Art & Archimedes

1200 words, 5 minute read

We celebrate the fact that we are children of the Renaissance and the Enlightenment. We look back on the Western world prior to 1500 CE and wonder, “How did people ever live like that, think like that?”

The particularly crass among us disdainfully refer to the 1,000 years immediately preceding the Renaissance as the Dark Ages. As if we, the authors of Hiroshima and the Holocaust, are in any position to denigrate others!

Truth is, the years around 1500 CE did witness the single biggest upheaval in the Intellectual History of Western Europe. Machiavelli  was both the catalyst and emblem of this change. In a nutshell, the theocentric Universe of the Middle Ages gave way to the modern, egocentric model.

Burgeoning science gave us the illusion of control. We no longer need respond to the actions of the Universe (the gods, fate); now we can impose our will (technology). We are no longer shaped by what happens around us; now we shape what happens to reflect our own image and likeness. Following the intermediation of the Reformation, Humanism, Moral Relativism and Secularism, Narcissism has replaced Roman Catholicism as the ‘universal faith’: I believe in me!

Imagine a new edition of the Baltimore Catechism: “Why am I? - To know, love, and serve myself.” Sound about right?

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Perhaps the best way to learn about any culture is to study its art. How do folks understand the world around them? How do they relate to ‘things’…and to each other? We should not be surprised then that the intellectual upheaval of the Renaissance was accompanied by a seismic shift in painting protocol.  

In Italy and elsewhere, artists ‘suddenly discovered’ perspective. Discovered? Feels odd to apply this term to something we simply take for granted. Consider Camille Pissarro’s Road to Marly (below). Notice how the trees and other figures pictured close to the viewer are huge compared to those in the background. The space is continuous, receding smoothly toward a fixed point on the horizon.  

This is a Big Bang/Big Crunch universe! Pissarro’s universe begins and ends at a singularity, a point, and from that point, like a cornucopia, it inflates to fill the canvas. 

This landscape appears to follow the laws of Euclidean Geometry, one of a much larger group of geometries jointly known as ‘Archimedean’. These geometries incorporate the rules and properties of real numbers (arithmetic, calculus). In our minds, this is not just a model, it’s Reality.

Would it surprise you then to hear that this representation of the world is virtually unknown outside of Western Europe,1500 – 1900 CE? The vast majority of human beings, past and present, see the world very differently. Take the famous Italian painter, Giotto (c. 1300). His painting of St. Francis receiving the Stigmata (below) was painted (below) without perspective. The sizes of the objects represented have little to do with their situ in space. Rather their dimensions are dictated by the roles they play in the narrative.

Francis, appropriately, is huge, as is the angelic manifestation of Christ. Two nearby houses-of-prayer are tiny by comparison. One is positioned forward of Francis, one behind. Both are depicted as same-sized, indicative of the characteristics of the buildings themselves rather than their placement in the frame or relative to each other or to the viewer. Imagine Francis, as illustrated, squeezing through either door in time for matins!

The landscape itself (mountain, trees) is merely sketched, and it dissolves into an undifferentiated, monochromatic background, symbolizing uncreated light. It does not recede from the main image but rises above it. (Giotto characteristically depicts distance as height.) Finally, the trees on the mountain side are all similarly sized (tree sized) despite their varying distances from Francis…and the viewer. 

For centuries, post-Renaissance critics attributed these anomalies to a lack of technical proficiency. How wrong can we be! Giotto was a master of his craft, but he was painting a different world, or rather, our same world seen and understood differently:

First, the size of an image is proportionate to its role in the narrative, not its position on the panel. Second, each object has its own intrinsic metric, unrelated to the metrics of the other objects. Finally, the frame does not contain the imagery; the images utterly overwhelm and overflow any edge.

We approach Pissarro’s landscape as voyeurs. We are looking at the scene through a knothole in a fence. Giotto’s images, on the other hand, are projected at us. There is no question of not looking; they are in the way. We have to deal with them before we can move on. 

Earlier we pointed out that Pissarro’s landscape presents us with a paradigmatic model of a Euclidean universe: a place for everything and everything in its place. Giotto’s landscape is radically non-Euclidean: nothing’s in its place. 

In fact, its geometry is not even broadly Archimedean, much less narrowly Euclidean! First, the sum of the parts can be greater than the whole; indeed a single part may be greater than the whole. Second, every object has its own metric, independent of other metrics, other objects, other images. Third, the value of any image is independent of its position in space or on the panel. Turns out, these are among the defining characteristics of non-Archimedean (non-A) Geometry. Check out this graphic:

Now at last we can consider the Byzantine icon at the head of this article: Madonna and Child by Duccio di Buoninsegna, c. 1300.

Much as I love Pissarro and Giotto and my own Claude assisted PowerPoint (above), this is clearly the masterpiece of our collection…and a clear exhibition of non-A Geometry in art. Mary and Jesus are the equivalent of our B and C circles (PowerPoint slide, above); A is the image as a whole. 

That leaves undefined that portion of A that is neither B nor C. Call it X. Our unknown artist, typical of the custom of his day, has colored this space a uniform, undifferentiated gold.

If B+C > A, then B+C+X = A if and only if X = A - (B+C): in other words, B+C > A, if and only if X is a negative number. One way to make ordinary arithmetic work in a non-A universe is to allow objects (B and C) to exist in negative space (X where X < 0). 

Got it? Are you a non-Archimedean mathematician now? Good for you! If not, take at least this away: “There are more things in heaven and earth, Euclid, than are dreamt of in your philosophy…(or ours).” 

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The Icon of the Virgin and Child (Hodegetria variant) centers on the theme of spiritual guidance, portraying Mary as "She who shows the Way" by gesturing toward Christ as the path to salvation. It reflects a unique intercultural synthesis, blending traditional Byzantine theological rigor with the distinct stylistic influences of 13th-century Crusader art. Finally, the icon emphasizes divine authority by depicting the Christ Child as a solemn, miniature philosopher offering a formal blessing.