Painting and mathematical anaylsis

Summary: Painting incorporates many mathematical concepts, and mathematics is also used to analyze paintings.

Human beings strive to comprehend their reality in a number of ways, including artistic expression and mathematics. Examples can be found in many cultures, such as the long history of interesting mathematical patterns in Islamic art and in the cave paintings of Paleolithic people. Many artists throughout history also have been mathematicians, such as fifteenth-century painter Piero della Francesca. Modern painter Michael Schultheis also worked as a software engineer. He and Mary Lesser, a painter and printmaker, both explicitly include mathematical elements like numbers, equations, and geometric objects in their work. Mathematical concepts, especially geometry, are embedded throughout the art of painting. Some that are most commonly used for analyzing paintings involve symmetry, perspective, golden ratios and rectangles, and fractals, as well as fundamental geometric forms, shapes, fractals, and abstraction. Mathematicians and scientists also use mathematical methods to determine whether or not unidentified paintings belong to a particular artist.

Symmetry

M.C. Escher, a graphic artist, used transformational geometry to create a variety of works that explored symmetry. His classic work Day and Night, a 1938 woodcut, transforms rectangular fields into flying geese and uses a black and white color scheme to emphasize the transition of a setting from day to night. While many artists explore symmetry and transformational geometry, Escher took it further by exploring and emphasizing mathematical concepts including Convex and Concave, a 1955 lithograph, Two Intersecting Planes, a 1952 woodcut, and Moebius Strip II, a 1963 woodcut. The use of symmetry as the catalyst for transforming the plane is one of the more pleasing aspects of his work. Navajo sand painting also offers many good examples of various types of symmetry. Four-fold symmetry is widely found in Native-American painting and other art forms, and it plays a role in some spiritual and healing ceremonies.

Perspective

Early paintings did not use perspective to show a three-dimensional world on a two-dimensional canvas. Giotto di Bondone, a thirteenth-century painter began to develop depth of field in some of his work; but the first artist credited with a correct representation of linear perspective is Filippo Brunelleschi (1377–1446), who was able to devise a method using a single vanishing point. An architect and sculptor, he shared his method with fellow artist Battista Alberti, who wrote about the mechanics of mathematical perspective in painting. Leonardo da Vinci used perspective in his paintings and explored artificial, natural, and compound perspective in his work. He examined how the viewer’s observation point changed the perspective, and how the perspective could be perceived by changing where the viewer was observing the painting. Notably, while perspective and the illusion of depth were widely used in Western painting from the 1300s onward, it was not universal. Painters from India rarely used this technique; rather, they tended to focus more on patterns and geometric relationships.

Golden Ratio and Golden Rectangles

Consider a rectangle with short side a and a long side that is a+b. A golden rectangle would be where the ratio a/b is equal to

In other words, the large rectangle is proportional to the smaller rectangle formed by side b and side a—this is the golden ratio. Some claim that this proportion influenced many artists and early Greek architecture, while others note the variability of picking points in a painting to have golden rectangles superimposed. It is, however, a way of considering the proportionality of a work.

Fundamental Geometric Forms or Shapes

Geometric forms and shapes are the basis for drawing and painting. For example, Piet Mondrian (1872–1944) explored cubism in his work from black and white lines and blocks of primary colors that divided the plane. Other cubists, such as Pablo Picasso, broke with the Renaissance use of perspective to provide an alternative conception of form. Cubists made it possible for the viewer to see multiple points of view simultaneously. Paul Cézanne ignored perspective in some of his work to construct color on the two-dimensional surface. Pointillism was used by Georges Seurat (1859–1891) to create Sunday Afternoon on the Island of La Grande Jatte. In pointillism, a series of small, distinct points of color are used to create a painting that relies on the viewer’s eye to blend them into a cohesive form. The brain uses the dots to create a solid space. The primary colors are used to create secondary colors for shading and create the impression of a rich palate of secondary colors.

Art deco is characterized by the use of strong geometric forms that are symmetrical. This style of painting was popular in the 1920s and 1930s.

Abstraction and Fractals

Abstraction is an important tenet of mathematics. In mathematical abstraction, the underlying essence of a mathematical concept is removed from dependence on any specific, real-world object and generalized so that it has wider applications. In abstract expressionism, the artist is expressing purely through color and form, with no explicit representation intended. However, that does not mean that abstract art is entirely unstructured. Fractals are one tool used to quantitatively analyze and explain what makes some paintings more pleasing than others. The argument is that, even in an apparently random abstract work, there is an underlying logic or structure that the human brain recognizes as fractal patterns and that it inherently prefers over other works that do not have these patterns. This preference is perhaps because such works are more reflective of the geometry of naturally occurring spaces. For example, physicists Richard Taylor, Adam Micolich, and David Jonas analyzed Jackson Pollock’s paintings and found two different fractal dimensions in his work that are mathematically and structurally similar to naturally occurring phenomena, like snow-covered vegetation and forest canopies. In addition to the application of fractals, mathematical concepts like open and closed sets have been used to compare and contrast the work of abstract expressionist artists like Pollock and Wassily Kandinsky to artists like Joseph Turner and Vincent van Gogh, whose works are among those credited with inspiring the expressionist movement.

Mathematical Analysis to Determine Authenticity

Sometimes, the painter of a particular artwork is unknown or disputed, which affects the study of art and the monetary valuation of paintings. Hany Farid and his team created a computer program that uses wavelets to analyze digital images of paintings and map the stroke patterns—some too small to be seen with the naked eye—that characterize an artist’s unique style. In one case, known drawings by Pieter Bruegel the Elder were compared to five drawings originally attributed to him. The analysis determined that the five drawings were different from the original eight and also from each other, suggesting multiple creators. Chinese ink paintings are an example in which brush strokes are critical to identification, since they do not have colors or tones to distinguish style. One successful method, tested on the work of some of China’s most renowned artists, used a mixture of stochastic models. In another case, fractal geometry was used to question the authenticity of some newly discovered Pollock works, based on his earlier patterns. Radioactive scans and X-ray analysis help to authenticate works by well-known and highly valued masters, such as Johannes Vermeer.

Additional Parallels in Painting and Mathematics

There are many natural parallels in the work of painters and mathematicians. In the same way that painters of different traditions and schools may represent the same scene in drastically different ways, mathematicians may approach the same problem from a variety of disciplines or perspectives. There are also varying degrees of connection to reality in both mathematics and painting. Applied mathematicians and realist painters may be primarily concerned with detailed and faithful representations of the real world in their work, while abstract painters and theoretical mathematicians often work in ways that are logically coherent and consistent, but that do not immediately or obviously connect to the real world. As with art, there is also subjective appreciation of the beauty of mathematics and arguments over what is or is not mathematically valid. Artist Michael Schultheis reported that he was often inspired by mathematical and scientific writing on whiteboards from his days as an engineer, and said, “I constantly revise equations with the Japanese calligraphy brush, rubbing out an area and thus creating a window into the equations. I draw and re-draw new ideas. All of these ideas are analytical. But they also live in the realm of beauty.”

Bibliography

Field, J. V. Piero della Francesca: A Mathematician’s Art. New Haven, CT: Yale University Press, 2005.

Jensen, Henrik. “Mathematics and Painting.” Interdisciplinary Science Reviews 27, no. 1 (2002).

Robbin, Tony. Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought. New Haven, CT: Yale University Press, 2006.

Taft, W. Stanley, and James Mayer. The Science of Paintings. New York: Springer, 2000.

Talasek, J. D. “Curator’s Essay—Blending the Languages of Mathematics and Painting: The Work of Michael Schultheis.” National Academy of Sciences. http://www.michaelschultheis.com/publications/talasek‗essay.pdf.