Math and Art Essays

We in the world divide time into periods to divide the advancements the world has. There exist lots of time periods that contain art advancements, but in this essay we are going to talk about the modern period. The modern period covers the period of time from 1,450 to the present day. Since 1,450 there has been a lot of art history and we are going to cover 5 highlights of art in the modern period that have a relationship with math and identifying and explaining the relationship. Art in the modern period saw a new discovery, paintings passed from being 1 dimensional to being 2 dimensional,also gave new advancements in architecture using new mathematical techniques, and the compositions present symmetry, proportion relations, perspective with a central vanishing point, the figures present volumetric treatments. There were and are lots of artists that make use of mathematics, so in the following paragraphs we are going to talk about the biggest highlights in mathematics and art. 

One of the highlights is “Las Meninas de Velásquez” that embraces the change between 1 dimensional and 2 dimensional “Las Meninas de Velásquez”. This painting was finished in the year 1656, it was made by famous painter Diego Velazques. During the 18 century this painting was also known as “La Familia de Felipe VI” This painting portrays the infant Margarita from Austria surrounded by her maids, and even at the left in the painting you can even see the painter Diego Velzques presumably painting the couple that can be seen in the far through a mirror . 

The painting uses a vanishing point to give an illusion of the size of the subject and the size and distance of the room. 

The vanishing makes it possible to differentiate the distances between our point of view and the distances that our subjects have. The vanishing point also makes it possible for us to see subjects that are very close to us but also subjects that are more distanced from us. The painting also has an advanced use of perspective, in the painting Diego Velazques is looking at the middle presumably looking at the same couple we can see at the back of the painting in a mirror, giving the piece an advanced new look at perspective. It is clear that the painting has a big influence on mathematics that isn't looked upon.

A very famous building known as “La Sagrada Familia”. This building started construction in 1882, designed by Antonio Gaudi this is known as the masterpiece of Gaudi. It is formed by 18 towers dedicated to the 12 apostles, the four evangelists, Jesus, and virgin Mary due to the long construction of the piece it has seen the works of a wide variety of artists starting with Antonio Gaudi, Lau Feliu, Joan Flotats, Etsuru Sotoo,NUria Tortras , and lots of others. I The building is not complete but when it is finished its highest tower will measure 172.5 meters, the scheduled date for the construction of the last tower is in 2026. 

Although the building encompasses a lot of mathematical thinking, one that is highlighted is in the iconic elevated conical towers. The inside of the towers are projected with a spiral staircase making use of the archimedean spiral which is a geometrical point that moves at a constant speed on a line that only rotates to a fixed point.

La Sagrada Familia also uses a rather new take on the parabolic curve using it in most of the temple but it being discreetly placed meant that it wouldn't be looked upon as much and thanks to that the math behind this curve wasn't found until years later.

Another highlight is the “Vitruvian man” by Leonardo da vinci in 1490. Leonardo da Vinci made this painting when he was an apprentice in Andrea del Verrocchio's workshop, where Da Vinci learned about architectural and technological design. The Vitruvian Man is a study of the human form visually completed by the application of mathematics and it was also Leonardo da Vinci’s we can also see the Da vinci application of pi in the attempt to solve the geometric problem of “squaring a circle.” 

Da Vinci viewed mathematics as a universal constant whose proportions repeat throughout the universe. Vitruvian Man is da Vinci's study of human physiology. It was meant to be a perfectly proportionate rendering of the human form, as determined by the application of geometry and mathematics. 

Vitruvius granted that a compass could be used to draw a perfect circle around a person with outstretched arms and legs, taking the person's navel as the center point. Vitruvius also noticed that for most people arm width and height match almost perfectly, and a person fits perfectly in a square. 

Da vinci used proportion because in the painting from the breasts to the top of the head is a quarter of the height of a man. The distance from the elbow to the tip of the hand is a quarter of the height of a man. The distance from the elbow to the armpit is one-eighth of the height of a man. the length of the hand is one-tenth of the height of a man. The proportions 1:3, 1:4, 1:6, 1:8, and 1:10 are proportions appropriate to man.

The “The Last Supper” painting also done by Leonardo Da Vinci has an interesting take on perspective. This piece was done in 1495 to 1498. The painting is rather large, being 460cm x 880cm or 4.6m by 8.8m . 

“The Last Supper” tells the story of the last supper but specifically the moments after Jesus tells the apostles that he is about to be betrayed by one of them, because the painting having a great handle on human emotions made it one of the most decorated paintings done by Da Vinci. 

The painting, like others from this time period, uses a vanishing point to give the piece a new level of perspective. Leonardo used a linear perspective creating an illusion of depth in a flat surface like the canvas he used. You can see perspective on all of the faces and a depth between the background and the subjects.

On another painting by Leonardo Da Vinci “Mona Lisa” in 1503 A portrait of a seated woman set against an imaginary landscape. We can appreciate the math in the golden ratio. The golden ratio is a natural pattern that repeats itself. That is why it is so fascinating and cherished by many Renaissance artists who wished to resurrect the ideals of Antiquity while also grounding their art on scientific truth. Mona Lisa displays the golden proportion in the face as well as the neck-head ratio, indicating that the ratio between these components is 1.618. This is owing to Da Vinci's interest in both anatomy and mathematics.

Conclusion 

All art uses math and it is connected to math in different ways. Math is what makes art complex and perfect. It is really amusing all the relationships you can find in art with math in all the periods of time. Something truly beautiful is that we can see math in places that are susceptible to the eye, meaning you have to look for it even if it is subtly shown. And we think that that is what makes this relationship beautiful

 

Mathematics is a fundamental part of our lives and it helps us in the areas of science, technology, and engineering. The Modern times of mathematics was characterized by the systematic synthesis of mathematical knowledge. It started from 1450 - present time. They are really complicated and sophisticated, but understanding them really makes a change in our world and lives. The characteristics about Modern Math are Applicability and Effectiveness, Abstraction and Relativity, Simplicity, Logical derivation, Axiomatic Arrangement, Precision, Correctness, and Evolution through Dialect. 

The purpose of this investigation is to find the relation between non euclidean geometry and art. So, what’s non euclidean geometry? 

Non Euclidean Geometry is one in which there's geometry on surfaces which are not flat, this means there's always movement or different shapes, unlike other types of art where there is a flat surface and it looks more steady. Since the surface is normally not flat and more curved like, there are basically no straight lines in it, but these distance minimizing curves known as geodesics will play the role of straight lines in these new geometries. 

There can be very visual art in this type of math , and it can be seen at clear sight when it's really noticeable. An example of this can be the famous artist that is known as MC Escher, who has paintings which are known for having this type of art. This artist's work is a combination between realism and fantasy, and it is known for making “impossible constructions”, which utilize mathematical shapes, perspectives and architectures, which sometimes demonstrate Non Euclidean Geometry. 

According to this context we can analyze some artists of this modern times and learn how they linked mathematics with art. 

Escher “Relativity”, 1953. 

Escher is known for having different geometric perspectives, like sometimes you can see how in his paintings there are three dimensional forms and two dimensional images, sometimes there's even different planes to convert a space two dimensional to three dimensional, he sometimes even plays with the images, which is a way of demonstrating geometrical perspective.

Yayoi Kusama ”The infinity room”, 2012 

Yayoi Kusama is well known for having very popular art that has a lot of geometry perspective, it is basically a room with mirrors, this art is called “the infinity room.”It is known for the explanation of infinity when light reflects on the mirrors, and it is shown as if the light that bounces off is infinite, not only that, but Yayoi Kusama is also known for using patterns, this patterns are known as polka dots, and this patterns also help give the illusion that the lights within them are trailing off into the abyss, even though the room is actually quite small and people believe that it is big because of the illusion, this is a clear example of math shown on art, in this case, the reflection is a big part of this room.

Santiago Calatrava “Oculus”, 2016. 

Santiago Calatrava is mostly known for his architecture but also for his sculptures. He has worked in the Lyon Airport Train Station and has sculptures in The Museum of Modern Art in New York. He likes to use perspective virtuosity. Santiago Calatrava uses folding geometry and you can see it in his work. “The L’Hemisfèric” is a good example as it shows a complete eye only having half of the structure actually built and the other half being only a reflection in the water. 

Here we have the Oculus, which is located in the new World Trade Center, made by Santiago Catrava. It is a really tall structure, and it has a unique design. It is a transportation hub and a shopping mall made out of steel ribs. Something interesting is that it has 2 semicircles on the top that look like wings, and one is larger than the other. In this structure, we can see the use of a hyperbola at the front, and if you take away the semicircles on the top, we can see a semicircle on the bottom.

Remi Coulon-“Non Euclidean Geometries in Virtual Reality” 2018-2020 

Remi Coulon is a talented mathematician who has also made plenty of works of art using non euclidean arts, one of the works of art is called “Non Euclidean Geometries in Virtual Reality”. It is very popular since it uses three important two dimensional geometries which are hyperbolic, euclidean, and spherical. The reason he created this was to develop accurate, real time geometries. An explanation of one of the major math is hyperbolic, this type of geometry in art replaces the parallel postulate with a type of statement that is two non-intersecting lines drawn through the point. Spherical geometry is the study of figures on the surface of a sphere, all these are examples of what Remi uses in his art.

Anish Kapoor, “Cloud Gate”, 2004. 

Sir Anish Kapoor is a British-Indian sculptor. He uses geometry and biomorphic in all of his sculptures. He has done many famous sculptures like Sky Mirrors, Temenos, Cloud Gate, ArcelorMittal Orbit, etc.. 

He became very popular after he made the Cloud Gate in Chicago, which also looks like a bean. It reflects the city’s skyline and the green surroundings. It is made of stainless steel and weighs 80 pounds. In this structure we can see the use of analytic geometry. The kind of sphere below can be seen in the bean, except that it has kind of like a hole below. 

In conclusion, mathematics can be found all over the world, and this is why it is necessary, even in art, in fact, it is possible that all art has something to do with mathematics. Some examples of this can be mainly the perspective, when changing the geometry or the figures in any painting or sculpture, the perspective changes, and gives you a different way of seeing it. Another example of this can be the analysis of symmetry and all of its ways. Finally, there may be patterns, balance, perspective, and much detail in each one. Art and math both need a balance, and when together, they can form amazing things, this is why many artists use both of them, and even try to play with them a bit. 

The world is divided into time periods that are widely known. We divide history into periods to measure events, to find importance and to have a sense of control of what happens. Art and mathematics have changed through history; they developed on their own and taught us many things, but did they ever intersect? We can assume a division from the 16th century to now, which we know as the modern period. Artists consciously or unconsciously connect math with the great passion of art. Which means that many mathematical concepts are used in art during this period of time. We'll talk about two of them, the widely known golden ratio and the vanishing point. 

The golden ratio’s concept is based on the Fibonacci series' use on art and architecture to create “visually appealing proportion” (Jonathan Cleveland, Expertise). What are both of these things, and how could we implement them in art? The golden ratio is the ratio between two numbers which is approximately 1.618 in decimal numbers. The golden ratio has been used by artists to get points and lines of interest in their work, the interest is in function of certain beauty or aesthetic standards. Artists divide their art in lines and rectangles through the golden ratio. The golden ratio was first mentioned by Ecluid around 300 BCE, but it was just recognized as a proportion. Through the years the golden ratio was investigated more thoroughly and praised. The ratio is associated with the Fibonnaci series or sequence. 

The Fibonacci sequence is an infinite succession of natural numbers. This pattern starts in 0 and 1 following a sequence in which you add the past and new number infinitely; they end up in a spiral. (0, 1, 1, 2, 3, 5, 8, 13…) The Fibonacci sequence was discovered in the XIII century by an Italian mathematician called Leonardo de Pisa (also known as Fibonacci). It was first a formula/solution for a rabbit breeding problem which was later related to nature, and is now used in art. With this knowledge, we can now analyze different art pieces more thoroughly.

Golden Ratio - Starry Night by Vincent van Gogh. 

Van Gogh painted the Starry Night while being in an asylum in 1889. He had a wide freedom in the asylum while he showed signs of recovering from the paranoia. He even had his own studio to paint. Sadly he relapsed and his work took a big change. As he did before, he began using darker colors on his paintings, the Starry Night being an example of this. How is math applied here? On the picture on the right, with enough attention, we can visualize the fibonacci spiral. Though we must note that the spiral does not cover the whole painting, just the clouds. The biggest cloud on its own is a spiral. The fibonacci spiral with help from the colors brings the attention towards the right top of the painting. The rule of third is also used, placing the ‘tree’ on the first third of the painting.

Golden ratio - Mona Lisa by Leonardo da Vinci 

The fibonacci spiral can be found in many paintings. One of the most famous examples is The Mona Lisa, painted 500 years ago by Leonardo Da Vinci. Leonardo worked intermittently on the piece over the course of several years. Francis I, a french king, acquired his work after his death and it became a part of the royal collection. It was claimed during the french revolution and then hung up in Napoleon's bedroom. Then it was installed in the Louvre museum during the 19th century. Leonardo Da Vinci knew a lot about math, as he was interested in whether it could help him find ideal proportions. The tip of her nose fits in the middle of the spiral, the bottom of her chin fits the spiral and her head, shoulders, and hands also fit the spiral.

Golden ratio - Cartier Brensson 

This picture was taken in 1932 by Cartier Bresson. He took it on a holiday while he was walking around his town. Cartier is a famous photographer who is well known for several reasons. One of them is using the beauty of the golden ratio. In the given photograph, on the left side, we can appreciate how Cartier applies the golden ratio to add beauty to his pictures. In the ‘center’ of the spiral, we can see the man in the bicycle while the rest is in the ‘background’.

The next concept is the vanishing point. The vanishing point refers to the space in a painting/picture which is supposed to be furthest or closest to the viewer. It's the point in which all receding parallel lines meet. 

It is a single point on the horizon line of an image where parallel lines appear to intersect as they move apart and give the illusion of depth to the image. The architect Filippo Brunelleschi was the one with the most radical breakthrough with the vanishing point. In 1415, Brunelleschi made the first known drawing using the mathematical system of linear perspective to create the illusion of a building receding towards the horizon line. He devised a simple and easy system of creating a vanishing point on a horizon line and drawing diagonal lines towards it. He identified that objects and spaces are smaller when they are further from the eye.

Vanishing point - Raphael and the School of Athens by Raphael Sanzio. An example of this fascinating phenomenon is the masterpiece of ‘Raphael and the School of Athens’ by Raphael Sanzio. This painting was made between 1509 and 1511. It is from the finest examples of the linear perspective, we can see how the mathematical system is used causing an effect of depth in the painting. The vanishing point is between the two people in the middle of the painting, with the horizon line right above their heads. As an interesting fact, the two people in the center are Plato and Aristoteles. You can see how all of the people are below the horizon line, bringing attention to them. The painting has this semicircle which creates a shadow so the attention is not only on the people, but on the part with more light.

Vanishing point- The Last Supper by Leonardo Da Vinci 

Another example of this phenomenon is the painting “The Last Supper”. It was painted somewhere between 1495 and 1498 for a Dominican convent in Milan. In this painting we can assume that the vanishing point was used because we can see Christ at the center of the table. The vanishing point is right in the middle of Christ’s head, and the horizon line is passing through the head of every apostle. It’s also visible how the diagonal lines match the imaginary line that the doors follow.

To conclude, math does have a purpose in our lives, especially in art. We might not notice it, but mathematics can—in certain ways be present everywhere. These forms of art are clear examples of this. We admire the Mona Lisa from the artistic point of view, and now we can admire the mathematical thinking process that went through Da Vinci’s mind while making it. While we don’t know if other artists used math intentionally, we can see different math concepts in their art pieces. This project was a way to open up our minds, and integrate a new way of viewing the world.

 

The relationship between math and art seems unlikely by nature since one is a more critical thinking and logical oriented discipline and the other considered as a form of expression based on aesthetic, feelings and passion. 

But in reality math is everywhere in art! From the measurements in proportions to more in depth concepts like analytical geometry, art and math have been intertwined throughout history. Art in the renaissance period was a very big part of history since different artists created revolutionary art manifestations and expressions, in this essay we will highlight the works of Davinci, Arnolfo di Cambio and more. That is why their influence in the renaissance period was crucial to the development of mathematics in art.

Florence Dome 

This dome was started in 1296 and finished in 1436. 

It was designed by Arnolfo di cambio, then he died and many other persons continued and made some modifications in the designs, but builded and finished the project. Between them was Fillipino Bruneleschi, who made and designed models, he worked with the construction and managed to implement a linear perspective on this and many other buildings that he did. Linear perspectives in the Florence dome : This math technique is used in the dome in the Top part (the dome), the way it works is by making lines that at the end get together in a space and this creates the vanishing point.

Another detail is that the bricks had a specific pattern to form the top part, the pattern was that they are in diagonal, this part is half sphere.

Mona Lisa / Girl with a pearl earring 

In the beginning of the 14th century the renaissance period was a big contributor in the advancements of mathematics and art since more artists started to use mathematical concepts in their work. They incorporated math in their artworks because they realized that it could make their artwork more appealing and pleasing to the eye, this could be referenced with a popular math concept called the Fibonacci sequence or more famously known as the golden ratio. 

This sequence was made and discovered by the mathematician Leonardo Fibonacci or Leonardo of Pisa and he is recognized as one of the greatest mathematicians of the middle ages. The golden ratio is an irrational number that is represented by the greek letter ϕ or τ, it was used by the greeks to represent the ratio of a line segment cut in two pieces of different lengths. The ancient Greeks recognized this sectioning or dividing as important since it was useful in geometry; later this golden ratio would be implemented by many artists of the renaissance period such as Davinci. 

An example of the use of the fibonacci sequence is in the portrait of “The Gioconda” or best known as the Mona Lisa made by Leonardo Davinci, in this painting the fibonacci spiral made with the sequence aligns perfectly with the Gioconda's face.

Nicolaus Copernicus 

Nicolaus Copernicus was a polish mathematician, astronomer and lawyer, he was born in 1473 and passed away 1543. At that time most people believed in the geocentric model of the universe which means that they thought the earth was the center of the universe. Copernicus created a model where the sun was the center and all planets moved in orbit around the sun. He studied the math of everyday objects just like games and knots, he contributed to group theory. He helped with the development of trigonometry necessitated by his sophisticated computations of the orbits of the planets. Years before Copernicus had made his discoveries, astronomer Ptolemy said the earth was the center of the universe and everything revolved around earth, he had made the best assumption of what earth is surrounded by.

Years later Copernicus came up with the theory and started investigating if he was correct. The Copernicus model allows calculation of planetary orbital periods around the sun. It shows the calculations from the distance of the sun, more distant planets have longer periods and move slower.

Perspective - The last supper & egyptians 

In ancient Egypt the paintings that were made were usually made 2 dimensional, this means that the subjects in the paintings were facing at the side and there was no perspective between the background and other elements within the painting. It was until perspective in paintings was a concept introduced at the end of medieval ages by an italian painter called Giotto di Bondone and he started to introduce the concept and practice of perspective in paintings. 

In the renaissance period perspective was used much more often in works of art, one of the most important concepts within perspective is the vanish point. 

The vanish point is a mathematical notion that focuses on the point or points to which the extension of parallel lines meet and create perspective in a drawing; this is mostly applied in geometrical practices. Between these two paintings we can recognize the advancements in the use of mathematical concepts in art.

 

The Codex, Leonardo Da Vinci 

The codex are ancient manuscript books created by Leonardo Davinci, these were used and created as a form of notes so he could theorize about geometrical images and it's relation with drawing of the human anatomy or even weapons. The process to create these ones is that first the animal skin is washed, hair is removed, the skin is attached to a frame, and then depending on the sheets for the final piece is the quantity you will use and take from the animal. The codex are of machines or new inventions that could be used for sketches. 

For example the codex Leicester is a perfect example between the mathematical relation with art concepts, in this codex Da Vinci analyzes the relation between how the eye works and its range of vision by using geometrical figures and lines of perspective on top of eachother.

Conclusion 

In conclusion, mathematics has been a concept used for many centuries for different purposes and people have always been a bit closed minded when it comes to learning about math, however many don't realize how much it is used in our day to day life and in our surroundings. 

The ability to learn how to think critically, be rational and develop problem solving skills are some of the things math can teach you as well as having an open mind to learn new things. When it comes to art with mathematical concepts ( in this case in the renaissance period ) math played a very crucial role in the development of many artistic abilities such as perspective and the analysis of symmetry in portraits which is a concept used and studied in art very frequently today so thanks to mathematics it has given many opportunities for creativity in art and in other disciplines. 

The world is currently changing at a phenomenal pace and the knowledge, tools and application of math has never been more advanced and many more ways to think critically and solve problems are bound to appear and change, so to be able to understand concepts of math and rationalize them in your day to day life even if they are not literal could be a great improvement and create a big impact in society.