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THE MÖBIUS BAND AND ITS PRESENCE IN ART( MARA SOTÉS AND CARMEN MARTÍNEZ)

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THE MÖBIUS BAND AND ITS PRESENCE IN ART( MARA SOTÉS AND CARMEN MARTÍNEZ)

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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.

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The golden number: aestheticism (Angelo Cristian Sarvu, Samuel Mackeron Pastora and Rodrigo Ochagavía Cabezón)

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The golden number: aestheticism (Angelo Cristian Sarvu, Samuel Mackeron Pastora and Rodrigo Ochagavía Cabezón)

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The golden number is an irrational number which denotes a mathematical ratio called the “Golden Ratio”. It is represented by the greek letter phi (φ) and it is approximately equal to 1.618033988749… Mathematically, it is defined as the ratio of two quantities so that the ratio of the sum of the two quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. Explained in a simple manner, you separate a line in a short segment and a large segment, so that the long segment divided by the smaller segment is equal to the whole length divided by the long part. It is also related to the Fibonacci sequence as the further you go into the sequence, the closer the ratio between each number and its previous one gets to the golden ratio.

The first known mention of the ratio is from around 300 BC in famous Greek mathematician Euclid’s book called “Euclid’s Elements”. Multiple other famous mathematicians such as Abu Kamil, Fibonacci or Luca Pacioli have studied this mathematical concept. This number can be found everywhere from art (Leonardo Da Vinci or Salvador Dalí) and architecture (Le Corbusier) to nature (plants, animals, etc.).

"Peacock Skirt" is a work of art created by Aubrey Beardsley, a British illustrator and writer who lived during the late 19th century. The piece was published in 1893 and was part of a larger collection of illustrations that appeared in the Yellow Book, a literary magazine that Beardsley contributed to. Despite his early success as an artist, Beardsley's career was cut short by his untimely death from tuberculosis at the age of 25. Nevertheless, his influence on the Art Nouveau movement and on subsequent generations of artists has been significant, and his legacy continues to be celebrated today.

The artwork depicts a woman wearing a long, flowing skirt adorned with peacock feathers. The feathers are intricately drawn and appear to cascade down the skirt, creating a sensation of movement and texture.

This work of art integrates the golden ratio in the girl and her flower crown, giving a sense of overall balance and harmony to the design. It is actually easy to see this golden ratio spiral which is hidden in the string that comes off the girl’s crown. It also seems to contain a fractal structure inside the girl's dress, although it cannot be confirmed as fractals weren't discovered until some time later, but this artist is well known for inspiring from the natural shapes.

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THE MÖBIUS BAND AND ITS PRESENCE IN THE ART (Miguel GONZÁLEZ and Iván RUBIO)

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THE MÖBIUS BAND AND ITS PRESENCE IN THE ART (Miguel GONZÁLEZ and Iván RUBIO)

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The Möbius band is a mathematical concept that has been studied for centuries. It is a two-dimensional surface with only one side and one edge. To mathematically explain the Möbius band, we need to use the concept of topology.

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.

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THE GOLDEN TRIANGLE AND ITS PRESENCE IN ART (Bilal Benmoussa Larbi-Aissa and Juan López Larumbe)

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THE GOLDEN TRIANGLE AND ITS PRESENCE IN ART (Bilal Benmoussa Larbi-Aissa and Juan López Larumbe)

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The Golden Triangle, also known as the Golden Ratio, was a central concept in the aesthetic movement of the late 19th century. This art and design movement emphasized aesthetic values and beauty over social and moral issues. The Golden Triangle was believed to be a key factor in creating aesthetically pleasing works that conveyed a sense of harmony and balance.

The Golden Triangle is obtained by dividing a line into two unequal segments so that the ratio between the longest and shortest segments is equal to the ratio between the original line and the longest segment. The resulting ratio is approximately 1.6180339887.

Many artists and designers in the aesthetic movement, such as William Morris and Aubrey Beardsley, incorporated the Golden Triangle into their work. Morris, one of the leaders of the Arts and Crafts movement, used the golden ratio in his wallpaper and fabric designs, such as in his work "Tulip and Willow" where the design is based on the golden ratio.

Beardsley, for his part, used the golden ratio in his illustrations, such as in his work "The Climax", in which the figure's body is positioned according to the golden ratio. Other artists and designers of the time, such as James McNeill Whistler, also used the Golden Triangle in their works.

In summary, the Golden Triangle was a key concept in the aesthetic movement of the late 19th century and was believed to be an essential factor in creating balanced and aesthetically pleasing works of art and design. Examples of its use in art and design include the works of William Morris, Aubrey Beardsley, and James McNeill Whistler.

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THE REULEAUX TRIANGLE ANT ITS PRESENCE IN ART (Claudia Gómez and Lucia Martínez)

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THE REULEAUX TRIANGLE ANT ITS PRESENCE IN ART (Claudia Gómez and Lucia Martínez)

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The Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the other two.

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.

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THE GOLDEN RATIO AND ITS PRESENCE IN ART  (Lucía Donoso and Daniela Teixeira)

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THE GOLDEN RATIO AND ITS PRESENCE IN ART  (Lucía Donoso and Daniela Teixeira)

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The golden ratio, also known as the divine proportion, is a mathematical concept that has fascinated artists, architects, and scientists for centuries. It is a ratio that is found in nature, art, and architecture and is believed to have aesthetic appeal. The golden ratio is approximately 1.61803398875 and is denoted by the Greek letter phi (φ). It is constructed by dividing a line segment into two parts, such that the ratio of the smaller part to the larger part is the same as the ratio of the larger part to the whole. This ratio can be seen in the spiral patterns of seashells, the arrangement of leaves on a stem, and the proportions of the human body. The golden ratio is a fundamental concept in mathematics and has had a profound impact on art, architecture, and design.

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.

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THE CORDOBAN NUMBER AND ITS PRESENCE IN THE ART                                                             (Víctor González and Gonzalo Navaridas)

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THE CORDOBAN NUMBER AND ITS PRESENCE IN THE ART                                                             (Víctor González and Gonzalo Navaridas)

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"The Cordoban number", also known as the "Cordoba golden number", is an irrational number approximately equal to 1.6180339887. This number arises from the ratio of two segments of a line that are divided in such a way that the ratio between the larger segment and the smaller segment is equal to the ratio between the sum of both segments and the larger segment.

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:

1. Start with a circle of radius 1.

2. Draw a straight line from the center of the circle to the circumference, and mark the point where the line touches the circumference.

3. Draw a line perpendicular to the first line, from the point marked in the previous step to the circumference.

4. Mark the point where the perpendicular line touches the circumference.

5. Divide the length of the perpendicular line by the length of the line drawn in step 2. This ratio is precisely the Cordoban number.

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.

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The hero's journey in "The Picture of Dorian Gray"

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The hero's journey in "The Picture of Dorian Gray"

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The Picture of Dorian Gray is the story of one beautiful, innocent young man's seduction, moral corruption, and eventual downfall.

Dorian Gray is a young man living in late-19th century London. While wealthy, charming, generally intelligent and very handsome, he is naive and easily manipulated. These faults lead to his spiral into sin and, ultimately, misery.

Dorian Gray personifies the aesthetic lifestyle in action, pursuing personal gratification with abandon. Yet, while he enjoys these indulgences, his behavior ultimately kills him and others, and he dies unhappier than ever.

Rather than an advocate for pure aestheticism, which claims that Art should be beautiful and pleasure its observer without further implications, then, Dorian Gray is a tale in which Wilde illustrates the dangers of the aesthetic philosophy when not practiced with prudence. Aestheticism, argues Wilde, too often aligns itself with immorality, resulting in a precarious philosophy that must be practiced deliberately.

The Aesthetic Movement in Britain (1860 – 1900) aimed to escape the ugliness and materialism of the Industrial Age, by focusing instead on producing art that was beautiful rather than having a deeper meaning – 'Art for Art's sake'. Artists drew inspiration from a variety of cultures and periods. They found beauty in Renaissance painting, ancient Greek sculpture and East Asian art and design, especially Japanese prints.

With this project we intend to highlight the beautiful math patterns hidden behind these works of art.