing or cutting it. This property is related to the concept of homotopy in topology.

In summary, the Möbius band is a fascinating mathematical object with many interesting properties that make it a topic of study in topology, geometry, and physics.

Max Bill, a Swiss artist and designer, created a fascinating sculpture in 1957 called the Möbius Band. The sculpture is a three-dimensional representation of a Möbius strip, a mathematical object that has only one side and one edge.

The Möbius Band sculpture consists of a single strip of metal twisted and looped in such a way that it creates a continuous surface that is both inside and outside. The strip appears to have no beginning or end, and it has a unique topological property in that a line drawn down the center of the band would end up on the opposite side without crossing over.

The sculpture is made of steel and measures approximately 28 inches in diameter. The surface of the band is smooth and reflective, giving the sculpture a sleek and modern look. The form is elegantly simple, yet visually striking, and it demonstrates Max Bill's interest in the intersection of art and mathematics.

The Möbius Band sculpture is a testament to Bill's innovative and avant-garde approach to art and design. He believed that art and science were intertwined, and that mathematical principles could be used to create beautiful and functional objects. The Möbius Band sculpture is a prime example of this philosophy, as it combines mathematical rigor with aesthetic elegance.

Today, the Möbius Band sculpture is considered a masterpiece of modern sculpture and is on display in several museums around the world, including the Museum of Modern Art in New York City. Its enduring appeal lies in its ability to seamlessly merge art and science, and in its ability to challenge our perceptions of space and form.

The möbius band is one of the strangest and most curious geometric objects that exist.

It was discovered independently by the German mathematicians Johann Benedict Listing (1808-1882) and August Ferdinand Moebius( 1790 - 1868). It is a two- dimensional surface that is projected into three- dimensional space, having only one face and one edge. In mathematics it is known as a non-orientable object because the inside, outside, top, bottom, right, and left have no meaning on this object.

This enigmatic object is used in mathematics specifically in the study of topology, which is the study of those properties of geometric objects that do not change when they are subjected to continuous transformations. 

The möbius band is represented by the following formulas: 

x = ( a+v•r•sen(u/2))•cos(u)

y=( a+v•r•sen(u/2))•sen(u)

z=v•r•cos(u/2)

 where a allows us to modify the length, r allows us to modify the width within the right intervals, and the variable u takes values in the interval [0, 2 Π] and v in [-1, 1].

An example of the presence of this möbius band is Endless surface by Max Bill who is a Swiss architect, artist, painter, graphic designer, typeface designer and industrial designer.

He was born in 1908 in Switzerland, and died in 1994 in Berlin. 

If you want to see it you have to go to the Foundation Juan March museum, there you will be able to see more of him. 

It was made of metal and forging with an approximated height of 15’5 x 24 x 13 cm

This structure wants to represent liberty and also freedom. It is a way to pursue beauty, like most of the art of that age.