Mathematics in European Renaissance Art

Math 3010-1 



    The European Renaissance was an era in the fifteenth and sixteenth centuries, centered in Italy.  It was considered the humanistic rebirth of ancient Roman and Greek cultural values, reintroducing the canons of perfection and achievement, and expanding upon them through enthusiastic, multidisciplinary research.  As such, artists during this time period relied on science and observation to build their perfect studies rather than crafting the identical and ideal ancient Greek figure.  An  example of such a traditional Greek figure is  the sculpture “The Doryphoros of Polykleitos” from the fifth century BCE, which portrays a soldier sculpted in a stable, balanced, and dynamic stance known as controposto, and emphasizes human beauty, perfection, and potential.  This sculpture became the canon of ideal male proportion at the time.
    The Renaissance reintroduced these values of beauty, perfection, and potential using rational observations and innovations in mathematics and sciences to develop  more sophisticated techniques, resulting in a stunning realistic style.  For example, Leonardo da Vinci was obsessed with this intersection of art and math, and wrote in his notebook, ““Non mi legga chi non e matematico”, or “Let no one read me who is not a mathematician.” He utilized his observational and technical talents to create numerous ink studies of the world around him, such as the famous “Vitruvian Man.” This piece, inspired by architect Vitruvius, measures how the body moves in relation to itself by way of observation, dissection, and measurement. It is one of the first of its kind, and served as both medical and painting guides.  Here, da Vinci depicts a figure in the center of a square and circle, accurately demonstrating how the human body exists in space.  He noted above his drawing that, “If you open your legs enough that your head is lowered by one-fourteenth of your height and raise your hands enough that your extended fingers touch the line of the top of your head, know that the center of the extended limbs will be the navel, and the space between the legs will be an equilateral triangle.”  Such enthusiastic mathematical inquisitions regarding the natural world further developed Renaissance art in terms of balance, composition, realistic accuracy, and more.
    The innovative paradigm of the European Renaissance also led to the cementation of artistic perspective by way of geometry.  Throughout history, the scaling of figures and objects in art was commonly hieratic, where size correlated to importance of the subject, rather than their relative position.  If it was the latter, it wasn’t achieved accurately, and therefore the secrets behind true perspective were coveted.  As J J O'Connor and E F Robertson wrote in their article, “Mathematics and Art - Perspective,” the problem was “how does one represent the three-dimensional world on a two-dimensional canvas? There are two aspects to the problem, namely how does one use mathematics to make realistic paintings and secondly what is the impact of the ideas for the study of geometry.”  It is believed that the solution to said problem was first found by artist Brunelleschi, which eventually led to what was known as Alberti’s veil (where scenes were traced onto a transparent canvas - plexiglass is today’s substitute).   Since Renaissance artists observed that a straight line in perspective always remains straight and parallel lines either remain parallel or converge to a single vanishing point, artist Alberti devised a method to portray a tiled floor in perspective where one set of floor lines is always horizontal. The non-horizontal floor lines touch the bottom of the foreground evenly spaced, and all converge to the vanishing point at the artists eye level, which is also the horizon line of the art. The new horizontal floor lines are next found by creating diagonals through any one foreground tile, so that this new line also intersects the horizon line.  Where this diagonal intersects the other non-horizontal lines marks where the further horizontal lines should be placed, which works because it is true on a flat tile floor.  This was the birth of true perspective, and was the perfect mathematical tool for ambitious Renaissance artists to use in order to achieve groundbreaking accurate pieces that satisfied the artistic ideals of the era.
    An example of such a piece is Masaccio’s “Holy Trinity,” circa fifteenth century CE.  This fresco demonstrates the reimplementation of anatomical correctness, along with accurate perspective, and lighting, and as such is a critical Renaissance piece.  There is a stable, triangular composition formed by the figures, and the light appears to emanate directly from Jesus, illuminating the scene.  The vanishing point is below the crucifix, where one’s eye level would be if they were inside the painting, and all perspective lines converge to this point perfectly.   This is especially obvious when looking at the coffered ceiling squares, which are the same shape in reality, but appear to converge downward the further away they are because they are above the horizon line (forming a perfect halo-like arc around the figures). The momento mori is below the horizon line, and therefore converges up.  This perspective is measured so accurately, that “current researchers have been able to create a 3D representation of the imaginary space described in the picture using digital technology.”
    In all, the European Renaissance was a time of explosive inquisition and creation, resulting in a wealth of cross-disciplinary knowledge for the world.  The art work was incredible, in part due to the lofty ancient Greek and Roman canons, but mainly because of the new ways of meeting those standards through scientific and mathematic mindsets, resulting in new tools to elevate the art such as mathematically accurate proportions and linear perspective.  These discoveries, in many ways, shaped society for centuries to come.


Works Cited
Artincontext. “The ‘Vitruvian Man’ Da Vinci - Why Was the ‘Vitruvian Man’ Created?” Artincontext.org, 31 Mar. 2022, https://artincontext.org/the-vitruvian-man-da-vinci/. 
Artincontext. “‘The Holy Trinity’ Masaccio - an Analysis of the Iconic ‘Trinity’ Artwork.” Artincontext.org, 23 Jan. 2022, https://artincontext.org/the-holy-trinity-masaccio/. 
DEBORAH, ROBERTSON JEAN HUTTON. History of Art: A Global View. THAMES & HUDSON, 2022. 
“Leonardo and Mathematics.” The Mona Lisa Foundation, 9 Aug. 2018, http://monalisa.org/2012/09/12/leonardo-and-mathematics-in-his-paintings/. 
Stillwell, John. Mathematics and Its History. Springer, 2020.