One might tend to think that math and art are vastly different domains, yet upon closer examination, their realms of inquiry and approaches exhibit striking similarities. Mathematics and the arts both attempt to represent the world, each using its own set of tools. The artist and the scientist are both in search of explanation, both bearers of questions. One expresses their feelings through visual tools, while the other seeks to answer using abstract tools. They converge on a vital plane: intuition, which is an essential element in their respective pursuits. The mathematician requires a significant amount of intuition to advance their research, and the artist employs intuition in their creative process.

The combination of art and mathematics thus does not appear to be incompatible; mathematics possesses its own form of “beauty.” We speak of a “beautiful” mathematical proof, a “beautiful” reasoning, a “beautiful” geometric figure. Throughout history, the relationship between art and mathematics has traversed various stages. In antiquity, the construction of pyramids and temples, the creation of friezes and intricate patterns, as well as mosaics, required the application of mathematics. Perspective, the foundations of which were already explored in antiquity, evolved into a mathematical theory during the Renaissance. Then, for several centuries, painting became highly academic, adhering to stringent rules.

Finally, in the 18th century, the advent of photography shifted the focus of artists away from the “academic” representation of the surrounding world. They sought new avenues of creation by attempting to break free from the constraints of perspective. The 20th century witnessed the emergence of numerous artistic movements, many of which drew inspiration from mathematics. Cubism marked a significant milestone as it aimed to circumvent mimesis (imitation or representation of reality) by embracing the geometrization of forms. Subsequently, geometric art or Constructivist art appeared, fully integrating geometry as its primary subject. Minimalist artists aimed to move away from overly emotional subjective art, turning instead to the representation of streamlined geometric forms. With the advent of fractal theory and, finally, digital art, the redefinition of the relationship between these two disciplines ensued.

We will now attempt to define the role that mathematics plays in art. There are two distinct ways in which mathematics intersects with the realm of art: as a tool aiding in the creation of a work, as seen with the use of perspective, or conversely, when the artist chooses mathematical objects as subjects, as is prevalent in geometric and fractal art. In the former case, mathematics serves as a tool for artists, while in the latter case, mathematics becomes a subject of art. We will observe that, before the 20th century, mathematics was primarily used by artists as a tool.

## The Golden Ratio

This definition: ‘a number equal to (1+√5)/2, approximately 1.618, and corresponding to a proportion considered aesthetic.’ More precisely, the golden ratio is the relationship between two lengths, a and b, such that the ratio of the sum a + b to the larger length a is equal to the ratio of the larger length to the smaller one, in other words,

( + )/ – / . From this relationship, it can be deduced that the golden ratio is the positive solution of the equation

2 x – x – 1, which is equal to

(1+√5)/2

approximately 1.618. It is also said that divides a segment in

‘mean and extreme ratio’.

For centuries, unique aesthetic virtues have been attributed to this number, leading to its moniker as the divine proportion. In 1498, the monk Luca Pacioli penned a work titled “De divina proportione,” published in 1509, in which he detailed the effects of dividing a length according to this divine proportion. The golden ratio has found significant use in architecture and has been “discovered” with varying degrees of rigor in numerous paintings, where artists are believed to have either intentionally employed these golden proportions or used them intuitively, with these proportions being thought to align with a universal aesthetic ideal.

As far back as 2800 BC, the dimensions of the Great Pyramid of Giza spotlighted the significance its architect placed on the golden ratio.

In the 5th century BC (447-432 BC), the Greek sculptor Phidias used the golden ratio to adorn the Parthenon in Athens. This historical trajectory highlights the golden ratio’s enduring presence and influence throughout the annals of art and architecture.

In her work, The Golden Ratio: Radiography of a Myth, Marguerite Neveux examines the studies of Adolf Zeising, a philosophy professor, and Gustav Theodor Fechner, a physics professor, who explored the connections between the golden ratio and aesthetics. Zeising attributed a universal aesthetic quality to the golden ratio, while Fechner nuanced these findings by noting that symmetry is preferred over the golden section as a division ratio, but that the golden rectangle holds significance in terms of dimension ratio.

It turns out that these proportions appear “balanced” and allow for practical and utilitarian formats, such as certain paper sizes. However, if this golden proportion represented universal beauty standards, why do photo papers or drawing paper sheets in A4, A3, A2, etc., formats not adhere to this golden ratio? For instance, the A4 format was chosen to maintain the same aspect ratio when the sheet is halved, and to achieve this result, the ratio of length to width is equal to √2 ≈ 1.414.

Thus, the role of the golden ratio must be realistically assessed. While it indeed boasts several intriguing properties, its demystification is essential. Perhaps this systematic exploration of the golden ratio’s role corresponded to a need to bring order and organization to the world, a way to better comprehend it. It served as a shield against the unconscious intuition that the world is governed by chaos. The golden ratio becomes a guarantor of a certain harmony, allowing for the rationalization of specific aesthetic criteria. This explains its significant role in the history of art. During the 20th century, painters such as Dali and Picasso, as well as architects like Le Corbusier, turned to the golden ratio.

From a mathematical perspective, the golden ratio is profoundly interesting; it appears in the construction of the pentagon, Penrose tilings, the Fibonacci sequence, and more. Its presence across these diverse mathematical contexts underscores its pervasive influence and the interconnectedness of mathematics, art, and the inherent beauty found within both domains.

**Perspective**

An essential mathematical tool for artists is perspective, which has been employed by many artists over numerous centuries. While the concept of perspective was explored in antiquity, its systematic study took root in the 15th century. When Alberti penned his treatise “De Pictura” in 1435, he utilized mathematical notions to formulate his theories. He made it clear that his approach was that of a painter, not a mathematician. Within this treatise, he defined the concepts of point, line, and surface, and then proceeded to establish his theories on perspective.

Paolo Uccello (1397-1475), an Italian primitive artist, is highly illustrative of the emergence of perspective in painting. In some of his works, perspective is entirely absent, as seen in “Saint George and the Dragon” (1458-60).

In contrast, it is prominently featured in “The Miracle of the Desecrated Host” (1465-69). Uccello’s artistic journey reflects the evolving relationship between mathematical concepts and their application in artistic representation. While some of his works might lack perspective, others demonstrate a profound understanding of how mathematical principles can imbue a two-dimensional surface with a sense of depth and realism.

Piero della Francesca, an Italian painter of the Quattrocento (15th-century Italy), was not only renowned for his artistic talents but was also recognized as a geometer and mathematician of his time. He was a master of perspective and Euclidean geometry. One of his notable works, “La Città Ideale” (The Ideal City) created between 1480 and 1490, exemplifies his adept use of perspective

Piero della Francesca’s multifaceted expertise underscores the intimate connection between art and mathematics during the Renaissance. His mastery of perspective allowed him to convey depth and spatial relationships within his works, giving viewers the illusion of peering into a realistic, three-dimensional world. “La Città Ideale” stands as a testament to his ability to transform mathematical principles into visual expressions, creating a harmonious balance between the mathematical precision of geometry and the emotive power of art.

In this painting, Piero della Francesca’s meticulous attention to the vanishing points, the convergence of lines, and the geometric proportions contributes to a sense of grandeur and depth. The perspective employed in “La Città Ideale” not only serves an aesthetic purpose but also reflects his deep understanding of the mathematical underpinnings of spatial representation. The fusion of his mathematical acumen with artistic sensibility is a hallmark of the Renaissance era, when artists sought to elevate their craft through the incorporation of scientific knowledge.

Axonometric perspectives or parallel perspectives are not frequently employed in art, but they find daily use in industrial settings. In cavalier perspective , which is used in spatial geometry at the high school level, one face of an object is depicted without distortion in the plane of the paper. Parallel lines remain parallel, there are no vanishing points, and the size of objects does not diminish as they move farther away. In other words, the ratio of lengths is preserved. While this mode of representation is useful for mathematical demonstrations, it doesn’t accurately depict reality.

Conic perspectives are the most commonly used by artists, as they closely resemble the images formed on the retina of the eye. They involve one or more vanishing points.

The distinction between these types of perspectives reveals the intricate interplay between artistic representation and mathematical principles. While axonometric perspectives serve more practical purposes, conic perspectives align more closely with human visual perception, enabling artists to create compositions that resonate with viewers’ experiences of the world around them. This dynamic relationship between mathematical concepts and artistic techniques highlights the multifaceted nature of the creative process, where precise geometry can give rise to emotionally evocative visual narratives. The choice of perspective, whether axonometric or conic, allows artists to craft their works in ways that best communicate their intended messages and evoke specific emotional responses from their audience.

**Anamorphosis**

When considering the challenge of representation in art, certain geometric tools like anamorphosis, for instance, can illustrate the difficulty of portraying reality by distorting it. In such cases, the viewer is compelled to exert effort in order to discern the intended image from what they see before them.

Anamorphosis involves transforming an object, rendering it unrecognizable, and then restoring the initial figure through an optical or geometric process involving a curved mirror or an examination from a perspective outside the transformation’s plane. This technique presents a fascinating exploration of perception and interpretation, as it demands the viewer’s active participation in deciphering the intended image from its distorted form.

One notable example is found in the work of Hans Holbein the Younger, a German painter. In his painting “The Ambassadors” from 1533, he ingeniously concealed an anamorphically transformed skull. When viewed from a specific angle or reflected upon a curved mirror, the hidden skull reveals itself, inviting viewers to engage in a visual puzzle that blurs the lines between artistic representation and viewer interaction.

In the 20th century, Georges Rousse harnessed anamorphosis to paint geometric shapes within abandoned spaces. When viewed from a specific angle, these shapes transform into recognizable forms such as squares or circles. One of his notable works, “Saint Cloud, 2004” , exemplifies his approach. Notably, Rousse’s art is never directly visible to the public. Instead, his work involves photographing the achieved image, which is then disseminated through books or exhibitions.

Similarly, Felice Varini employs the same technique, but his works are visible within exhibition spaces. For instance, his installation at the Grand Palais in 2013 showcases his ability to merge artistic innovation with spatial engagement.

## Paradoxical images: transformation of the visible world

Impossible or paradoxical images challenge the laws of perspective. They may be the result of chance or an error but may also have been created by the artist who thus questions the place of representation in art.

The representation of space on a two-dimensional medium has given many artists the idea of representing paradoxical objects that would not have a possible material realization in our three-dimensional space. Examples of impossible drawings before the 20th century are very rare,

we can cite a 15th century wall fresco in the Grote Kerk in Breda, William Hogarth’s False Perspective in 1754 with a small text at the bottom of the engraving: “He who executes a picture without any notion of perspective will quickly fall into similar absurdities to those present in the frontispiece. In the 20th century, many artists became interested in this issue of the creation of paradoxical images. We then oscillate between paradoxical images from a geometric point of view and surrealist images as with Dali or Magritte.

It was by chance that the Swedish artist Oscar Reutersvärd (1915-2002) drew one of his first impossible objects in 1934 (fig.18). He writes: “In high school I had no math or biology classes, but I had Latin and philosophy classes. During class, while our Latin teacher was making uplifting remarks about Romans, almost every student was scribbling something on the blank pages of their Latin grammar. I myself tried to draw four, five, six, seven or eight pointed stars as evenly as possible. When one day I surrounded a six-pointed star with cubes, I discovered that these cubes formed a strange constellation. Driven by an inexplicable impulse, I added three more cubes to this configuration to obtain a triangular shape. I was smart enough to recognize that I had thus drawn a paradoxical figure.

An interesting work in this field of paradoxical images is that of the artist M.C. Escher. The mysterious and absurd side of his works is fascinating, as in Relativity, 1953 . There are a number of reproductions of his engravings in school mathematics textbooks. His work is poorly known and little appreciated by the art world, and he is often considered a hard-working artist whose productions seem out of step with contemporary art. His work mainly interested a scientific and especially Anglo-Saxon public.

He is an artist who used mathematics a lot, even if he did not have great knowledge in this field. To create his fantastic universes, he relied on the rigor of the mathematical tool without really knowing it, but he was able to get help from the mathematician H.S.M. Coxeter1 (1907-2003), whose geometric work inspired him a lot and he used the multiple transformations that this tool can apply to the real universe. About Ascent and Descent, 1960 M.C. Escher says:

We imagine ourselves climbing; about 20 cm high, each step is very tiring and where does it lead us? Nowhere ; we do not take a step forward and we do not go up either. And descending, letting ourselves deliciously roll down, is just as impossible for us.2

The theme of infinity often recurs in the works of Escher, we find it in Perpetual Movement lithography from 1961 , where we can see an impossible current of water.

In the depiction of paradoxical images, geometry is employed but subverted, leveraging the challenge of representing the three-dimensional universe on a two-dimensional surface. Strict rules and conventions are essential to uphold perspective and create the illusion of reality. Through the creation of paradoxical images, artists express a desire for transgression. These works signify a questioning of the focal point emphasized by Alberti.

The art of paradoxical images highlights the intricate relationship between mathematical principles and artistic expression. By defying conventional visual norms, artists venture into uncharted territory where perception and geometry coalesce in perplexing and captivating ways. Paradoxical images disrupt the expected interplay of dimensions, inviting viewers to explore the boundaries of their own visual understanding. This defiance of conventional norms showcases the artist’s ability to manipulate geometry to evoke emotions and challenge cognitive frameworks.

As artists engage with paradoxical images, they confront the very nature of representation and perception. By pushing the limits of perspective and geometry, they lead us to question our assumptions about the visual world. In doing so, they perpetuate a lineage of artistic innovation that constantly redefines the relationship between mathematical concepts, artistic techniques, and the human experience.

## What are the links between math and art?

Mathematics and art are two seemingly disparate disciplines, but they are in fact closely linked. Both fields require creativity, problem-solving, and the ability to see patterns. Artists use mathematics to create works of art that are visually appealing and meaningful, while mathematicians use art to illustrate mathematical concepts.

One of the most fundamental links between math and art is the use of **abstraction**. Abstraction is the process of simplifying and removing unnecessary details from something in order to focus on its essential features. Artists use abstraction to create paintings, sculptures, and other works that are more expressive and evocative than they would be if they were completely realistic. Mathematicians also use abstraction to create mathematical models of the world around us.

Another link between math and art is the use of **patterns**. Patterns are repeated sequences of shapes, colors, or other elements. Artists use patterns to create visual interest and harmony in their work. Mathematicians study patterns in order to understand the underlying structure of the world.

The use of **symmetry** is another shared characteristic of math and art. Symmetry is the repetition of elements on either side of a central axis. Artists use symmetry to create balance and order in their work. Mathematicians study symmetry in order to understand the properties of shapes and objects.

Finally, both math and art can be used to **express emotions**. Artists can use their work to express a wide range of emotions, such as joy, sadness, anger, and love. Mathematicians can also use their work to express emotions, such as the beauty of mathematical patterns or the power of mathematical logic.

The links between math and art are complex and varied, but they are undeniable. Both fields are essential parts of human creativity and expression, and they can be mutually beneficial when combined.

Several specific examples of how math and art have been combined in different cultures and time periods:

- The ancient Greeks used mathematics to create the perfect proportions for their sculptures.
- In the Renaissance, artists used mathematics to create the illusion of depth in their paintings.
- In the 20th century, artists such as Piet Mondrian and Wassily Kandinsky used mathematics to create abstract paintings that were based on geometric shapes and patterns.
- In the 21st century, mathematicians and artists are using computer graphics to create new forms of art that are both beautiful and mathematically interesting.

The links between math and art are constantly evolving. As new technologies emerge, artists and mathematicians will continue to find new ways to combine their disciplines to create innovative and inspiring works of art.

## Why do we integrate math and art?

Math and art are two essential parts of human creativity and expression. They have influenced each other throughout history, and they can be combined in many different ways to create beautiful and meaningful works.

**Integrating math and art can help students to:**

**Develop critical thinking and problem-solving skills.**Both math and art require students to think creatively and solve problems in order to achieve their goals.**Improve their spatial reasoning skills.**Math and art both involve understanding the relationships between shapes and objects in space.**Develop their creativity and imagination.**Math and art can help students to think outside the box and express themselves in new and innovative ways.**Increase their appreciation for beauty and design.**Math and art can help students to see the beauty in the world around them and to appreciate the power of design.**Engage their emotions.**Math and art can both be used to express a wide range of emotions, from joy to sadness to anger to love.

**When math and art are integrated, students are able to see the connections between these two disciplines and to appreciate the power of each one.** This can lead to a deeper understanding of both math and art, as well as a greater appreciation for the creative potential of the human mind.

Several specific examples of how math and art can be integrated in the classroom:

**Students can use math to design and create their own works of art.**For example, they could use geometric shapes to create a mosaic or use mathematical principles to create an optical illusion.**Students can use art to represent mathematical concepts.**For example, they could create a painting that shows the relationship between different shapes or a sculpture that represents a mathematical formula.**Students can use math to analyze works of art.**For example, they could measure the proportions of a painting or calculate the number of colors used in a sculpture.**Students can use art to solve mathematical problems.**For example, they could create a diagram to help them solve a geometry problem or use art to represent a mathematical equation.

The possibilities for integrating math and art are endless. By combining these two disciplines, we can help students to develop their creativity, critical thinking skills, and appreciation for beauty.

## How can maths help me improve my art skills?

If you are interested in using mathematics to improve your art skills, there are a few things you can do:

**Study the work of other artists**

Pay attention to how they use math in their work. This can be done by visiting museums, galleries, and online art resources. Look for patterns, shapes, and other mathematical elements in their paintings, drawings, sculptures, and other works of art.

**Take art classes**

This is a great way to learn about the fundamentals of art and to get feedback from your teacher. Art classes can teach you about perspective, proportion, composition, and other essential art concepts. They can also help you develop your drawing, painting, and sculpting skills.

**Read books and articles about the relationship between math and art**

There are many resources available that can teach you more about this topic. These resources can help you understand the mathematical principles that underlie art, and how artists have used math to create their work.

**Practice regularly**

The more you practice, the better you will become at using math in your art. This means setting aside time each day to practice your art skills. You can practice drawing, painting, sculpting, or whatever art form you are interested in.

**Don’t be afraid to experiment**

Try new things and see what works for you. There is no one right way to use math in art. Experiment with different techniques and see what you can create.

**Use math to solve problems in your art**

If you are struggling with a particular problem in your art, try to think of a mathematical solution. For example, if you are trying to create the illusion of perspective, you can use math to calculate the placement of objects in your painting.

**Use math to create patterns and designs**

Mathematics can be used to create beautiful and intricate patterns and designs. For example, you can use Fibonacci numbers to create a spiral pattern, or you can use golden ratio to create a symmetrical design.

**Use math to express your emotions**

Math can be used to express a wide range of emotions, such as joy, sadness, anger, and love. For example, you can use geometric shapes to create a sense of order and balance, or you can use color theory to create a mood or atmosphere.

With practice and dedication, you can use math to create art that is both beautiful and mathematically sound.

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