Artful Equations

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Joshua Caleb Weibley, Excerpts from Engineering Forms, 2011, ink on paper. Courtesy of the artist.

In High School, I loathed math. I was obviously very interested and invested in art and music, and didn’t realize how artistic discovery relates to principles of mathematics (and vice versa). If I had been introduced to mathematical concepts via visual art, performance and music, perhaps it would have made a significant difference in my enthusiasm and effort in my math classes. I might have ended up challenging myself with numerical equations and problems, if artists like Jennifer Bartlett, Agnes Denes, Piero della Francesca (whose day job was as a mathematician) and Sandro Botticelli were discussed in relation to the content we were learning in math class.

The confluence of art and math should have been a forgone conclusion, because the mathematics we know today has its foundations in art. The practice of synthetic geometry, which was discovered by the Greek mathematician Euclid, in the 4th Century (BCE), is still taught in schools and utilized by graphic artists and architects. Euclidean geometry uses tools like compasses, rulers and protractors to visualize optical dimensions in a physical and tangible manner. In the 15th Century, Filippo Brunelleschi’s concept of linear perspective (inspired by Euclid’s optics) changed both the disciplines of math and art in a monumental fashion. Linear perspective directed the way artists, such as dell Francesca, realized and depicted three-dimensional space within a flat picture plane. The resulting aesthetic explorations with linear perspective led to enormous breakthroughs in the fields of architecture, science and engineering. STEAM learning was a huge component of the Renaissance and its lasting influence, which is why it is so shocking that the arts have largely been left out of the equation in educational curricula until recently (Gunn, 2017).

The cultural impact of linear perspective and other aesthetic mathematical revelations is the subject of Lynn Gamwell’s book, Mathematics + Art: A Cultural History. Gamwell lays out the formulas and shows her work, in order to make the case that art and math are intrinsically linked and have progressed nicely together through time. Gamwell doesn’t solely focus on Western culture; she traces the topic of mathematics within human development back to prehistoric times and our early explorations with counting systems and pattern design. During the modern and contemporary eras, both mathematicians and artists have been concerned with more abstract ways of defining what space is and can be. Non-euclidean geometry gave way to theories regarding the relationship between space and time, which artists of the 19th and 20th centuries sought to visualize in their artwork.
When you look at Jackson Pollock’s drip paintings, it is not a stretch to think about fractal geometry. There is mathematical theory testing to prove the correlation between Pollock’s chaotic splashes of paint and complex fractal patterns which are self-similar over different dimensions. As Jennifer Ouellette (2001) recounts, “the physicist Richard Taylor was on sabbatical in England six years ago when he realized that the same analysis could be applied to Pollock’s work. In the course of pursuing a master’s degree in art history, Taylor visited galleries and pored over books of paintings. At one point in his research, he began to notice that the drips and splotches on Pollock’s canvases seemed to create repeating patterns at different size scales—just like fractals.” In fact, Taylor did the math and revealed that Pollock’s painting Number 14 (1948) has a fractal dimension of 1.45, which is very similar to the fractal dimension of many natural coastlines (Taylor, Micolich and Jonas, 1999). In November 1945, Pollock and Lee Krasner moved to the town of East Hampton on Long Island, so he was definitely attuned to the natural seascapes nearby his home and studio.

The integration of math and aesthetics can also be deciphered within the work of artists such Dorothea Rockburne, Jennifer Bartlett, Agnes Denes, Joshua Caleb Weibley and Nick Naber.

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Dorothea Rockburne, Egyptian Painting: Basalt, 1981, oil, glue, pencil on gessoed linen. Photograph by Nick Naber.

Dorothea Rockburne fulfills her academic interest and passion for math via her creative practice as a studio artist. While studying at the renowned Black Mountain College in the 1950s, she was influenced by a professor named Max Dehn, who was a leading practitioner and scholar in the mathematics of geometry, topology and geometric group theory. She is also intrigued by the scientific and astronomical explorations of Leonardo da Vinci and Piero della Francesca, who she references in her painting Piero’s Sky (1991-92). The painting alludes to the ‘natural’ starry night skies that della Francesca depicts in paintings like The Dream of Constantine (1464), which reinforces his expertise as both an artist and astronomer (see: Valerio, 2011). The sublime and serene character of Renaissance humanism and the elongated forms Mannerism, are evident in many of Rockburne’s contemporary abstract paintings. She connects 15th and 16th century painting to topology, by creating geometric forms that retain their essence under material deformations that include bending, stretching and twisting. This mathematical treatment of her imagery also makes them feel as if they are in motion, akin to the avant-garde choreography of her friends from the Judson Dance Theater. Rockburne personally describes her painterly process, which results in very fluid and accurate geometric compositions, as “visually solving equations” (Hoban, 2015). In a 2013 article Rockburne wrote for the Brooklyn Rail, she elaborates on her studio process and its connection to math:

“During the ’60s and ’70s I struggled to find a new geometry, something beyond the grid and Euclid. Excited by topology and set theory I began to look at transitive geometry, always envisioning concepts in different, possible materials that could be made into art, but which were outside of art materials. Carbon paper seemed a perfect choice. My intuition demanded that previously unseen, invisible structures and proportions be made visible through a transitive process.” – Dorothea Rockburne (Sept. 2013)

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Jennifer Bartlett, House: Dots, Hatches, 1998, enamel on 81 baked enamel plates. Photograph by Adam Zucker.

Jennifer Bartlett makes paintings that are inspired by systems based processes, sets, proportions and ratios. She presents these self-imposed mathematical elements via a highly expressive painterly style that comments on painting’s narrative history and its roots in geometry (see: Artful Arithmetic for further analysis of Bartlett’s math infused art practice).

Agnes Denes is also drawn to mathematical systems, ratios and proportions. She utilizes complex equations and improvises on the work of mathematicians like Pascal and Whitehead and Russell, in order to address social, political and ecological concerns. Her oft-environmentally themed artworks employ geometric structures such as pyramids and sets of flora planted to form patterns inspired by natural rhythmic and evolutionary phenomena (see: Differentiation and Multiple Intelligences for more about Denes’ work).

Joshua Caleb Weibley utilizes synthetic geometry to create very intricately hand rendered drawings that discerningly provide insights into the evolution of technology, game theory and programming language. Many of his drawings parallel the ideological process of Minimalist art, the language of play and the optical mechanics of Op art. Weibley’s critical analysis of technology, presents it within the framework of time and space. His major focus is the coordinated obsolescence of technology, a process which is consistently stimulated by new technological advances and machine based learning. By replicating digital ephemera using an analog technique, Weibley’s art melds the fields of fine art, industrial engineering and computer science.

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Nick Naber, Facility 23, 2019, marker and graphite on watercolor paper. Courtesy of the artist.

Nick Naber’s technically stunning paintings and drawings adopt a personalized mathematical process that highlights line, geometry, and repetitive gesture to make commentary on architecture’s affect on the human psyche. Naber’s geometric structures, which largely resemble archetypal modern and post-modern buildings, impose upon one another to form implied three-dimensional compositions. These structures are drawn to scale and often based on odd numbers, often sets of three. They are like a contemporary version of Giovanni Battista Piranesi’s Carceri d’invenzione or ‘Imaginary Prisons,’ because they similarly form fantastical architectural labyrinths, which are Kafkaesque in nature. Through Euclidean geometry, Naber’s works envelop the viewer with the illusion of feeling trapped, alienated and/or imprisoned within the confines of overarching forms.

The aforementioned artists represent a few examples of how mathematical processes and aesthetic concepts inform one another. With mathematical knowledge and tactile skills, artists continue to probe, explain and expound upon the phenomena of our lived experiences. For the people like myself who struggle with didactic math (i.e. studying baseline formulas), analyzing works of art that combine math, science and technology, can open inquiring minds into developing a better understanding and application of these fields.


References, Notes, Suggested Reading:

Hoban, Pheobe. “Works in Progress,” T Magazine, 15 May, 2015. https://www.nytimes.com/interactive/2015/05/15/t-magazine/17older-female-artists-agnes-dene-herrera-rockbourne-farmanfarmaian.html

Gamwell, Lynn. Mathematics + Art: A Cultural History, New Jersey: Princeton, 2016.

Garner, Mary L. ‘The Merging of Art and Mathematics in Surface Substitution on 36 Plates’, in Kirsten Swenson (ed.), In Focus: Surface Substitution on 36 Plates 1972 by Jennifer Bartlett, Tate Research Publication, 2017, https://www.tate.org.uk/research/publications/in-focus/surface-substitution/art-and-maths, accessed 17 March 2019.

Gunn, Jennifer. “What is STEAM Education?” Room 241, A Blog by Concordia University, 8 Nov. 2017. https://education.cu-portland.edu/blog/leaders-link/importance-of-arts-in-steam-education/

Ouellette, Jennifer. “Pollock’s Fractals,” Discover Magazine, 31 Oct. 2001. https://www.discovermagazine.com/the-sciences/pollocks-fractals

Rockburne, Dorothea. “Points of Change; A Painter’s Journey,” Brooklyn Rail, 4 Sept. 2013. https://brooklynrail.org/2013/09/criticspage/points-of-change-a-painters-journey

Taylor, Richard, Micolich, Adam and Jonas, David. “Fractal analysis of Pollock’s drip paintings,” Nature 399, 422,

Valerio, Vladimiro. “Piero della Francesca’s Sky in The Dream of Constantine,” The Inspiration of Astronomical Phenomena VI. ASP Conference Series, Vol. 441. San Francisco: Astronomical Society of the Pacific, 2011, p.161, http://adsabs.harvard.edu/full/2011ASPC..441..161V, accessed 11 Dec. 2019.

Artful Arithmetic

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Jennifer Bartlett, Air: 24 Hours, 5 P.M., 1991-92, oil on canvas. 84 x 84 inches. Collection of the Metropolitan Museum of Art. Purchase, Lila Acheson Wallace Gift, 1993. © Jennifer Bartlett

When confronted with a mathematical problem, have you ever thought to yourself ‘if only I could see an image (instead of numbers and symbols), this equation might make more sense?’ If so, then you are someone like me, whose method of learning is more inline with visual-spatial abilities than logical-mathematical modalities (see: Gardner, 1983).

That is not to say that if you are more inclined to perceiving things visually/spatially then you can’t also be logical. In fact, these two ways of thinking and reasoning (along with six other multiple intellegences, explained by Gardner, see: ibid) are actually complimentary to logical reasoning and are both bolstered through artistic engagement.

Through employing the theory of multiple intellegences, learners are empowered to combine and/or hone in on problem solving methods by utilizing one or more of eight modalities. The eight modalities are: musical-rhythmic, visual-spatial, verbal-linguistic, logical-mathematical, bodily-kinesthetic, interpersonal, intrapersonal and naturalistic.

The systems-centered artwork of Jennifer Bartlett is a great example of how art can combine multiple intellegences in order to arouse responses from a diverse array of viewers, who each bring different abilities and prior knowledge to the viewing experience.

Bartlett’s paintings are inspired by systems based processes, proportions and ratios. She presents these self-imposed mathematical elements via a highly expressive painterly style. For example, within her series titled Air: 24 Hours, Bartlett created twenty four paintings to represent each hour of the day. She arranged her square canvases by painting a grid-based system that always adds up to the number sixty. While she has implemented the structure of a grid, a comment on a trope within Modernist painting, Bartlett contrasts the logical-mathematical system by overlaying imagery and formal elements that are at once absurd, mysterious and intimate. Bartlett makes logical structures more personal by including symbols and vignettes from her personal life. The scenes, while not overtly telling, represent moments and happenings around Bartlett’s house at a specific hour of the day.

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Jennifer Bartlett, Squaring: 2; 4; 16; 256; 65,536, 1973-74, Enamel over silkscreen grid on 33 baked enamel on steel plates, 77 inches x 9 feet and 8 inches. Collection of the Metropolitan Museum of Art. Purchase, Alex Katz Foundation Gift and Hazen Polsky Foundation Fund, 2018. Photograph by Adam Zucker

Another work of art by Bartlett, which combines mathematical systems with formal aesthetics is the painting Squaring: 2; 4; 16; 256; 65,536 (1973-74). This painting consists of black enamel paint applied over a silkscreen grid on 33 baked enamel on steel plates. The title is a literal description of Bartlett’s self-imposed mathematical formula for cumulatively squaring the number two. The mathematical function was also Bartlett’s artistic process, because for each solution, she composed the precise number of hand-painted dots within the grid to represent the whole numbers: 2, 4, 16, 256 and 65,536. The resulting painting juxtaposes logic with subjectivity. The perspective changes depending on how you view the painting (i.e. from closer up you can clearly see the dots within the grid, but from afar they seemingly amass into an abstract form or blend together into obscurity).

The work of Jennifer Bartlett is an exemplary intermediary between mathematical and aesthetic thinking and doing. Incorporating visual art with mathematical systems is a great way to gain a well-rounded grasp on math formulas, while also expressing a personal element to problem solving, which makes overcoming challenging tasks efficacious and relevant.


References, Notes, Suggested Reading:

Gardner, Howard 1983. Frames of Mind: The Theory of Multiple Intelligences , New York: Basic Books

Gardner, Howard. 1999. Intelligence reframed: Multiple intelligences for the 21st century. New York: Basic Books.

Garner, Mary L. ‘The Merging of Art and Mathematics in Surface Substitution on 36 Plates’, in Kirsten Swenson (ed.), In Focus: Surface Substitution on 36 Plates 1972 by Jennifer Bartlett, Tate Research Publication, 2017, https://www.tate.org.uk/research/publications/in-focus/surface-substitution/art-and-maths, accessed 17 March 2019.

Zucker, Adam. “Differentiation and Multiple Intelligences.” Artfully Learning. 11 Jun. 2018. https://theartsandeducation.wordpress.com/2018/06/11/differentiation-and-multiple-intelligences/