While there are many different fields of mathematics, one thing they all have in common is the value of a good proof. Proofs use a variety of techniques to explain logically why a statement is true. Good proofs have an aesthetic quality and can even be considered beautiful. Although artists and art critics talk about expressions of emotion more often than mathematicians, it is not uncommon for a mathematician to have emotional response to their work in similar ways that an artist does. But is mathematics “art,” and can it really be “beautiful”? Like art itself, the issues of beauty, expression, and emotions are complex subjects, but then so is mathematics. This entry will look at what is mathematics in art as well as what is art in mathematics. Specifically, it will look at examples of the explicit and implicit uses of mathematics in well-known great works of art, contemplate if mathematics itself can be considered art or beautiful, and critique Birkhoff’s mathematical equation for judging the beautiful.

Math can be found in all forms of art, throughout history, across the world and cultures. An early example is the emergence of sacred geometry in Ancient Greece, where the belief that God created the universe according to a geometric plan inspired many works of art. The Old Testament says, “When He established the heavens I was there: when He set a compass upon the face of the deep” (Proverbs 8:27). This proverb inspired medieval manuscript illustrations depicting God drawing the universe with a compass.^{1} Another example of sacred geometry art is William Blake’s painting of Newton taking God’s place as a geometer, which shows the contrast between the mathematically perfect spiritual world and the imperfect physical world. There is Salvador Dali’s Crucifixion, which depicts the cross as a hypercube, representing a divine perspective with four dimensions and The Sacrament of the Last Supper, where Christ and his disciples are pictured inside a dodecahedron. Astronomer and mathematician Galileo Galilei believed artists who drew nature must first fully understand mathematics because, “[The universe] is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures.”^{2} These examples show how connected mathematics and art are and how mathematical concepts enhance the meaning of art.

Mathematics also had a heavy influence on modern artist M. C. Escher who used structures such recursions, logical paradoxes, polygons, glide reflections, tessellations, polyhedral, and the shaping of space is his artworks. Escher’s talent showed that art can be created using geometrical objects to set up contradictions between perspective projection and three dimensions. Much of Escher’s drawings were inspired by conversations with the mathematician H. S. M. Coxeter on hyperbolic geometry.^{3} His drawings looked possible by perception, but were mathematically impossible, an example of the sublime. Ascending and Descending is a prime example of a sublime Escher masterpiece. In this drawing, Escher created a staircase that is perceived to continue to ascend and descend infinitely, a mathematically impossible situation although the drawing makes it seem realistic. When looking at this, the imagination and reason are strained as they try to grasp the meaning and infinite recursion of the artwork.

The golden ratio, known to be extremely aesthetically pleasing, is an example of a mathematical concept inexplicitly used in a variety of artworks. Leonardo Da Vinci’s Mona Lisa is painted according to the golden ratio. Her face can be inscribed by a golden rectangle, a rectangle with dimensions that reflect the golden ratio. Dividing that rectangle with a line across her eyes gives another golden rectangle, meaning the proportion of her head length to her eyes is golden. There is another golden rectangle from her neck to the top of her hands. In fact, there are golden rectangles all over her body.^{4} Da Vinci’s The Last Supper and The Vitruvian Man were also drawn according to the golden ratio. The Vitruvian Man, the drawing of a man inscribed in a circle, illustrates the divine proportions of the human being. The height of the man is in golden proportion from his head to his navel and from his navel to his feet.^{5} Da Vinci used the golden ratio to portray divine aesthetic beauty in his art.

Not only is mathematics seen in art, but some argue that art is seen in mathematics. Artist Richard Wright believes images from the Mandelbrot set and Chaos Theory, images generated by a cellular automaton algorithm, computer-rendered images, and other pure mathematics concepts can be considered art.^{6} Hungarian mathematician Paul Erdős agreed that mathematics possessed beauty but considered the reason beyond explanation. He said, “Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful.”^{7} I believe that when mathematicians talk about the beauty of concepts (numbers, theorems, relations, etc.), they don’t always mean that the concept itself is beautiful, what is beautiful is how the concept relates to so many other things in mathematics and the physical world.

For me, math can be beautiful and even sublime. For example, this is true of the Mandelbrot set. Looking at a computer-generated image of the Mandelbrot set, such as the cover photo of this entry, I experience a moment of the beautiful. It inspires feelings of disinterest, without depending on my desires nor creating any desire. It is necessarily universally valid. It creates harmony within the free play of my imagination and understanding. It has purposiveness without a purpose. I have an openness to the image in a contemplative and reflective relation to myself. The Mandelbrot set has conditioned beauty. It is beautiful for me because I understand the mathematics, history, and culture behind the set. The Mandelbrot set is not only beautiful, but mathematically sublime. Its infinite recursive nature creates a tension in my faculties of imagination and reason when I try to grasp its whole. Watch this video.

Although mathematics can be beautiful and sublime, I do not consider it to be art because it does not possess the same kind of creativity and inspiration that art does. Math does involve creativity in writing proofs, solving theorems, and identifying patterns and relationships, but this creativity is different than that to produce art. Math is logical and systematic; art is fluid and dynamitic. They are not one in the same.

On this point, I strongly disagree with American mathematician Geroge Birkhoff’s quantitative metric of the aesthetic quality of an artwork. In his book, *Aesthetic Measure*, published by Harvard University Press, Birkhoff described a formula that encapsulated aesthetic value.^{8} Birkhoff put a high aesthetic value on orderliness and a low value on complexity, believing beauty increases as complexity decreases. His created the formula M = O/C, where M is aesthetic measure or value, O is aesthetic order, and C is complexity. For polygons, Birkhoff’s formula for aesthetic measure is applied such that, O = V + E + R + HV – F, where V is vertical symmetry, E is equilibrium, R is rotational symmetry, HV is the relation of the polygon to a horizontal-vertical network, and F is a general negative factor. C is the number of distinct straight lines containing at least one side of the polygon. Birkhoff claimed that in any work of art, imaginary lines can be drawn following the principal lines of composition and light and dark areas (example 1, example 2). These lines define geometric areas to which his formula can be applied. Birkhoff explained, “There should be a natural primary center of interest in the painting and also suitable secondary centers. Such a primary center of interest is often taken in the central vertical line of the painting or at least near to it. The elements of order are of course taken to be the same as in the three-dimensional object represented. Finally, there are the connotative elements which play a decisive part; a good painting requires a suitable subject just as much as a poem requires a poetical idea.”^{9}

Birkhoff’s proposal has been criticized by many. This biggest criticism being that his equation does not account for subjective entities like inspiration, feelings, emotions, and expressions. This misses necessary characteristics of the beautiful. Beauty, by nature, is something that cannot be quantified. It is a feeling, an expression of emotion, the harmonic free play of the faculties. It is an internal understanding that cannot be assigned a numerical value. Birkhoff’s formula suggests that a piece has a beautiful measure always, but the beautiful is an experience in a moment, not always. When the janitor sweeps around the David, it is not beautiful in that moment, but the first time he walked into the room with David and experienced all of its glory, it was beautiful. Assigning one numerical value of beauty to the David to account for both situations is unreasonable.

For Kant also, the beautiful is a subjective feeling in a moment, not an objective systematic process that can be inputted into an equation. Birkhoff’s theory is far removed from Kant’s four moments of the beautiful. Kant says, “there is no science of the beautiful or beautiful science, but only beautiful art.”^{10} He argues that a science of the beautiful would have to determine scientifically whether a thing was to be considered beautiful or not, and hence the judgement of beautiful would, if belonging to science, fail to be a judgment of taste. Hence Kant would fundamentally disagree with Birkhoff’s theory. In fact, most artists, art critics, and laymen would find Birkhoff’s equation unsatisfying and inadequate.

In conclusion, although mathematics and art are connected and complementary in many ways, and share some of the same properties (beauty and sublimity), mathematics is not art, nor can mathematics describe art.

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