He is someone for whom mathematics features a lot in his works. These include such works as The Last Supper, the Vitruvian Man and the Adoration of the Magi to name but a few.
It is, however, the book ‘The Da Vinci Code’ by Dan Brown where many will have come across Da Vinci’s ideas about mathematics and art.
Maths and Art
When it comes to mathematics, it has a link with art that goes back a long way. The view is that mathematical ideas such as linear perspective, symmetry, the golden ratio and geometric shapes have a direct influence on art. Such ideas can shape how we view things as they enhanced the works. It was during the Renaissance period and artists like Leonardo Da Vinci that ideas about mathematics in art gained attention.
Leonardo Da Vinci’s knowledge of mathematics did not take off until he was in his early forties. Much of his education came through gaining knowledge by following an ongoing program of self-study. In his forties, he studied Mathematics with the Franciscan friar named Lucca Pacioli. Following this, Da Vinci used what he had learnt by going on to study engineering and anatomy.
The Golden Ratio/Divine Proportion
In art, the golden ratio is a well-known concept in mathematics. It is a special number that has been around since the time of the Egyptians and the Greeks. It is special as it is an irrational number. Known as Phi, the golden ration has a value that is about 1.618. Over the years the golden ratio has been known by many names – the Golden mean, the Golden section and the divine proportion amongst many others. As well as appearing in nature and science, the Golden ration also appears in art. Its connection with art comes at the end of the 1400’s from Leonardo Da Vinci and a Franciscan friar named Lucca Pacioli.
Pacioli was a mathematician who wrote the book ‘De divina proportione’ (Divine Proportion). The text discusses the idea of perspective as well as architecture’s use of mathematics. It also discusses what the ancient Greeks considered to be the five perfect solids. The friar was a friend of many of the artists of the time. One of those was Leonardo Da Vinci. Having trained as a painter, Da Vinci worked with Pacioli to illustrate the book. Leonardo Da Vinci supplied around sixty drawings for the book. Some of these included geometric shapes to illustrate mathematical concepts.
Having studied with Pacioli, Da Vinci made use of the idea of the golden ratio in some of his paintings and drawings. Its use appears in works such as the Last Supper and the Mona Lisa.
When it comes to mathematics and art, Da Vinci used the mathematical principles of linear perspective in his works. The three elements needed for linear perspective are parallel lines, the horizon line, and a vanishing point. He was able to make it seem as if objects were further away from the viewer. Da Vinci was able to create the illusion of depth on a flat surface such as a painting or drawing.
Having studied the principles of linear perspective, Da Vinci put it to good use in some of his earlier works. In his painting of the Annunciation, he uses perspective to bring out various features that enhance the works. Using linear perspective, he includes features such as the corner of a building, a walled garden and a path. Taken together such features contrast enclosure and openness.
Of his other works, the unfinished Adoration of the Magi was meant to show the extent of Da Vinci’s knowledge of figure drawing and perspective. In his work on the picture, he produced a number of studies. This included an in depth study of the perspective. One of those involved the left of the picture where he was planning to illustrate it with a complex background of ruined Classical buildings. To achieve this, he included a thorough study of the perspective.
In his use of perspective, Leonardo Da Vinci is recognised as the first to use anamorphosis. This is a technique whereby an image can only be seen using a curved mirror or from a particular position.
Mathematics and Da Vinci’s Works
Two of Leonardo Da Vinci’s best-known works that employ mathematics are the Last Supper and the Mona Lisa. Each applies mathematical principles of perspective, golden ratio and proportions in its composition. In his painting of ‘The Last Supper,’ he uses the golden ratio to position the different elements in the painting. He uses the ratio to place people in the table to the proportions of the walls and windows in the background. By using mathematical principles, Da Vinci helps bring together the different elements in the painting.
Da Vinci’s painting of the ‘Mona Lisa’ uses the golden ratio in its structure. Some see the beauty of the picture as being its use of the golden ratio. Central elements of the composition use a golden rectangle all over the painting. While the golden rectangle is a rectangle, the difference comes from shapes dimensions. The dimensions reflect the golden ratio. The golden ratio is visible in Mona Lisa’s face. If a rectangle bounds the face and this rectangle is divided by drawing a line across her eyes, it creates another golden rectangle. The result is that the ratio of the length of Mona Lisa’s head to her eyes is also that of the golden ratio. On the rest of the body, it is possible to draw other golden rectangles.
Ideal Human Proportions
There was a belief that man was the perfect creation that which mathematical law could describe. There was a firm belief that mathematical principles lay behind all forms and governed by divine proportion. For many, man was thought to be God’s most perfect creation. Many believed that the use of proportions shapes the human form.
Leonardo Da Vinci was someone who used his knowledge of mathematics to create a number of drawings that show what were thought to be the ideal human proportions. One of his best-known works on human proportions was the ‘Vitruvian Man’. It is a visual image of the perfect human form through the use of mathematics. To people like Da Vinci mathematics, was a universal constant that allowed proportions to be seen everywhere.
Da Vinci’s Vitruvian Man is a pen and ink drawing on paper that shows a man with arms and legs outstretched. The man appears in superimposed positions fitting inside a circle and square. Da Vinci uses the picture to show us the meaning of the different proportions of the human body. By doing so, he is using mathematics to describe what the perfect physical features of a man are. In illustrating the ideal proportions of the human body the text that appears around the drawing is just as important. The text provides the details of the different proportions of the human body as described by the Roman known as Vitruvius on which the Da Vinci bases his drawing.
In addition to the Vitruvian Man, Leonardo Da Vinci also produced a number of drawings that show the proportions of a man’s head. Keen to advance his art and improve his knowledge, Da Vinci created a number of illustrations of people and their proportions.
It was through his involvement with Luca Pacioli on his manuscript ‘De divina proportione’ that he gained a keen interest in geometry. It saw him devote a lot of time to the subject. Da Vinci’s interest in geometric shapes such as the polyhedra are visible in the many sketches scattered throughout his drawings.
For the book ‘De divina proportione’ Da Vinci created sixty illustrations of geometric shapes basing them on Pacioli's writings and models. Some of the illustrations were the first ever of polyhedra where solid edges were visible. By drawing shapes in this way, it was possible to view those edges at the front from those that were at the back. It removed any obvious confusion that might come from a simple line drawing. Having hollow faces on the shape meant it was possible to see through the structure from the front to the back of it. The geometric illustration provided a lot of useful geometric information. In addition to the polyhedra, Da Vinci created similar drawings for other geometric shapes. The illustrations were mostly produced in pairs. One showed the model with solid faces while the other made use of his solid edge technique.