Tuesday, 2 July 2013

Circling the Square

Vitruvian Man, by Leonardo da Vinci,
c. 1490, pen & ink on paper

In book three of his series, Vitruvius wrote:

“since nature has designed the human body so that its members are duly proportioned to the frame as a whole, it appears that the ancients had good reason for their rule, that in perfect buildings the different members [sections/structural elements] must be in exact symmetrical relations to the whole general scheme.[1]

Leonardo da Vinci’s Vitruvian Man was based on this study, but it was also the basis for floor plans and decorative features in Ancient and Medieval architecture. The overlapping of the square and the circle in Vitruvian Man is also seen at Notre Dame Chartres (NDC), and very often called a mandala. In some instances, the mandala maybe better called circling the square. In this post, I wish to define these two terms and examine Notre Dame Chartres’ geometric features to distinguish whether the intent of Medieval design is best labelled mandala or circling the square.

Before examining NDC, I would first like to more clearly define my terminology. A mandala is the overlapping of squares and circles to symbolize the universe. The mandala is created from a religious standpoint. Its origins lay in Buddhism and Hinduism and the term was brought to the west in the 20th century to describe the western use of the overlapping shapes. Mandalas represent "cosmic truth" in the east, and in the west came to represent the intermingling of heaven and Earth and the creation of God. The square, Earth, represents limited space, whereas the circle represents the boundlessness of the heavens. This could be interpreted as expounding upon the boundlessness of God and the limitations of man. The circle and the square are also used because they are symmetrical and balanced and like God in their uncomplicated forms. Cardiff University psychologist David Fontana states that mandalas symbolic nature aid the viewer in reaching a" mystical sense of oneness with the ultimate unity from which the cosmos in all its manifold forms arises."[2] The mandala is therefore a form of religious design used for meditative purposes. Circling the square, like a mandala, is a symbol depicting the relationship between the heavens and Earth. Unlike the mandala, circling the square is created with scientific and mathematical interest. The overlapping shapes are used to measure proportions, ratios and distances like in da Vinci's Vitruvian ManMiranda Lundy, author of Sacred Geometry, states that when the circle (the heavens) overlaps the square (Earth) and they are of equal area or perimeter that we call this circling the square, and that “if the Earth fits inside the square, then the equal perimeter circle defines the relative size of the Moon to 99.9 percent accuracy.”[3] Both terms refer to the formatting and design of the universe, yet they derive from differing intentions. I feel that although these terms do label one for science and one for religion, the two fields  were very much married in the Middle Ages, making the identification of circle/square art difficult.

“if the Earth fits inside the square, then the equal perimeter circle defines the relative size of the Moon to 99.9 percent accuracy.”

When I began my research, I was determined to pick one term over the other to describe the NDC window. Mandala seemed appropriate being that it is a religious setting, but the incredible understanding of math by Gothic architects argues for the use of circling the square and the use of modules to create a sound structure. I have come to believe that circling the square and the mandala are united at NDC. In my previous post about NDC, I discussed the numerological qualities of the NDC West Rose Window which represents both biblical and scientific ideas. The West Rose Window depicts the Second Coming of Christ, but also the twelve months of the year, the four seasons (as represented in the arms of the cross), and the Aristotelian elements. With the emphasis on religion from the interior the pattern seems more mandala-like. From the exterior where these images are unavailable to the viewer, we are dependent on circling the square to communicate the idea of heaven and earth intermingling that is seen from the inside.The window is symmetrical, and divine in the eyes of Vitruvius for simply that reason [5]. But with no imagery, the numbers and the format are more cosmological, and therefore better labeled as circling the square from the exterior view. Knowing that from the interior the viewer sees both God looking down at them and a calendar, the circling the square and mandala are really one in the same at NDC.

West Rose Window NDC, Interior
Circling the Square, NDC Exterior

Circling the square can also be applied to the NDC Labyrinth. The Labyrinth, a symbol for life’s journey, is overlooked by the West Rose Window with Christ in Judgement. With this in mind, the circling the square concept could represent the heavens around the earth and God watching the lives of creations from above.

NDC Labyrinth Circling the Square

So is circling the square synonymous with term mandala? Yes and no. Circling the square and mandalas are both the overlapping of circles and squares in the creation of a symbol of the universe. Yet mandalas are created from a religious background and circling the square from a mathematical. This mathematical means of creation focuses on proportions and symmetry, and once completed can be associated with cosmological and religious meanings. The existence of circling the square at NDC acknowledges the creators recognition of the importance of geometry in academia and rather that they were academics themselves. 


1 Lundy, Miranda. "Circling the Square." Sacred Geometry. New York: Walker &, 2001. 14-15. Print. 
Fontana, David (2005). "Meditating with Mandalas", pg 10. Duncan Baird Publishers, London.
3 Lundy.
4 Hemenway, Priya. The Secret Code: The Mysterious Formula That Rules Art, Nature, and Science. N.p.: Evergreen, 2008. 92-96. Print.
5 "Geometry in Art & Architecture Unit 9." Geometry in Art & Architecture Unit 9. Dartmouth University, n.d. Web. 30 June 2013


  1. I would caution you against assuming that *all* religious uses of this form can be qualified as "mandalas". The term is of eastern origin and is entirely foreign to the western Middle Ages. Furthermore, its use in contemporary culture is intimately connected with New Age adaptations of eastern religion for the West.

    Thus, it is probably inappropriate to use it to describe this form in western medieval art, because it comes with cultural implications that are not present in medieval art and whose use may actually distort our understanding of the medieval context of such art.

    (This is a pet-peeve of mine, because the circular visual designs in the illustrations that accompany Hildegard of Bingen's works are so often called "mandalas". The problem is that Hildegard didn't even know what a mandala was -- and I, at least, am suspicious when we try to force modern assumptions onto medieval thinkers.)

  2. Thank you for your feedback, Nathaniel. I agree with you—mandala is an eastern term. It was not known in the western Middle Ages and as I mentioned in my post, was a term adapted by scholars in the 20th century to describe the western patterns so closely resembling those in the east. I have yet to find a western or medieval term to describe what I have called a mandala in this post. I understand your dislike for the ‘forcing of modern assumption onto medieval thought’ but that is not my intention at all, nor is it not my intention to ‘distort the medieval context.’ I am a contemporary scholar of medieval art and the contemporary word to describe this form is mandala. I would very much prefer a western or medieval term to better represent the cultural content of the form, but being that there is none, I have resorted to using what has become the common vernacular. I do not have the authority on Hildegard of Bingen like you do, but were her mystic designs not an overlapping of shapes representing the cosmos like a mandala? Circling the Square (also called Squaring the Circle), however, is in fact a western concept dating to Antiquity. According to Professor Robin Wilson of Gresham College, circling the square is part of a series of mathematical equations that the Greeks deemed unsolvable. It became synonymous with the Greek phrase for “vain undertaking” by the fifth century BCE, but was originally contemplated by Euclid in the 4th century BCE. According to Professor Paul Calter of Dartmouth University, circling the square was used in the construction of the pyramids, Hindu temples, and eventually medieval architecture like NDC. Greek literature, we know, made its way to the Middle East, and perhaps these mathematical studies were contemplated in the creation of Eastern mandalas. Although the term mandala may not be universal, the math is. I do wish there was a term to differentiate between the circle/square patterns of the west from the east, but until such a word is created I do feel comfortable using mandala despite its eastern origins.

  3. I didn't mean to imply that your use of the word "distorted the medieval context" -- you seem quite aware of that! It's more that I am wary of the loss of context that occurs when non-specialists adopt the terminology -- which is what has happened with Hildegard, prompting a popular misconception that Hildegard's images are equivalent in form and function to the eastern mandala.

    While it is true that she would overlap squares and circles, she does not appear to have ever linked that compositional strategy with the mathematical tradition to which you refer. Despite the frequent comparison of one of the illustrations in the Lucca MS of her "Liber Divinorum Operum" to Da Vinci's "Vitruvian Man", there is no evidence that Hildegard was working from a Vitruvian or Euclidean model -- though she is operating within the general conceptual space that humanity is a microcosm in element, form, and proportion to the macrocosm of all of creation. (I've written a bit about the relationship between text and image in this particular illustration here.)

    The real problem with designating Hildegard's images as "mandalas" is that they do NOT perform the same function as centerpieces for meditative prayer practices. Rather, their function is really more didactic, in the sense that they are actually meant to display and convey information about the relationship between the Creator and the creation, between God and humanity. (This distinction is related to the problematic identification of Hildegard as a "mystic" -- while she did experience visions, they were didactic and prophetic [in the literal sense] rather than "mystical" in the proper sense.)

    On terminology: would it be possible, perhaps, to adopt the term "mandorla" (which is not related, despite homophony, to "mandala", coming rather from the Italian word for "almond")? After all, that is already a technical term of western art used to designate the almond-shaped frame that encloses images of Christ in Majesty (the Maiestas Domini theme).

  4. I am familiar with mandorla. After reading your post (wonderful!), I must agree that mandorlas can have cosmological qualities, but I hesitate to use the term in this case being that there is a lack of almond shape present. Although the two terms do have several similar features, including the use of vesica piscis, I still feel mandala is a more appropriate in my discussion. Although a contemporary application of the Eastern word on Western art, it does best fit the pattern represented at NDC.
    Thanks again for all of your feed back!