Beauty is in the eye of the beholder, but did you know it can be created mathematically, too? Learn how math and science influence perception, and how to use them to create effective visual art—even if you’re starting from level 1.
There is a widespread belief that innate talent is required to succeed in any endeavor. But should talent be considered the main condition for achieving mastery? Let’s explore this question in the context of computer graphics for the computer game industry.
To begin with, it’s important to note that denying the existence of talent or downplaying its importance would be a mistake. There are certainly people who can just feel the nuances of the drawing process at their fingertips and masterfully execute the most complex techniques, guided by their intuition. Having talent is a winning starting position. But with enough diligence, determination, and practice, you can achieve the same results using mathematical principles in drawing and painting.
Let’s split approaches to drawing into two categories. We’ll call the first sensory and the second mathematical.
The Sensory Approach
The sensory approach is characterized by its emotional component. Importance is placed on motivation (the inner need and willingness to draw) and inspiration (a state of creative excitement), which encourage us to create.
This approach can be useful at the first stage of conceptualizing the work and searching for references, when the artist is inspired to discover new meanings and images for further work.
In this case, the process itself will often be subordinated to the emotions and mood of the artist. For this reason, light exposure, color choice, form, composition, and other aspects become more subjective, reflecting the inner state of the creator.
The Mathematical Approach
The mathematical approach, in turn, is expressed through more consistent, rational methods. The principles on which the mathematical approach is based are the fundamental rules of fine art.
An artist following the mathematical approach does not rely on intuition to solve the problems they face. Instead, they use existing methods that allow them to handle the complexities of the project.
The more consistently your workflow is organized, and the less sensory work involved, the more stable the result will be, and the less likely emotional burnout will become!
Of course, the sensory component plays an important role in the creative process, especially when it helps to add depth and integrity to the work. However, in an industry environment, well-designed and well-defined processes are key. The more detailed they are, the easier it is to ensure consistent quality, minimizing the impact of fickle inspiration or negative emotions caused by, for example, bad weather outside.
The Role of Science and Math in the Arts
It’s no secret that the human brain is essentially a complex computer that processes incoming information into accumulated experience. Billions of neural connections are formed every moment in order to comprehend our surroundings. Each new neural connection, as well as each one lost, triggers complex chemical processes in the body that seeks to make sense of what is happening. Our emotions, perceptions, and ability to experience reality are shaped by amino acids and hormones produced by the brain. Science is constantly developing, and the deeper understanding we have of the brain today allows us to more precisely define what criteria shape beauty. Of course, the statement “beauty is in the eye of the beholder” is still true, since the perception of beauty is subjective and depends on each person’s unique way of processing information—what seems beautiful to one person may seem ugly to another. However, if a wide enough sample is taken, universal criteria can be identified that determine the attractiveness of an object for most people. This means that humanity is capable of using this knowledge to create a true masterpiece that will be understood and recognized as such by the majority.
Since this time has not yet come, let’s turn to the principles and rules underlying the drawing process that we already know:
1. The Golden Ratio.
It is a mathematical proportion that expresses the harmonious relationship of two parts of a whole. It occurs when the ratio of the parts is equal to the ratio of the larger one to the whole. In numerical terms, it approximates 1:1.6181:1.6181:1.6181:1.618, denoted by the Greek letter ϕ (phi).
- In Ancient Egypt and Babylon, proportions close to the golden ratio were used in the design of pyramids and other architectural structures.
- In Ancient Greece, the golden ratio is mentioned in the works of Euclid (“Beginnings,” 300 BCE) and was used in the architecture of the Parthenon.
- In the mathematics of the medieval Islamic world and in Gothic architecture, the golden ratio was used to create aesthetically perfect structures.
- During the Renaissance, Leonardo da Vinci popularized the concept of the golden ratio, calling it the “divine proportion.” He used it in his paintings, including The Last Supper and The Vitruvian Man.
- German mathematician Martin Ohm coined the term “golden ratio” in the 19th century, and in the 20th century, it was actively studied in biology, art, design, and music theory.
- Nowadays, the golden ratio is used to create harmonious compositions in, for example, painting and photography.
Pic. 1. Da Vinci’s Anatomical Man and Mona Lisa. Pic. 2. The Parthenon Built in 447–438 BCE
2. The Rule of Thirds.
It is a simplified version of the golden ratio, providing a way to quickly construct a harmonious composition. It is one of the basic principles of composition, widely used in fine arts, photography, cinematography, and design. The rule involves the visual division of a frame into nine equal parts using two horizontal and two vertical lines. The intersection points of these lines are called power points or crash points and are considered optimal areas to place key compositional elements.
Principles similar to the rule of thirds began to be used as early as the Renaissance to create balanced compositions in painting.
In 1797, English artist and theorist Joshua Reynolds described a similar principle in a treatise on art, mentioning the division of space to achieve harmony.
This rule helps artists balance their composition by drawing attention to the main parts of the painting.
Calm Seas, Ivan Konstantinovich Aivazovsky, 18853. Perspective.
It is a method of depicting three-dimensional objects and space on a plane in such a way as to convey their volume, depth, and relative position.
Elements of perspective can be seen in the art of Ancient Egypt, Greece, and Rome, where simplified ways of conveying depth were used, such as making distant objects smaller in size.
In the Gothic era, conventional ways of representing space were used, but without strict rules of perspective.
In the 15th century, Italian architect and art theorist Filippo Brunelleschi developed linear perspective. Leon Battista Alberti systematized it in his treatise “On Painting” (1435). This was revolutionary in the art world.
Artists refined the technique, adding complex multipoint perspectives and experimenting with distortion.
In modern times, alternative approaches have emerged, such as isometric perspective in technical drawings and experimental perspectives in cubism.
Types of perspective:
- Linear perspective: Uses vanishing lines that converge at one or more points on the horizon. Main forms:
○ Single-point (one vanishing point, such as a view of a road).
○ Two-point (two vanishing points, such as the corner of a building).
○ Three-point (a third point is added for height, creating a more dramatic effect). - Aerial perspective: Uses changes in color and contrast to convey depth—distant objects appear more blurred and cold.
- Reverse perspective: Used in iconography when lines diverge to the sides, creating a sense of supernatural space.
- Isometric perspective: All lines remain parallel, used in technical drawing and graphic design.
Relationship to math: Building perspective requires precise calculations of angles, proportions, and vanishing points.
4. Symmetry and Asymmetry.
Symmetry is the balance and harmony created by the repetition of elements or the uniform distribution of form around an axis or center. It is a fundamental principle of aesthetics and is widely used in art, architecture, and design, as well as found in nature.
Since ancient times, symmetry has been considered the embodiment of order and beauty. It was used in architecture (the Egyptian pyramids, Greek temples) and decorative art.
Symmetry became the basis for classical architecture and painting, emphasizing proportion and balance.
In geometry and science, symmetry was described mathematically, which influenced design and engineering.
Asymmetry is a lack of balance, where the elements of a composition are unevenly distributed, yet create dynamics and interest. It highlights the contrast and attracts attention, and is often used to create emphasis and uniqueness.
Artists and architects have long used asymmetry to create expressive and innovative forms.
Asymmetry is popular in graphic design, fashion, and interior design as a way to create dynamic and unusual visuals.
Symmetry is described by geometry, while asymmetry is based on the distribution of the weight of objects.
The Traveling Companions, Augustus Leopold Egg, 1862
The Dream, Pablo Picasso, 1932
Asymmetry5. Meter and Rhythm.
In the context of visual art, meter can be seen as the organization of compositional elements into regular, repetitive structures or groups that create order and consistency in a work of art. It helps the viewer navigate the space of the image, giving it predictability and regularity.
Rhythm in visual art is associated with dynamics and movement. It is a principle in which the elements of a composition are arranged in such a way as to create a sense of the flow of time, movement, and change. Rhythm can be either linear (in direction) or created through repetitive shapes, colors, light accents, or textures.
Sportsmen, Kazimir Malevich, 1930-1931
Rhythm of Labor, Nikolai Zagrekov, 19276. Color and Tone.
They are two key elements in the visual arts that are closely related and play an important role in how art is perceived. Both encompass aspects of light perception and influence the composition and emotional expressiveness of a piece.
Color is a visual perception that occurs when light interacts with surfaces. It depends on the spectrum of light that is reflected from the object. Color can be described using three basic characteristics: hue, saturation, and brightness.
Color is often used to convey emotions. For example, warm colors (red, orange, yellow) can evoke feelings of warmth and excitement, while cool colors (blue, green, purple) can evoke feelings of calmness or sadness.
In composition, colors are used to highlight accents, create contrast or harmony, and control the viewer’s attention. Bright colors, for instance, can attract attention, while neutral colors can serve as a background.
Tone is a characteristic of color that defines how light or dark the color is. Tone is closely related to brightness and is used to create depth and volume in an image.
In painting and drawing, the use of different tones allows the artist to create a three-dimensional effect on a flat surface. For example, a light tone on one side of an object with a darker tone on the other side will indicate the light source and shadow, creating a sense of volume.
The contrast between light and dark tones helps to emphasize certain parts of the composition for dynamism and focus.
Changes in tone associated with light sources and shadows help to create realistic images. This allows the artist to capture not only the shape of objects, but also their texture, light reflection, and degree of illumination.
Color and tone are always intertwined because tone is a property of color. For example, two colors may have the same hue but differ in tone: one may be light, while the other is dark.
- Color + tone = depth and volume: By changing the tone, you can create a realistic image even in a monochromatic hue. Changing tone helps you create contrast, separate objects from the background, and control the viewer’s attention.
- Color tints: When a color changes in tone but retains its hue, it’s called a shade (e.g., dark blue or dark red). The use of tones in shades helps convey more complex visual effects.
- Harmony and contrast: Compositions often use harmonious combinations of colors with different tones. The contrast between light and dark tones can emphasize form, texture, and highlights, while harmony of tones helps create softer and more balanced compositions.
Rouen Cathedral: The Portal (Sunlight), Claude Oscar Monet, 1894If you’re looking to start a career in the industry as a digital artist, don’t be afraid to take the first step. Mastering drawing is easier than it might seem. Art is accessible to almost everyone. It’s not the privilege of the gifted, but the result of persistence, patience, and practice. Everyone can succeed, regardless of starting level or position. It’s important to remember that true success depends not on innate ability, but on a willingness to learn and grow. Mathematical principles serve as an indispensable tool for artists, helping them to create pieces that harmoniously combine structure, emotional expression, and depth. So don’t be afraid to try, learn, and create!
