The Vitruvian Man, a famous sketch by Leonardo da Vinci, is one of the most iconic images from the Renaissance era. But what makes this simple pen and ink drawing so significant? Let’s explore this question through the lens of mathematics and philosophy.
To understand the Vitruvian Man, we first need to delve into a classic mathematical problem: “squaring the circle.” This involves creating a square with the same area as a given circle. To find the area of a circle, we multiply pi by the radius squared. For a square, we multiply the base by itself. However, due to the nature of pi, creating a square with the exact area of a circle is mathematically impossible. This ancient problem was revisited during the Renaissance, sparking renewed interest and exploration.
Leonardo’s drawing was inspired by Vitruvius, a Roman architect who believed that the human body could be perfectly inscribed within both a circle and a square. According to Vitruvius, the navel is the center of the human body. By placing a compass point on the navel, one can draw a circle around the body. Additionally, Vitruvius noted that a person’s arm span is nearly equal to their height, allowing the body to fit neatly inside a square.
Leonardo da Vinci used Vitruvius’s ideas to metaphorically address the problem of squaring the circle. He placed a human figure at the center of both shapes, symbolizing the connection between mankind and the universe. But Leonardo’s thinking went beyond Vitruvius. He was influenced by Neoplatonism, a philosophical movement in Italy that revisited ancient ideas from Plato and Aristotle.
Neoplatonism introduced the concept of “The Great Chain of Being,” a hierarchy that starts with God and descends through angels, celestial bodies, and all forms of life, ending with demons. Initially, humans were thought to occupy the center of this chain, bridging the mortal and the divine. However, a Neoplatonist philosopher named Pico Della Mirandola proposed a different view. He believed that humans have the unique ability to choose their place in the universe.
Pico argued that God created humans with the capacity to understand the universe’s complexity and beauty. He suggested that humans could ascend the chain to act like gods or descend to behave like animals. This flexibility is reflected in Leonardo’s sketch, where the human figure can adapt to fill both the circle and the square. It symbolizes humanity’s potential to exist within all elements of the universe, both geometrically and philosophically.
Leonardo’s Vitruvian Man masterfully combines mathematics, philosophy, religion, architecture, and art. It encapsulates the spirit of the Renaissance, a period marked by the fusion of diverse fields of knowledge. This is why the Vitruvian Man remains an enduring symbol of that transformative era.
Engage in a hands-on activity by attempting to “square the circle” using geometric tools. Although mathematically impossible, this exercise will help you understand the historical challenge and appreciate the complexity of pi. Reflect on why this problem fascinated Renaissance thinkers.
Conduct an experiment by measuring your own body proportions to see how closely they align with Vitruvius’s observations. Measure your height and arm span, and try to inscribe yourself within a circle and a square. Discuss how these proportions relate to the concept of ideal human form.
Participate in a debate on Pico Della Mirandola’s idea of human potential within the Great Chain of Being. Consider the implications of humans having the ability to choose their place in the universe. Discuss how this idea is reflected in the Vitruvian Man and its relevance today.
Create your own artwork that combines mathematical principles with artistic expression, inspired by the Vitruvian Man. Use geometric shapes to represent philosophical concepts, and present your work to the class, explaining the symbolism behind your design.
Work in groups to develop a project that explores the interdisciplinary nature of the Renaissance. Choose a topic that integrates mathematics, philosophy, art, and science, and present your findings. Highlight how these fields influenced each other during the Renaissance and their impact on modern thought.
This image of the Vitruvian Man, taken from Leonardo’s sketches, has become one of the most recognizable symbols of the Renaissance. But why? It’s a simple pen and ink drawing, right? Wrong! Let’s start to answer this question with a math problem.
I know how to calculate the area of a circle. I take the value for pi and multiply it by the radius squared. I also know how to calculate the area of a square by multiplying the base by itself. But how can I take the area of a circle and create a square with an equal area? This problem, often called “squaring a circle,” was first proposed in the ancient world. Like many ideas from that time, it was given new life during the Renaissance. As it turns out, this problem is impossible to solve because of the nature of pi, but that’s another story.
Leonardo’s sketch, influenced by the writings of the Roman architect Vitruvius, places a man firmly at the center of a circle and a square. Vitruvius claimed that the navel is the center of the human body and that if one takes a compass and places the fixed point on the navel, a circle can be drawn perfectly around the body. Additionally, Vitruvius recognized that arm span and height have a nearly perfect correspondence in the human body, thus placing the body perfectly inside a square as well.
Leonardo used the ideas of Vitruvius to solve the problem of squaring a circle metaphorically, using mankind as the area for both shapes. Leonardo wasn’t just thinking about Vitruvius, though. There was an intellectual movement in Italy at the time called Neoplatonism. This movement took an old concept from the 4th century developed by Plato and Aristotle, called “The Great Chain of Being.” This belief holds that the universe has a hierarchy resembling a chain, starting at the top with God, then traveling down through angels, planets, stars, and all life forms before ending with demons and devils.
Early in this philosophical movement, it was thought that mankind’s place in this chain was exactly in the center. Because humans have a mortal body accompanied by an immortal soul, we divide the universe nicely in half. Around the time Leonardo sketched the Vitruvian Man, however, a Neoplatonist named Pico Della Mirandola had a different idea. He proposed that humans have a unique ability to take any position they want. Pico claimed that God desired a being capable of comprehending the beautiful and complicated universe He had created. This led to the creation of mankind, which he placed at the center of the universe with the ability to take whatever form he pleases.
According to Pico, mankind could crawl down the chain and behave like an animal or crawl up the chain and behave like a god; it’s our choice. Looking back at the sketch, we can see that by changing the position of the man, he can fill the irreconcilable areas of a circle and a square. If geometry is the language the universe is written in, then this sketch seems to say we can exist within all its elements. Mankind can fill whatever shape he pleases, geometrically and philosophically as well.
In this one sketch, Leonardo was able to combine the mathematics, religion, philosophy, architecture, and artistic skill of his age. No wonder it has become such an icon for the entire time period.
Mathematics – The abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics). – In the study of mathematics, university students often explore complex theories that form the foundation of modern technology.
Art – The expression or application of human creative skill and imagination, typically in a visual form such as painting or sculpture, producing works to be appreciated primarily for their beauty or emotional power. – The art of the Renaissance period is renowned for its emphasis on realism and human emotion, which can be studied in university art history courses.
Philosophy – The study of the fundamental nature of knowledge, reality, and existence, especially when considered as an academic discipline. – Philosophy often intersects with mathematics in exploring the logical foundations of mathematical concepts.
Circle – A round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center). – In geometry, the circle is a fundamental shape that is used to explore concepts such as pi and circumference.
Square – A plane figure with four equal straight sides and four right angles. – The square is a basic geometric shape that is often used in art to create symmetry and balance.
Vitruvius – An ancient Roman architect and engineer, known for his multi-volume work entitled “De Architectura,” which has influenced architectural theory and practice. – Vitruvius’ principles of symmetry and proportion have been integral to the study of architecture and art history.
Renaissance – A period in European history marking the transition from the Middle Ages to modernity, characterized by an interest in classical scholarship and values. – The Renaissance was a pivotal era for the development of both art and mathematics, as scholars sought to harmonize these disciplines.
Neoplatonism – A philosophical system that originated in the 3rd century AD, based on the teachings of Plato and later philosophers, emphasizing the spiritual and metaphysical aspects of reality. – Neoplatonism influenced Renaissance artists who sought to express the ideal forms and divine beauty in their works.
Humanity – The quality of being humane; benevolence, or the human race collectively. – In art, the depiction of humanity often involves exploring the human condition and emotional experiences.
Potential – Having or showing the capacity to develop into something in the future. – The potential of mathematical theories to solve real-world problems is a key focus in advanced university courses.
