Week 2: Math and Art

I've always considered mathematics and art to be, to some extent, two sides of the same coin. Mathematics is not absolutely art, nor is art absolutely mathematics, as the more abstract and theoretical one wishes to delve into either subject, the more insular they will find it to be. Despite how much I value widespread appreciation of math through meaningful application, I must confess that I despise "pop math". What I consider to be pop math I will not strictly define at the risk of sounding elitist, but I believe that the image it evokes within the reader will suffice to convey my distaste.

The idea of a relationship between maths and art is hardly new. As Vesna expressed in her video, an example of mathematic concepts employed in art is through the use of perspective. She discusses the history of the the development of optical references to create a sense of perspective and depth in paintings by using an imaginary vanishing point in the distance (Vesna). A piece that exemplifies this is Parmigianino's Self-portrait in a Convex Mirror, an aptly titled Late Renaissance piece in which an artist paints a self-portrait using a convex mirror, causing his hand in the foreground to be elongated, the objects in the background to be compressed, and the perspective of the piece as a whole to be distorted. 

Self-portrait in a Convex Mirror
https://en.wikipedia.org/wiki/Self-portrait_in_a_Convex_Mirror#/media/File:Parmigianino_Selfportrait.jpg

Even more representative of the relationship between mathematics and art is the medium of sculpture, as the introduction of a third dimension tremendously increases the depth of this interplay. Some of my favorite examples come in the form of bodies of constant width, or as coined by Euler, orbiforms. The most obvious example of this polygon is a circle, but Reuleaux polygons, shapes formed by the intersection of three circles, are much more interesting examples of shape that meet this criteria (Martini). I find this object particularly fascinating, as it not only fulfills a mathematical property in an unconventional way, but because it also defies expectations in practice and function.

A 2-D Reuleaux Triangle
https://en.wikipedia.org/wiki/Curve_of_constant_width#/media/File:Reuleaux_supporting_lines.svg.

But, in the pursuit of attempting to see art through a mathematical lens, I believe that many people overlook the inherent beauty and elegance within mathematics that makes it an art form in and of itself. There exists an innate beauty within many mathematical concepts, from the ability of fractals to be infinitely self-similar, to the Ricci flow's tendency to transform geometry to more uniform shapes (Boehm). Some less monumental discoveries, such as the fraction 987654321/123456789 being an estimator for the integer 8, or the Gelfond-Schneider theorem, despite having no practical applications in the creation of art, could themselves be considered "art". 

For the sake of argument, we can also present a more formal argument as to how maths and art are intuitively, fundamentally related. What society considers to be "art", though admittedly largely subjective, can be reduced to criteria which have basis in anthropology, psychology, and sociology. All three of these studies are fundamentally dictated by biological processes, which themselves are dictated by underlying chemical attributes. Proceeding even further, we understand chemistry as a form of applied physics, which is itself applied maths. This elaboration obviously holds little water should one hold it up to real logical methods, but I find it a valuable tool to demonstrate how not just maths and art, but all disciplines are connected. 

xkcd webcomic about different fields
 https://xkcd.com/435/

Ultimately, despite my dislike for "pop math", I fully support artists going the extra mile to incorporate scientific, mathematic, and geometric principles in their art, as it not only provides real-world applications of otherwise abstract formulae, but exposes viewers to the inherent beauty of maths that they would otherwise have never known. As such, I firmly believe that people will begin to appreciate the interplay between all disciplines in larger numbers. 

Oh, and as an afterthought, I also hate the notion that the Golden Ratio is the end all be all for design, both artificial and natural. Very few things in nature actually adhere to the Golden Ratio, and though it has valid underlying design and mathematic principles, I'm confident that the public perception of what it actually is is completely overblown and was perpetuated by quack scientists and marketing teams to con people into believing that products are aesthetically superior just because they approximate a 1:6 ratio. 

References

Böhm, Christoph, and Burkhard Wilking. “Manifolds with Positive Curvature Operators Are Space Forms.” Annals of Mathematics, vol. 167, no. 3, 2008, pp. 1079–97. Crossref, doi:10.4007/annals.2008.167.1079.

Gelfond, A., and Leo Boron. Transcendental and Algebraic Numbers (Dover Books on Mathematics). Dover Publications, 2015.

Richeson, David. Euler’s Gem: The Polyhedron Formula and the Birth of Topology. Edition Unstated, Princeton University Press, 2012.

Martini, Horst, et al. Bodies of Constant Width: An Introduction to Convex Geometry with Applications. 1st ed. 2019, Springer, 2019.

“Mathematics-pt1-ZeroPerspectiveGoldenMean.mov.” Youtube, uploaded by UConline, 9 April 2012, http://www.youtube.com/watch?v=mMmq5B1LKDg.

Mazzola, Francesco. Self-Portrait in a Convex Mirror. 1524. Kunsthistorisches Museum, Vienna, Austria.

"Measuring the width of a Reuleaux triangle as the distance between parallel supporting lines. Because this distance does not depend on the direction of the lines, the Reuleaux triangle is a curve of constant width." Wikipedia, Wikimedia Foundation, 1 Nov. 2015, https://en.wikipedia.org/wiki/Curve_of_constant_width#/media/File:Reuleaux_supporting_lines.svg.

Munroe, Randall. "Purity." xkcd, https://xkcd.com/435/.