Amy has a master's degree in secondary education and has taught math at a public charter high school.
The Golden Ratio
Leonardo da Vinci painted the Mona Lisa, while M.C. Escher made some visually intriguing drawings. Both of these artists used mathematical principles in their work.
Leonardo da Vinci used what is known as the golden ratio, which is approximately 1.618 and is represented by the Greek letter phi. This ratio is achieved when you divide a line in such a way that the longer part divided by the smaller part is equal to the whole line divided by the longer part. Mathematically, the golden ratio can be expressed as:
a / b = (a + b) / a
For example, the ratio of the length of Mona Lisa's head (top to chin) and the width of her forehead form a perfect rectangle, thereby fulfilling the golden ratio. The placement of her arms also fit the golden ratio.
Contemporary photographers also use the golden ratio when composing their images. Instead of positioning a skyline in the middle of an image, they'll position it so that it follows the ratio.
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At first glance, M.C. Escher's paintings appear normal. However, upon a closer look, the elements of the composition are not as they seem and may in fact be mathematically and physically impossible. To achieve this effect, Escher drew tessellations. A tessellation is a group of shapes tiled to form a continuous pattern on an infinite plane. Tessellations aren't always simple grids, they can also be very complex, like this one:
Notice how there are no overlaps and no gap between each individual component of the drawing.
Another art form that relies on math is anamorphic art, a visual illusion that requires the distorting of proportions so that a two-dimensional image looks three-dimensional when observed from a certain perspective. Once a point of reference is chosen, subsequent points are mapped from the three-dimensional object, the flat two-dimensional plane, to produce the illusion. The aim is to emulate the path of light from the human eye to a three-dimensional object. If you achieve it, you'll know because when you stand at a certain spot, the image will look three-dimensional.
A good example of anamorphic art is Andrea Pozzo's painted ceiling in the Church of St. Ignazio. When you stand directly under the ceiling and look up, the ceiling appears to continue upwards. Stand somewhere else, though, and you'll clearly see where the ceiling ends. The anamorphic approach is not used only in painting, but also installations, photographs, sculpture, and the special effects found in film.
Let's review. Some famous artists that use mathematical principles in their art include Leonardo da Vinci and M.C. Escher. The Mona Lisa by da Vinci involves the use of the golden ratio, which is approximately 1.618 and is represented by the Greek letter phi. In his famous drawings, M.C. Escher played with perspective as well as tessellations. A tessellation is a group of shapes tiled to form a continuous pattern on an infinite plane.
Another art form that relies on math is anamorphic art, a visual illusion that requires the distorting of proportions so that a two-dimensional image looks three-dimensional when observed from a certain perspective. Examples include the painted ceiling in the Church of St. Ignazio by Andrea Pozzo, as well as those used in installations, photographs, sculpture, and the special effects found in film.
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How Mathematical Models are Used in Art
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