Wednesday, July 13, 2016

Fractal Geometry in African Villages: Lessons from an Outsider

In the 1960s, the Italian architecture firm Superstudio proposed the eradication of all architectural difference under a ubiquitous grid that they called a "Model for Total Urbanization." This is one of their posters.

"The exception proves the rule." How many times have we heard that without really understanding it? It was Cicero in ancient Rome who first said it, and what he implied was that exceptions presuppose the existence of a norm (to which they are the exception). We always look to outsiders, the exceptions to the rule, to help define ourselves as normal. Outsiders have always defined insiders. An old military adage says that you need a great enemy to create a great army. For ancient Greece, it was the Spartans and Macedonians who provided the "savage" exception that proved the superiority of the civilized Athenian polis, or city-state. For the West, a monstrous "other" occupying the rest of the world provided evidence of the dubious superiority of its own worldview.

The Western gridded "ideal city," as imagined by Fra Carnevale

In popular imagination, the exceptions to Western civilization still represent everything uncivilized, chaotic and savage in the world. When Edward Said and Claude Levi-Strauss came along in the '60s and pointed out that, hey, just because other cultures are different doesn't mean that they are devoid of their own internal logic and structure, we began to open our eyes to the fact that maybe, in the end, we had something to learn from "them." We'd gotten so blind to the veil we've fabricated as a frontier between city/country, order/chaos, insider/outsider, West/Rest, that we'd lost sight of the fact that ours is just one of many ways of being in the world, and that the veil is an illusion.

Somewhere in America

Case in point: the gridded and bordered modern city that sprang up in Europe towards the end of the Roman Empire - with its straight lines and corners retained by a circumscribing wall - quickly became the template for Western life. They were initially walled defensive positions (called Oppida), but the Romans soon became aware of their greater (in the long run) symbolic significance: The walled city is a locus of power. Those outside the walls were the medieval "wildmen," or savages beneath consideration. It established a clear - if fictional - boundary between the order of Man and the chaos of Nature.



The Universe is Euclidean, its rigid geometry tells us. The very linearity of modern cities came to represent the mythic Western advancement from barbarian to citizen, from chaos to order. The more squared-off the space, the more civilized its occupant. The barbarians lived outside the polis amid the chaos of nature, and "Nature," as Katherine Hepburn reminded Bogart in African Queen, "is what we are put in this world to rise above." Consequently, linear gridded space became the Western standard.

Hardly a coincidence that Star Trek automatons, "The Borg," occupy a cube.

By the 15th Century, Linear Perspective gained favor as a way of reifying what had already become a pervasively geometric worldview, literally set in stone by the Romans.  Perspective appealed to rich patrons because it backed up their notions of the hierarchy of social power (think majestic cathedrals with impossibly soaring trompe l'oeil ceilings inspiring awe among the plebs). Artists of Europe clamored to learn the rules of Linear Perspective as a way of codifying a "civilized" worldview, which flattered their clients by portraying them as higher-order citizens. Even if anyone had been aware of another way of seeing the world, they wouldn't have cared for it. West is best, and all that.

Every day I walk the streets of New York City, I'm aware that to get across town means zig-zagging at right angles across an artificially imposed grid that by its very inorganicness was designed to position the works of man as superior to those of nature.

What would an alternative even look like? We're so used to what we've got that it's hard to picture it, but we don't have to: When the fractal geometry of African villages was "discovered" by the West, it provided the exception that exposed the tenuousness of the norm we've come to accept. Certain villages in Africa (such as Tiébelé on the Ghanian border) have been organizing themselves for centuries according to mathematical principles that were only discovered in the West in the 1900s.

The classic Mandelbrot fractal set

Whereas the Western gridded polis denies Nature by proposing space as a system of stackable finite blocks, fractal architecture suggests an unfolding of space according to principles of organic growth. Each unit of fractal geometry relates intrinsically to its neighbor, regardless of scale. There is no frontier, no hierarchy of space as in the West.

Fractal sets are everywhere in Nature

Fractal architecture could never produce a Versailles, for example, that so self-consciously set itself apart from the populace that surrounded it. The "self-similarity" of fractal geometric modules would preclude it. Western architecture relies on the grid as an exclusionary device - you are either inside or outside the square - but fractal architecture seems to suggest that we all have the same potentiality.



I always worry that I'm writing too much in blog posts, that they're too long (because who the hell reads blog posts?), so I won't be getting into the weeds about what fractals are, except to say that they began as an outlier set of mathematical rules once considered to be useless oddities - Euclid's outsiders.

The fractal architecture of Ba-ila village, in Southern Zambia

The realization that whole communities live according to a spatial map that is entirely different to ours was an eye-opener. Watch this TED talk (and buy the book) by this fascinating mathematician who traveled Africa on a Fulbright scholarship, standing on rooftops and recording the fractal geometry he saw all around him. And next time you're sitting at a traffic light, imagine a world that has no right angles.

Ron Eglash mapped the fractal set at the heart of community life for Ba-ila villagers

Tiébelé, Ghana

Monday, March 14, 2016

How to Paint Reflections on Water

Sommarnoje, Anders Zorn
showing lengthened and broken reflections beneath the boat and pier.


The post title is a tad misleading, as it is more about seeing and understanding than painting. Why do reflections appear longer in rippled water compared to smooth water? Why do reflections not appear at all in very rough water? Is there any quantifiable way for painters to measure the correct length of reflections, besides just painting a bunch of wiggly lines?

And if you feel like a little musical accompaniment while you wrap your head around some math, try Claude Debussy's "Reflections in Water."

I'm going to let my illustrations do most of the talking in this post. I hope that they're clear enough that you will be able to understand the principles just by studying them.

Figure 1

In Figure 1 above, there is a pole [AB] standing out of the water. You can imagine looking at it along sight lines that run through an imaginary picture plane. The top of the pole intersects the picture plane (the canvas you're painting on) at point A1. The visible bottom of the pole above the water would appear on your canvas at point B1. If the surface of the water was as smooth as a mirror, the reflection of your pole on the water would be painted on your canvas extending down to point C1. Notice too that the angles x, y, and z are the same when on a perfectly flat surface.

Figure 2

If there was a concrete pier as in Figure 2, the reflections would end up looking as they do in Figure 3, below. Note the reflection of the pole in the water below. If you study Figure 2 you will understand why the reflection simply peeks out a short length below the pier reflection instead of appearing to be the full length of the pole itself.

Figure 3


Fur Traders descending the Missouri, George Caleb Bingham

In the beautiful painting by Anders Zorn that opens this post, we see that the reflection of the boat extends downwards quite a bit below where one might expect. Why? Simply because water is never as smooth as it appears in our illustrations above. Despite what Caleb Bingham might have us believe (above), water almost always has some visible movement causing ripples or waves. These waves cause the reflections to distort and extend lower on the canvas.

The reflection of a pole in rippled water [Fig. 4] would thus appear to extend in a broken line all the way down our canvas to point D1. I hope that by studying the next couple of illustrations you can understand the effect of broken water conditions upon reflections.

In addition, rougher bodies of water appear to be brighter overall than smooth bodies of water. The fascinating reason has something to do with Galileo, and can be found in a separate blogpost, here.

Figure 4


The Champion Single Sculls (Max Schmitt), by Thomas Eakins


Thomas Eakins, famously mathematical when it came to constructing his paintings, drew an illustration [Fig. 5] in one of his notebooks that shows why the reflections in rippled water appear "wiggly."


Figure 5


I've created my own version of the same phenomenon [Fig. 6], with some added notes that might help clarify what's happening.



Sunday, January 3, 2016

The Quadratura of Francesco Natali



Francesco Natali created these wonderful baroque quadratura and wall frescos in a little church in Pontremoli, Italy. I'd hoped to write a lengthier post about this church, its fabulous history, and the charming work of this lesser known Italian artist, but I think you'd rather just see the photos. So here they are.

I uploaded a ton more photos of the Church on my Flickr page, here.

Remove the stone lintel in the small niche (above) and it reveals a deep crawl space used to secrete
dissidents and refugees during World War II.

Incidentally, this church weathered Allied bombardment and was even liberated by the Buffalo Soldiers, America's historic all-Black Infantry (originally Cavalry) division in World War II. They stormed the town only to free a group of jews hidden in a specially-designed crawl space disguised behind one of Natali's decorative niches in the church sacristy (photo above).



















Friday, October 2, 2015

The Black Paintings of Goya


"In 1814 Ferdinand VII posed for Francisco de Goya. There was nothing unusual in that. Goya, court painter for the Spanish Crown, was doing a portrait of the new monarch. But artist and king detested each other. 


The king suspected, and with good reason, that Goya's court paintings were disingenuously kind. The artist had no choice but to do the job that earned him his daily bread and provided an effective shield against the enmity of the Holy Inquisition. There was no lack of desire on God's tribunal to burn alive the creator of La Maja Desnuda and numerous other works that mocked the virtue of priests and the bravery of warriors. 

La Maja Desnuda, Francisco de Goya y Lucientes

The king had power and the artist had nothing. It was to reestablish the Inquisition and the privileges of nobility that Ferdinand came to the throne borne on the shoulders of a crowd cheering: "Long live chains! 

Sooner rather than later, Goya lost his job as the king's painter and was replaced by Vicente Lopez, an obedient bureaucrat with a brush. 

The unemployed artist then took refuge in a country home on the banks of the Manzanares River, and on the walls he created the masterpieces known as the Black Paintings. 


Goya painted them for himself, for his own pleasure or displeasure, in nights of solitude and despair. By the light of candles bristling on his hat, this utterly deaf man managed to hear the broken voices of his times and give them shape and color."

- Excerpted from Mirrors: Stories of Almost Everyone, by Eduardo Galeano

Tuesday, September 22, 2015

On Fragments



"Phidias, the most envied sculptor of all time, died of a broken heart after his insufferable talent landed him a jail sentence. 

Many centuries later, Phidias was punished again, this time by usurpation. 

His best works, the sculptures of the Parthenon, are no longer in Athens but in London. And they are called not the Phidias Marbles, but the Elgin Marbles.

Lord Elgin was not exactly an artist. As British ambassador a couple of centuries ago, he shipped these marvels home and sold them to his government. Since then, they sit in the British Museum. 

When Lord Elgin filched what he filched, the Parthenon had already been devastated by weather and war. Erected to the eternal glory of the goddess Athena, it endured the invasion of the Virgin Mary and her priests, who eliminated several figures, rubbed out many faces, and mutilated every penis. Many years later came the Venetian invasion and the temple, used as a powder house, got blown to pieces. 

The Parthenon was left in ruins. While the sculptures that Lord Elgin took were broken and remain so, they speak to us about what they once were: 

that tunic is just a piece of marble, but in its folds sways the body of a woman or a goddess, 

that knee walks on in the absent leg, 

that torso, decapitated, bears an invisible head, 

that bristling mane conveys the missing horse in full whinny, and those galloping legs how it thunders on. 


In the little there is, lies all that was."

- Excerpted from Mirrors, by Eduardo Galeano

Sunday, September 20, 2015

Life, The Matrix and Everything Part II


Orientalism and cultural appropriation at its best;
T. E. Lawrence striking a suitably romantic pose.
I've been prattling on about outsiders and the role they play in our conception of self and reality. For anyone who's still with me, here is the second story that illustrates my point. It's a fascinating example of how we manufacture reality and create our idea of what is normal.
It's recounted by T. E. Lawrence ("Lawrence of Arabia") in his classic book, Seven Pillars of Wisdom. While planning his next foray into what were the outer fringes of civilization from a Western standpoint, Lawrence was canny enough to know that he'd need to bring back a visual record of his travels among the wildmen who inhabited the uninhabitable desert.

Abd el Rahm, by Eric Kennington (1920)
One day, he came across an artist called Eric Kennington at an exhibition of paintings created during World War I, when the young artist had enlisted with the 13th Battalion London Regiment and been sent to fight on the Western Front. Kennington was a member of the Royal Academy and an official artist commissioned by the British Government in both World Wars, and was by all accounts a solid portraitist. Lawrence was so impressed by his work that he invited Kennington on his next campaign into the desert in 1920, where he painted numerous sensitive pastel and watercolor portraits of the characters they ran into. 

Another of Kennington's indecipherable scrawlings,
this time of Muttar Il Hamoud min Beni Hassan (1920)

To the Western eye, they're beautiful and naturalistic renditions, but when Lawrence showed the Arabs their portraits, most of them failed to recognize that they were images of men - let alone paintings of them! They blinked and stared blankly, spun them around, flipped them upside down and handed them back. One of them even hazarded a guess that it was a camel because the line of the jaw looked like a hump. Lawrence and Kennington were gobsmacked. The two dusty, sun-burned white men were utterly alien to their Arab hosts because the cultural signs of Western art were completely unintelligible, and therefore had no value whatsoever. 



The peripatetic DH and Frieda Lawrence, mid exile.


Outsiders may enable us to see ourselves, but our desire to experience alien-ness in the World is always conditioned by an instinct for self-preservation that keeps us within our comfort zone. It’s not really the void we’re after, but a safe approximation. It’s like standing on the edge of a cliff during a storm and watching the smashing of the waves against the black wall. All we really want is a quick selfie and a nice dry car to jump back into. Only a madman would throw himself off the edge. 

When the poet DH Lawrence went on his “savage pilgrimage” - a self-imposed exile from 1919 to the end of his short life at the age of 44 in 1930 - he was in search of fulfillment from a life outside industrial western civilization. His wanderlust took him to Sicily, Ceylon, Australia and eventually New Mexico. He found bits of what he wanted among the peasants of Germany, Italy Mexico and India, but towards the end he grew disenchanted with his “savages.” He even seemed genuinely surprised when it turned out that “their consciousness is so different from ours that there’s scarcely any possibility of communication.” [i]


          If the outsider cannot render nature tame in our eyes, cannot “soothe our imagination,” then she must at least be able to speak of its strangeness in a language we can comprehend. Van Gogh spent years vacillating between desiring acceptance from the Salon and railing against it. His search for a style meant tiptoeing the line between being originality and commerce.* He couldn’t know that the actual myth he was creating around himself centered on his tense temperamental dis-ease with the world, that the key to his fame in the public's eye was the fact that he could never fit in. Van Gogh would never be allowed to fit in because we need our outsiders to stay strange so that we can feel less so. He could never paint nature in a way that satisfied him, because being true to his vision would mean being incomprehensible to others, while seeking recognition from others would mean faking his art. When William Styron wrote DarknessVisible, his deeply personal memoir of madness, he said, “never let it be doubted that depression, in its extreme form, is madness.”[i]

Van Gogh's Starry Night, the quintessential image of a world on the edge of falling apart.
Attempts to fully describe the madness of depression necessarily fail to grasp its true nature because they rely on words, and words fall miserably short. It’s like trying to describe the color blue to a blind person. The madness of depression is “so mysteriously painful and elusive in the way it becomes known to the self – to the mediating intellect – as to verge close to being beyond description.”[ii] Any transcendent experience percolates down through the civilized consciousness before it’s capable of being communicated, but something gets lost in translation. People who experience profound states of being often find it difficult to locate words that accurately convey the intensity of what happened to them. It’s as if the words reduce the experience. The outsider who is capable of mediating between both worlds does not fully live in either one; she’s suspended in Dante’s purgatorio. Anyone communicating the depths of insanity cannot doing so from within its dungeon. Communication is only possible if we’ve stepped back from the void, otherwise its language is utterly incomprehensible.

Skrik, or The Scream, by Edvard Munch

Edvard Munch’s iconic painting known as The Scream, instantly recognizable to any highschooler, was originally titled The Scream of Nature (Der Schrei der Natur) and holds a special place of fascination, revulsion and horror in the minds of the public. Skrik is its Norwegian name; it describes more of a screeching banshee wail than a scream. But the title is unimportant; it’s a visual language of alienation that we can all understand. This is what it looks like when you cannot control your place in the world, we think. The sky bleeds and nature melts. 

On May 2 2012, it went to the auction block at Sotheby’s. Bidding started at $40 million and quickly shot up from there. At the 12 minute mark, a phone bid came in that brought down the auctioneer’s hammer and caused dumbfounded gasps around the room: Leon Black had given the final offer of $119,922,500.[i] Otherness is something we’re evidently willing to pay very good money to possess, or perhaps more correctly, to circumscribe.



With The Scream, Munch gave us a world that coincided with what our vision of strangeness is. If it was truly strange, we wouldn’t have been able to understand it at all. That Munch’s painting had an audience at all was a good sign on two fronts, not least for the man himself. Whereas many inmates in the asylums of France cowered incommunicably in darkened corners, Munch, fortunately, was still functional on some level. It also meant that society had progressed to the point where it was willing to at least imagine what a “madman” like him had to say, if only because his work subscribed to the stereotype of the Disturbed to a society that was newly and romantically inclined to want to investigate such extremes of character. That The Scream exists in no less than four versions – two pastels and two paintings – should suggest to us that Munch had enough wits about him to know he was on to a good thing. A few years previously and he’d have been discarded and left to rot in an asylum, and so-called inspired lunacy bedamned. Meanwhile, for the liberated societies of Paris and Berlin, wildmen, savages and the mad were being dragged along for the ride as contemporary representatives of our evolving conception of wildness, and as mute witnesses to the changing whims of Western fashion.

The Scream is a stunning painting to be sure, but is it really worth all that money? I believe its popularity derives at least in part from the same mechanism that was at play in the contemporary popularity of Medieval accounts of monsters at the outer edge of the map. The Scream is a first-hand account from the outer fringes, which lends it the weight of authenticity, but at the same time it is a madness that we can all understand.  Looking at the painting, we can see straight away that it conforms to standard Western notions of the landscape. There’s a clearly defined background, middle distance and foreground, containing the screamer in a paroxysm of emotion that verges on cartoonish. The sky is primary red, immediately recognizable in the West as the customary color of distress. Even Munch's choice of medium and standard portrait format are all signs that are utterly ordinary. In fact, the painting is striking precisely because it is so instantly readable. 


It conforms to a stereotype of the outer frontier of consciousness that we created to define us, the viewer gawping from the outside, as not mad. That’s not to question Munch’s motivation as anything but a genuine quest to express the anguish he was feeling at the time, but the stratospheric sum paid for his painting should alert us that there’s something deeper at play.


...and The Scream becomes a meme

In the end, Munch’s painting is never so different that it’s incomprehensible – as Kennington’s portraits had been to the alien society he'd found himself in. If Munch had been a true outsider, as Kennington had been to his Arab hosts, it would never have caused such a circus at Sotheby's. 

Munch’s colors might be inverted but his forms are still recognizable to us because The Scream is a visual idiom with which we are familiar. Its nightmarishness signifies this side of the boundary between reality and dreams, between order and chaos. Through it, we define the edge of normality by witnessing its transgression. But disturbing as it is, Munch’s world has failed to completely fall apart. It's popularity lies in its power to uphold our manufactured, pre-conceived stereotype of what is normal, what is real.



* Van Gogh's letters to his brother Theo are filled with the anguish it caused him to sit between two stools.



[i] Lawrence, DH, “Apocalypse,” from the foreword by Richard Aldington, Penguin (1976) Pxvii
[i] Styron, William, “Darkness Visible,” Vintage (1990), P47
[ii] Styron, ibid. P7
[i] Crow, Kelly "Munch's "The Scream" Sold to Financier Leon Black". The Wall Street Journal, (11 July 2012).