Transcript Slide 1
Mathematics and art
“Mathematics and art emanate from the most pure part of the human soul,only that art is the pure expression of the feelings and mathematics is the clear expression of the pure reason. “
• Mathematics and art have a long historical relationship. The ancient Egyptians and ancient Greeks knew about the golden ratio, regarded as an aesthetically pleasing ratio, and incorporated it into the design of monuments including the Great Pyramid, the Parthenon, the Colosseum. There are many examples of artists who have been inspired by mathematics and studied mathematics as a means of complementing their works.
• Galileo Galilei in his Il Saggiatore wrote that “The universe is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures.”Artists who strive and seek to study nature must therefore first fully understand mathematics. On the other hand, mathematicians have sought to interpret and analyse art through the lens of geometry and rationality.
Ancient times
Pyramids
• • Evidence of mathematical influences in art is present in the Great Pyramids, built by Egyptian Pharaoh Khufu and completed in 2560BC. Pyramidologists since the nineteenth century have noted the presence of the golden ratio in the design of the ancient monuments. They note that the length of the base edges range from 755– 756 feet while the height of the structure is 481.4 feet. Working out the math, the perpendicular bisector of the side of the pyramid comes out to 612 feet. If we divide the slant height of the pyramid by half its base length, we get a ratio of 1.619, less than 1% from the golden ratio. This would also indicate that half the cross-section of the Khufu’s pyramid is in fact a Kepler’s triangle.
• • Debate has broken out between prominent pyramidologists, including Temple Bell, Michael Rice, and John Taylor, over whether the presence of the golden ratio in the pyramids is due to design or chance. Of note, Rice contends that experts of Egyptian architecture have argued that ancient Egyptian architects have long known about the existence of the golden ratio.
The golden section is a line segment divided according to the golden ratio: The total length a + b is to the length of the longer segment a as the length of a is to the length of the shorter segment b.
• Critics of this golden ratio theory note that it is far more likely that the original Egyptian architects modeled the pyramid after the 3-4-5 triangle, rather than the Kepler’s triangle. According to the Rhind Mathematical Papyrus, an ancient papyrus that is the best example of Egyptian math dating back to the Second Intermediate Period of Egypt, the Egyptians certainly knew about and used the 3-4-5 triangle extensively in mathematics and architecture.
Possible base:hypotenuse(b:a) ratios for the Pyramid of Khufu: 1:φ (Kepler’s Triangle), 3:5 (3-4-5 Triangle), and 1:4/π
• While Kepler’s triangle has a face angle of 51°49’, the 3-4-5 triangle has a face angle of 53°8’, very close to the Kepler’s triangle. Another triangle that is close is one whose perimeter is 2π the height such that the base to hypotenuse ratio is 1:4/π. With a face angle of 51°50’, it is also very similar to Kepler’s triangle. While the exact triangle the Egyptians chose to design their pyramids after remains unclear, the fact that the dimensions of pyramids correspond so strongly to a special right triangle suggest a strong mathematical influence in the last standing ancient wonder.
Parthenon
• The Parthenon is a temple dedicated the Greek goddess Athena, built in the 5th century BC on the Athenian Acropolis. It is contended that Phidias, the main Greek sculptor in charge of decorating the Parthenon, also knew about the golden ratio and its aesthetic properties. In fact, the Greek symbol for the Golden Ratio is named Phi (φ) because of Phidias.The golden rectangle, a rectangle whose length to width ratio is the golden ratio and considered the most pleasing to the eye, is almost omnipresent in the façade and floor plans of the Parthenon. The entire façade may be enclosed within a golden rectangle.
• The ratio of the length of a metope and triglyph to the height of the frieze, as well as the height of the columns and stylobate to the entire height of the temple is also the golden ratio. Phidias himself constructed many Parthenon statues that meticulously embody the golden ratio.Phidias is also notable for his contributions to the Athena Parthenos and the Statue of Zeus.
• As with the Pyramids however, more recent historians challenge the purposeful inclusion of the golden ratio in Greek temples, such as the Parthenon, contending that earlier studies have purposefully fitted in measurements of the temple until it conformed to a golden rectangle.
De Divina Proportione
• The first printed illustration of a rhombicuboctahedron, by Leonardo da Vinci, published in “De divina proportione”.
• Leonardo da Vinci drew illustrations of regular solids in De divina proportione while he lived with and took mathematics lessons from Pacioli. Leonardo's drawings are probably the first illustrations of skeletonic solids, which allowed an easy distinction between front and back. Skeletonic solids, such as the rhombicuboctahedron, were one of the first solids drawn to demonstrate perspective by being overlaid on top of each other.
The first printed illustration of a rhombicuboctahedron, by Leonardo da Vinci, published in De divina proportione.
• It is in De Divina Proportione that the golden ratio is defined as the divine proportion. Pacioli also details the use of the golden ratio as the mathematical definition of beauty when applied to the human face.
• “The Ancients, having taken into consideration the rigorous construction of the human body, elaborated all their works, as especially their holy temples, according to these proportions; for they found here the two principal figures without which no project is possible: the perfection of the circle, the principle of all regular bodies, and the equilateral square.” from De Divina Proportione (1509)