Leonardo da Vinci: Geometric Advances of the Renaissance

Math is everywhere. Not only has it been embedded in our school system, but we use math to intuitively navigate through our world. Before we reach the age of one, we are given blocks of various shapes to match on an outlined board to learn to categorize tangible reality in terms of geometry. Because we are taught these concepts from a young age, people do not think twice about the origins of this methodology. Similarly to our current world perspective, individuals do not often consider the geometric applications to Renaissance paintings that formulated the realism that we admire. Leonardo da Vinci utilized mathematical concepts to perfect his art. In pursuing a new approach to visual artwork and scientific principles, da Vinci drew a connecting line between the two through the incorporation of the Golden Ratio, optical theories, and the dynamics of physical and metaphorical relations.

Figure 1. A copy of Leonardo da Vinci’s notebook

To begin analyzing Leonardo’s contribution to art and science through his study of geometry, we must first observe the fundamental mathematical themes that he recorded. He studied solid geometry and experimented with the relationships of different shapes taking up the same volume. For example, da Vinci sketched relationships between three dimensional shapes and developed equations to compare cubes and pyramids by altering their heights (Figure 1). This appeared similar to a sculpture on paper, and led to his perfection of depth in his art. In addition to solid geometry, he also explored the concept of spirals, which represented the “dynamism” of the physical and natural world (Shape and Volume: Leonardo’s Elegant Geometry - Google Arts & Culture, 2024). From basic investigation, Leonardo was able to lay the foundation for the Golden Ratio, optical illusions of depth perception, and dynamics.

Figure 2. Fibonacci’s sequence geometrically applied to the Mona Lisa

Figure 3. Study of the geometric application of the Fibonacci sequence

In his masterpieces, Leonardo da Vinci is praised for the balance of color, subject placement, and proportions, which make a painting seem like a realistic view of the world. One may ask how he achieved this, and the answer lies in geometry. The Golden Ratio, or “De Divina Proportione,” is described as the ratio of two numbers in a particular sequence and is presented as a spiral that is commonly seen in nature. The specific sequence begins with 1 and takes the sum of 1 and 2. Then, you add the last number in the summation with the result of the previous equation (such as 2 plus 3) (Figure 3). The farther along the sequence you go, the closer you the ratio within the sequence is to approximating the golden value of phi. Therefore, it produces a geometric spiral. The spiral that is physically witnessed in patterns all around us is mathematically described by Euclid as an “extreme and mean ratio” in the book Elements (Duvernoy, 2021). Leonardo was fascinated by spirals and even worked alongside mathematician Luca Pacioli in the early 16th century to develop de Divina Proportione. Through his studies, da Vinci created the first successful three-dimensional representation of the Golden Ratio in his illustrations. He was able to combine aspects of the physical world, including the attributes of nature, and a mathematical concept using visual art. For example, in the famous Mona Lisa, the geometric representation can be seen as the spiral is drawn from the bottom of the painting towards the subject’s face, where the focal point lies (Figure 2). Although the physical spiral is not included, it contributes to the natural balance and appeal of the masterpiece.

Figure 4. The “Vitruvian Man”

Figure 5. Architectural designs by da Vinci

Although Leonardo da Vinci’s architectural endeavors did not reach the same magnitude of admiration as his paintings, he applied similar ideas of balance and symmetry that became a common attribute throughout Renaissance buildings. Leonardo was fascinated by dome structures. Da Vinci understood that to produce the greatest visual and metaphorical effect, the dome must be placed either at the center of a Greek cross or at the center of a structure that has a symmetrical affinity to that of a circle (The Notebooks of Leonardo da Vinci, 2025). In many of his architectural designs for churches, Leonardo incorporates circular structures embedded within hexagonal or octagonal parameters, similar to the Vitruvian Man's design of a circle within a square (Figure 4). He places surrounding domes to formulate a symmetric appearance from the perspective of looking straight down onto the structure. In another example of an architectural design, he creates a portion of the church with large external circles that join the rest of the structure to emulate a natural spiral (Figure 5). This could have been the application of the Golden Ratio in da Vinci’s architectural designs, which would indicate another mathematical concept that travels between various types of visual art.

Figure 6. A geometric perspective of Leonardo da Vinci’s “The Last Supper”

Along with the incorporation of geometric symmetry and the Golden Ratio, optical theories were necessary to project realistic three dimensions on a two-dimensional surface. Geometric discoveries mentioned in Euclid’s Optica discuss the concept of an “angle axiom” and an “angle distance,” which suggests that objects of equal size but varying distances are proportional to the visual angles (Renaissance and Renascences in Western Art, 1972). This advancement in classical optics led to the formation of the vanishing point, which Leonardo da Vinci frequently utilized to project depth in his paintings. For example, in the “Last Supper,” a progression of lines of diminishing magnitudes is outlined in the ceiling itself (Figure 6). Mathematically, the strategy of diminishing lines to illustrate depth can be represented as a recursive formula. Interestingly, so can the Fibonacci sequence, which directly correlates to the Golden Ratio.

For the optical qualities to convey meaning, the time, space, thematic subjects, and narrative must be carefully considered in each painting. He used light and shadow to his advantage to perfect his work; however, as da Vinci described in his journals, mathematically using geometrical constructions leads to an “impressionist” portrayal of the real world. (Wildgen, 2009). From da Vinci’s perspective, his work was imperfect, but he was still able to illustrate the optical illusions that coincided with the themes of the Renaissance. Not only did he introduce the vanishing point in his work, Leonardo included the “linear perspective” to make static figures appear dynamic in a physical and metaphorical sense. The “linear perspective” refers to different shapes being placed at various distances and geometric rules are applied for proportioning. Most figures in Medieval paintings were static, but a painter who wishes to convey movement has to develop a “dynamic” balance. In doing so, the artist can illustrate the motion of the minds of the subjects. This was the incorporation of a new concept during the Renaissance: everything is in linear motion, including the intentions and feelings of individuals. Leonardo describes force as an “incorporeal agency, an invisible power, which using unforeseen external pressure is caused by the movement stored up and diffused within bodies.” In the “Last Supper,” Leonardo organized the geometric composition of the painting to create the interpretation of the flow of a force, along with the impacts of that force. In this case, the force is Jesus Christ predicting a betrayal and the effects of this can be seen through the orientation of the actors. As the force, Christ is the focal point of the painting and the movement is caught in the surrounding characters, similar to a ripple effect in a small body of water. In the geometric perspective, his motion is aligned with the guidelines for the vanishing point, indicating that his impact is infinite (Figure 6). Leonardo was able to convey symbolic meaning through mathematics. Through balance, geometric orientation, and proportion, da Vinci successfully portrayed the movement in nature in a static frame.

Leonardo da Vinci was able to achieve groundbreaking work through mathematical principles. He created close impressions of the natural world and human emotions by using optical illusions to convey depth and movement in a two-dimensional, still plane. In the transition from the Medieval period to the Renaissance, scientific theories introduced a way to connect dimensions of thought together and develop the idea of constant motion. Leonardo’s work is a prime example of this shift in philosophy. With geometric principles, people could see themselves as characters in a complex world that are not merely a passive part of a cycle, but rather an active participant in a linear projection of motion.

Works Cited

Duvernoy, S. (2021). Renaissance Architecture. In: Sriraman, B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-57072-3_10

Leonardo da Vinci’s Geometric Sketches. (2018, January 24). Life through a Mathematician’s Eyes. https://lifethroughamathematicianseyes.wordpress.com/2018/01/24/leonardo-da-vincis-geometric-sketches/

The Notebooks of Leonardo Da Vinci. (2025). Google Books. https://books.google.it/books?hl=en&lr=&id=kOYfDAAAQBAJ&oi=fnd&pg=PR11&dq=da+Vinci+architecture&ots=-oiD7XrWcc&sig=u23_mcxO6jVDTkhbcN1gRRT4yTM&redir_esc=y#v=onepage&q=da%20Vinci%20architecture&f=false

Panofsky, Erwin. Renaissance and Renascences in Western Art. New York: Routledge, 1972.

‌The Geometry of the Last Supper. (2023). Google Books. https://books.google.it/books?hl=en&lr=&id=UsjcEAAAQBAJ&oi=fnd&pg=PA13&dq=da+Vinci+and+geometry&ots=mPYrk13DqD&sig=zJjW-pvJb2AYBvxqLa6Retjcpgo&redir_esc=y#v=onepage&q=da%20Vinci%20and%20geometry&f=false

‌Shape and Volume: Leonardo’s elegant geometry - Google Arts & Culture. (2024). Google Arts & Culture; Google Arts & Culture. https://artsandculture.google.com/story/shape-and-volume-leonardo%E2%80%99s-elegant-geometry-the-british-library/_AVB0C95fzyjLQ?hl=en

Wildgen, W. (2009). Geometry and Dynamics in the Art of Leonardo da Vinci. Cognitive Semiotics, 5(fall2009). https://doi.org/10.1515/cogsem.2009.5.fall2009.66