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Leonardo Da Vinci(1492-1519) was born in Anchiano, Italy. His father was a notary named Ser Piero Da Vinci while his mother was a peasant. Leonardo later moved to Florence where he trained as a painter under the tutorship of Verrochio. As a painter, Leonardo's work was distinguished for quality. Every artwork he did was a masterpiece of a kind. He employed mathematical concepts in his work believing that the two are inseperable. Some of his most famous painting include the Last Supper, Mona Lisa and the Virgin of the Rocks. For the purpose of this paper, Leonardo's Last Supper painting will be used and the mathematical concept of Phi will be analyzed in the mural.
The Last Supper
The last supper is a famous painting that was done by Leonardo Da Vinci between 1495 and 1498. The painting depicts the twelve disciples and Jesus as they took the last supper prior to his crucifixion. In the picture are the twelve disciples divided into group of threes with Jesus at the centre. The painting was done by Leonardo on the wall of a dining hall at a monastery in Milan, Italy, called Santa Maria delle Grazia. Leonardo's painting utilized the mathematical concept of symmetry. In this painting the supper table fills the basic horizontal line and Jesus Christ marks the intersection with a vertical line of symmetry (Mehler & Köhler 429). The diagonals correspond to lines of perspective which produce the illusion of 3-dimensionality.
The last supper painting measured 15 by 29 feet. Leonardo started the artwork by placing a nail in the center of the wall on which the drawing was to be drawn. He then radiated strings outward from the nail so as to place the twelve disciples in a proportional way to Christ. Proportionality here implies that the men seem to be of same one to the other. The second purpose was accomplished by letting the wall, the ceiling and the main table converges to the focus point which is the forehead of Jesus. "An asymmetric ideal is defined y the golden proportion based on the irrational number ½ (square root of 5) = 1.61803399..."(Mehler & Köhler 429). By doing this, it makes Jesus the focus of the whole artwork so that from whichever angle you look the artwork from, it radiates from Jesus' forehead as the source of light. This was purposely done to lay the emphasis of the artwork on him as the main person in the artwork.
Phi which as shown above is approximately 1.618 is also referred to as the "Golden Ratio." This extraordinary number occurs everywhere; in paintings, in architecture, in pyramids among others. Thus:
"For of three magnititudes, if the greatest (AB) is to mean (CB) as the mean (CB) is to the least (AC), they therefore all shall be one (4). Therefore AB is to CB as CB is to AC or 1:1.618...(Mehler & Köhler 429).
Phi can also be obtained through Fibonacci series: 1, 2, 3, 5, 8, 13, 21, 34..... By dividing any number in the series by the number that immediately precedes it, the ratio obtained gets closer to Phi. Phi also be obtained algebraically as follows;
The formula for Phi is
The formula for Phi is
Leonardo Da Vinci's The Last Supper manifests Phi. Jesus being the center of the painting, the ratio of distance from the line above Jesus head, to the floor below and from the line to the top is Phi. Also the rectangle created around Jesus from left to right and from right to left is Phi. In a true symmetry, one side of the object is a mirror reflection of the other side. According to Pipes (198), symmetry can occur in any direction- horizontally, vertically or about a diagonal- "as far as the image is on either side of the axis". Phi is used to balance an artwork. It is actually the balance in a piece of fine art. Balance can be achieved in three ways: using symmetry using approximate symmetry and using asymmetry.
The head of Jesus is at the center of all the force fields. The disciples are divided into four groups each with three memberrs. The sub-centers of these disciples' groups lie in the intersection between the horizontal axis and the symmetrical spiral which finally ends at the head of Jesus. The rectangle of the whole artwork breaks the symmetry of the square. Mathematically "if the sides a, b of the rectangle fulfill the golden proportion, it can be subdivided into smaller rectangles fulfilling the same proportion ad infitum; under this aspect this rectangle is also ideal." The head of Jesus and to be more specific the ear is at the center of all force fields while the sub-centers of the four groups of apostles lie in the intersections between the horizontal axis and the symmetric spiral which ends at Jesus head (Mehler & Köhler 429). The rectangle of the whole fresco breaks the symmetry of the ideal square. The perspective in the artwork generates a subdivision of the background space into three equal zones. In the central zone are to be found Jesus, John, Thomas and James Major.
Leonardo used the concept of Phi in all his artwork. He also demonstrated the concept of phi on the human anatomy. The human head if measured from the top of the head to below the eye and from below the eye to the chin produces the ratio of Phi. The whole human body from the top of the head to the navel and from the navel to the bottom of the feet is also close to Phi (Lucertini, Millán & Nicolo 12). Leonardo was an artist who applied mathematics in his work to the extent that he is in many instances called a mathematician. His application of mathematics and other scientific principles enabled him to venture in engineering where he invented several machines for human use. Leonard was a typical renaissance man; curious and interested in any discipline there was to learn. However it is his paintings; the Last Supper and Mona Lisa that have made a name for him. The two artworks are still popular today and their influence on future artist was immense. Many of future artists did learn from him about balancing and other artistic concepts.