Gospel & Universe 🔭 The Sum of All Space

The Unknown Arcs of a Sphere

The Straight Line - The Circular Park - Andromeda & the Greeks - Línea Recta

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The Straight Line

The Daughter of Beauty is followed in the dark by a man named Mateo, who is a member of the Secret Society of Geometer Mystics. While it’s called a society, it’s actually a cult whose members believe that there’s nothing more frightening, and nothing more worthy of worship, than a straight line. Any straight line. To a Geometer Mystic, a straight line transcends any system imaginable. It travels beyond religion or philosophy, beyond Empire, beyond God Himself. The high priest of the sect is a Tejano from Austin who lives in a log cabin in Massachesetts, somewhere near Waldo Pond. His name is Granlineas the Mystic.

Whenever Mateo’s thoughts stray towards circles and revolutions, he yanks them back to what Granlineas calls The Fate of Infractional Trigonometry. By this he means that a circle is an abomination, and that it only exists as a function of the straight lines that radiate from its hypothetical centre. Granlineas notes in his treatise, Hidden Infinities (1967) that each radiating line is in fact infinite at all points, since mathematically even the smallest length can be sub-divided without end. Mateo has read Hidden Infinities over two thousand times. The treatise also plays continuously in his head, its various chapters and verses coursing through every sub-structure and minor relay junction of his brain.

Mateo is also proud of his Greek human ancestry, since Zeno was the first human to explore the ramifications of infinite divisibility. His fellow Greek-Argentinians took this as yet another sign that the Greeks invented civilization. Yet Mateo pointed out to them that, like the Hebrews, the Greeks borrowed more from the Mesopotamians than they’re prone to admit. Even the Sumerians divided the circle into 360 lines, each radius an infinity that can never be sounded. And because their number 360 dominated space, it also came to dominate time, with 60 seconds in a minute, and 60 minutes in an hour. Mateo laughed when he heard stories about The End of Time, the Prime Mover, and the God with a beard who controlled everything. He laughed even harder when so-called scientists talked about the finite universe that wrapped in upon itself. Didn’t they understand that a straight line has no end?

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While Mateo was a Geometer Mystic, he was also male. This meant that all of his passion for straight lines went up in smoke when he saw a beautiful female of his own species. Humans are intrinsically weak in this way, and he hated himself for it. And it was so frustrating! For instance, he often strolled along crowded Avenida Corrientes, dutifully pondering the unknown arcs of a sphere and devoutly calculating how many decimal points it would take to differentiate a planet in Andromeda from its moon, when all of a sudden he was brought down to earth by the gravity of dark eyes and by the ferric magnetism that has been created in this part of the world by the mixture of Spanish, Native, and Italian blood. He vowed to follow his theories into the Empyrean, yet he saw jeans so tight that he was forced into an abstraction of a different sort, and soon lost his way in the bright beauty of the Buenos Aires day.

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The Circular Park

Every fortnight or two, he strolled in the evening through nearby Parque Centenario. He challenged himself to walk around the circular park, daring its circularity to defy the fact of its radiant lines.

Although it irked Mateo to admit it, Parque Centenario is an almost perfectly round park 500 meters in diameter. It’s located in the district of Caballito or “Little Horse,” smack in the middle of Greater Buenos Aires. At night, dozens of joggers make their way through the semi-darkness, around crumbling pavement and hawkers selling coke. Mateo of course would prefer to walk in straight lines, yet he considered it his sacred duty as a Geometer Mystic to demonstrate not only that a circle is defined by its infinite radiant lines (as per Granlineas’ treatise), but also that the apparently round, outer circumference of a circle is also made up of an infinity of straight lines. Just as some zealous rectangularists try to square the circle, so Mateo was trying to straighten it.

The night was calm and dark, with streetlights casting an intellectual calm on the strollers and joggers that he joined on the foot path. Mateo loved this time of the evening, since it hid the faces far away and lit them up as they approached. He froze the moments, so that he could pinpoint the location of each face along a series of points that eventually flowed past him and into the darkness again. The faces were a benevolent mix of Spanish and Italian, and the svelte bodies belonging to these faces provided the perfect geometrical segments for his calculations.

Unlike the joggers, all the strollers walked at about the same speed. As a result, in theory at least, Mateo would never encounter another stroller going his way. He may have begun, in time, to believe that he was the only one walking in a clockwise direction. That he alone saw the beautiful forms of the women looming in and out of the shadows toward him. That he alone was moving in the right direction.

He imagined that others were making similar calculations in the astronomical observatory, located on the north end of the park, just past where zero meets 360 degrees. He imagined the lonely astronomer, trying to come to terms with the sensual geometry moving beneath him. How he strained to bring into focus the spherical and hyperbolic convexities! Yet unfortunately his giant telescope was directed at the stars. His yearning eyes met only a dark sky populated by random points that could only be connected like a paint-by-number Venus de Milo. The figures weren’t even remotely like the things themselves.

The lonely astronomer would have to content himself with lustful thoughts about spherical trigonometry, that arcane realm first explained in Al-Jayyani’s 11th Century treatise, The Book of Unknown Arcs of a Sphere.

Mateo, on the other hand, looked up into the night sky, superimposing onto it the visions below. He saw Diana with her bow, and Andromeda at the edges of a choppy sea:

Of all the Geometer Mystics on the planet, Mateo believed that he was the one who most understood the intercourse of terrestrial and celestial mathematics. Only he truly understood how the earthly and heavenly spheres orbited in perfect conjunction! 

Yet we live in a world of rare constants. Half way around the circle of the park, at 180 degrees, nel mezzo del cammin, Mateo ran into a snag, as if from a thin branch jutting out from one of the iron fences that separated the walkway from the trees. He saw up ahead of him that he wasn’t the only one walking in a clockwise direction. Fate, in the guise of a strolling Porteña, altered the constant according to which he made his celestial calculations.

Now only five meters ahead of him, with her body moving slowly backwards toward him, was a young woman in a loose sleeveless cotton shirt. The floating white shirt revealed smooth tanned shoulders. Her triangular shoulder-blades were massaged by a wavy mass of dark brown hair, lightly drifting in the ten o’clock breeze. As she got closer, he saw that she was wearing a pair of white sneakers and tight-fitting jeans.

Now, when Mateo thought to himself, tight jeans, he couldn’t be sure that those outside of Argentina, and possibly Brazil, would understand. Only those familiar with spandex yoga pants were likely to grasp the point. In vain, Mateo searched for the precise verbal equation that corresponded to the geometry he was trying to describe.

Sometimes, walking down crowded Avenida Corrientes, Mateo walked into lamp posts. Or, he went flying across a gulf where a series of brown tiles ought to have been. He often wondered if it wouldn’t be prudent for him to imitate the horses once found toiling in the fields, equipped with vertical blinds cutting off the ends of the horizon. Only in this case he would use horizontal blinds blocking the visual range from the neck (just above the fine spirals of dark hair) to the middle of the calf (or slightly lower if she was wearing an anklet).

The tightness of her jeans sent Mateo’s mind into a whorl of numbers and geometric extrapolations. He was tempted to get closer, yet he reminded himself that his point of observation must be as consistent as possible, so as not to disrupt the reassuring order of his calculations. It was bad enough contending with the variables of uneven pavement, canine excrement, and branches poking out from the selva oscura. He didn’t want to complicate matters by factoring in the decreasing distance between his body and hers.

He slowed down, so that they were moving around the park at exactly the same speed. He then drew an imaginary line from the back of one knee to the next, and followed the denim upward to the apex.

Now that they were in a synchronous orbit, Mateo was at leisure to make some fairly solid calculations. Yet a thought occurred to him: given that space can be divided infinitely, how can anybody ever make an accurate calculation? How can anyone even go from one number, or one point, to the next? He felt himself falling between the numbers, between one ten-millionth and one eleven-millionth of a millimetre. And he knew that the million segments between ten and eleven million had a million segments each. And that each of these millions had a million as well. He stopped himself in the middle of the abyss, remembering Granlineas’ warning that once one fell for the logic of division, there was nowhere to land.

All of this made him think about Zeno’s famous paradox, the one where Achilles lost the race against the tortoise the moment he gave it a head-start. For every distance Achilles ran to reach the point where the tortoise was, the tortoise would have advanced beyond that point. Pushing Zeno’s point even further, one could say that it’s impossible to travel from one point to the next because an infinite series of points open up mathematically between the point of departure and the point of arrival.

Mateo recalled that Granlineas had much to say about this in Hidden Infinities:

The matter of infinite divisibility is less a mathematical or practical problem than a philosophical one. Mathematically, there’s no end to the division of space. As a result, it’s clear that theoretically there’s an endless counting of segments of space, which makes it impossible to count from one number or point to the next. It’s equally clear that practically, that is, in real space, we can go from one point to the next. Therein lies the paradox: going from one point to the next is a mathematical impossibility yet a practical possibility. Yet this isn’t the real problem, which is instead philosophical, epistemological, and ontological: going from one single thing to the next in a coherent statement of meaning, or in a coherent state of being, isn’t a theoretical impossibility which becomes possible in the real world; rather, it’s a theoretical impossibility which remains impossible in the real world.

Mateo also recalled a 1939 essay by his fellow Porteño, Jorge Luis Borges: “Pierre Menard: Author of Quixote.” In it, Borges quotes a comment that Leibniz made on the problem of infinite divisibility:

Ne craignez point, monsieur, la tortue. / Don’t worry at all, sir, about the tortoise.

In his mind, Mateo drew a straight line from Leibniz to calculus, binary codes, and the earliest computer. He also drew a straight line from Leibniz to Voltaire, who admired the German polymath’s mathematical brilliance but was deeply skeptical of his quasi-theological extrapolations. In his 1759 story Candide, Voltaire attacked Leibniz’ notion that everything is connected in a meaningful way, and that we therefore live in the best of all possible worlds. Mateo imagined what type of French wit Voltaire might have used in some powdered salon when he heard his German idealist say, Ne craignez point:

Mais qu’est-ce qu’un point, Monsieur? / But what is a point, sir?

Dans le meilleur des mondes possibles, on peut repérer ce point et le suivre d’un point à l’autre. Mais, malheureusement, mon cher Gottfried, on ne sait point si ce monde est le meilleur des mondes possibles. Il ne le semble point. Nous ne pouvons même pas repérer un point àfin de commencer nos calculs, puisque chaque point est perdu dans l’abîme des points numérotés qui mènent a ce point. Et malgré le fait que peu de choses soient certaines, c’est clair qu’on ne peut pas traverser d’un point de certitude au suivant. Il n’y a pas de Grande Chaîne de la Vie, liant un point au suivant, en ordre prédéterminé. Réveillez-vous, mon vieux!

In the best of all possible worlds, one can locate this point and follow it from one point to the next. But, unfortunately, my dear Gottfried, we don’t have a clue if this is the best of all possible worlds. It doesn’t seem that way at all. We can’t even locate a point to begin our calculations, since each point is lost in the abyss of numbered points that lead to it. And despite the fact that few things are certain, it’s clear we can’t go from one point of certainty to the next. There’s no Great Chain of Being, linking one point to the next in a predetermined order. Wake up, my friend!

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Andromeda & the Greeks

Looking up into the cobalt firmament, Mateo saw Andromeda, that tragic beauty in chains, still visible in the late November sky. How he lusted after her, that form in the heavens that gave everything form, from Alpheratz to Saturn, and everything in between. Leibniz might have said that we live in the best of all possible worlds because we live in the best of all possible cosmoi. But the Geometer Mystics knew better than that.

Still, the Geometer in him pleaded with the Mystic to find the meanings that the Greeks and Italians had ascribed to myth. If only, like the early Babylonian astronomers, he could superimpose geometry and astronomy, and conjure more than just an aesthetic theory. But for him it would always remain a theory, beautiful in its abstraction and beautiful in its art, but not beautiful in the sense that it was true, however sorely that might disappoint John Keats.

But how he wept for Andromeda! How unjust it was that she was treated as a mere sacrifice! How he wished that he could take the chains from her wrists, and soothe her swollen body parts with olive oil and a soft sea-sponge.

Yet all of this lusting after heavenly connection just made it harder to connect the dots here on Earth. He looked ahead of him at the slim Porteña, with her perfect form. It was always beyond his reach. He was always just about to connect the dots, and locate the pinpoint of the apex, that moment of perfect bliss that lies between the legs, at the junction of X and Y. Yet there was always something getting in the way.

The problem was this: Assume for instance an isosceles triangle, the base running from the back of one knee to the back of the other. The lines that move upward from this base should meet at the apex, somewhere in the vicinity of Mirach, the giant red star whose name derives from the Arabic al-Maraqq, meaning the loins or the loincloth.

While practical logic insists that the two lines will meet (barring outside circumstances affecting their trajectory), in theory these lines can never meet. This may seem like nonsense, yet there’s no use railing against theoretical geometry and mathematics, unless you're prepared to argue with Pythagoras, Zeno, Euclid, Liebnitz, Newton, Einstein, and Bernhard Riemann. And unless you’re willing to read Granlineas’ twenty-thousand page treatise, Hidden Infinities, and fathom the depth of Space itself.

So Mateo got back to the concrete facts of the case. He looked again, very intently, at the imminent union of two lines. There was only one centimetre to go before the lines touched, which was their natural destiny. Granlineas went further: it was their universal destiny, just as when one draws a circle with a compass, it will at some point go from 359 degrees to 360. It will reach the point where 360 and Zero meet. Or else no one has ever drawn a complete circle before. Which, of course, they have. Yet Granlineas still forced his reader to contend with Achilles and the tortoise. Mateo chuckled to himself when he thought that even if he were to think of advancing a single ångström, some disgruntled Greek would find a way to divide that ångström into fifths.

At approximately 200 degrees, Mateo and the Daughter of Beauty reached the point where Kármán Street met the perimeter of Parque Centenario. Líneas called it Karman Vortex Street, after the repeating pattern of swirling vortices caused by the unsteady separation of liquid flowing over bluff bodies. And he called it the Karman Line, the point 100 kilometres above the Earth where the atmosphere meets outer space. Mateo entered a swirling vortex and was about to lift off.

At 270 degrees he felt like he was making headway, and would soon arrive at a conclusion, which would allow the two lines to converge at a single point. Yet such is the tight, agonizing power of logic in the city of Borges, that the point where the two lines met still eluded him. At 300 degrees his eyes began to strain and his vision became extremely fine. The lines were nearing each other, yet there was still a gap between them. Not a gap so much as an abyss! And yet he felt so close to the hypothetical point of unity that he could almost touch it with the tip of his finger!

At 330 degrees, where the perimeter of the park touches Calle Angel Gallardo (‘Gallant Angel Street’), he saw the answer to the problem! But at the exact moment he formulated it to himself in the clearest terms possible she stepped onto the outer edge of the walkway. She looked left as if to cross the street. As if she followed in her mind’s eye its northeastern trajectory toward Corrientes. As if she were about to leave the periphery of the circle, just before the point where 360 meets Zero.

She turned left at the observatory, yet he kept circling. Perhaps if he circled the park once more, he’d see the sequence of events more clearly. Perhaps if he continued to circle the park, eternally returning to the starting point, he would arrive at some sort of understanding about what it all meant. If he got to the very centre of the park, and sat at the edge of the lake, as if he were sitting on a ghat in Benares, would the Truth appear to him? Maybe Nietzsche and Shankara could see eye to eye after all.

The fact that the lake had no official name might even begin to take on some sort of meaning. Yet this seemed a precariously Hindu way of thinking, and was likely to lead to red dots on the forehead and the worship of gods with more than one arm. How could he consider reincarnation, the eternal circling from one life to another, when it was hard enough to believe in even one afterlife?

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Línea Recta

Leaving behind the perfect circle of Parque Centenario, the Daughter of Beauty stepped into a world of lines and tangent in which she was free to choose her own direction. She was free to go backward or to never go back there again. Indeed, she was free to never go back anywhere she had ever been. But what kind of freedom would that be, to never return to where she grew up on quiet Calle Panamá, where she learned to balance religion and doubt, and where she first fell in and out of love with an angel called Gabriel?

The choice was up to her. Wasn’t that what humans meant by freedom of choice, even if it led to a Great Big Nowhere? Even if it meant that she was free to go some place new, where she could find new myths to live by. Even if it meant she could keep on travelling, till there was no myth to live by. Was this what Buddha meant when he talked about travelling beyond the farthest shore?

Yet this hardly seemed reassuring. Indeed, it sounded alot like the atheist’s concept of death, ridding us of a world that made too much sense, made sporadic sense, or made no sense at all. Religious people say that the universe makes sense, that there’s a divine Plan, yet what they say is hard to believe. They claim to understand the highest Truth, as if height somehow meant depth of understanding. What is the truth 300 miles above the Earth, at minus 270 degrees celsius? What does the grey eel know eight thousand metres in the dark water of the Mariana Trench? Some people say that God speaks in Arabic. Others say that Vishnu would chant in Sanskrit if he felt like it. Perhaps it’s no surprise Buddha left the Hindu lake to became one with rivers and eternally receding shores. Perhaps Laozi and Zhuangzi were right that as soon as you think you understand the highest Truth you’ve missed It.

Not surprisingly, the only theologies that made any sense to the Mystic Geometers were Daoism and Buddhism. And even then they wondered, If you have a Straight Line, why do you need a religion? It was a precept with Granlineas, as it was with Buddha, that the sage, however much others may want to erect golden statues of him, is only a human being. And what applied to human beings also applied to philosophical and religious systems: none of them could match the simplicity of a single Straight Line. This Straight Line could even be a stick, held by an idiot boy about to beat a dog. Or it could be a magic wand in the hand of a Wizard who was about to save all the elves and donkeys in the world. It was still — never only — a Staight Line.

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The Daughter of Beauty, meanwhile, crossed the street, and directed her footsteps toward Corrientes, The Current, the five-lane thoroughfare leading straight into the heart of Buenos Aires.

Mateo asked himself, Where’s she going? He assumed she’d continue in her present direction. After all, she chose it among 359 other possible directions. Yet once she reached Corrientes, the single degree or ‘minute’ that she chose would start to divide outward. Minutes would multiply like the fine spiderwebs of a mariner’s chart. Arcminutes would divide outward into arcseconds. Arcseconds into milliarcseconds, milliarcseconds into microarcseconds, the spaces between the spaces dividing into an infinite number of radiating lines and spokes on a cosmic wheel. Zeno’s paradox writ large.

Perhaps she was now at a bus stop, breathing in the heavy smoke of the buses as they lumbered down the running avenue. Mateo imagined her with her arm outstretched, waving for the Number 24 bus, which had Once written above the driver in an ominous black script. Once referred to Train Station Once or Train Station Eleven, yet there was no mistaking its meaning in English. If that wasn’t a sign, what was?

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The Daughter of Beauty is flying down Corrientes for the last time, dancing past Medrano and Carlos Gardel, only once. She stops for a brief moment, for a last time, to savour the charm of the old wooden coffee shops. She looks into her favourite cafe on Santa Fe and Pueyrredon, yet there’s a Canadian sitting in her usual seat and he’s staring at her with such a demented look that she literally runs back to Corrientes. She takes a last loving look at the theatres and the milongas where she once tangoed late into the night.

She never returns to Angel Gallardo or to the lake that has no name. She just keeps going, in the wake of Bs, and Cs, Ds and Es, toward the infinite unknown crossroad X, never reaching Z, travelling onward toward everything that she never knew.

What if there’s no return? What if nothing in our lives were repeatable, verifiable, coherent? More disturbing than the question, What if we never step in the same river twice? is the question, What if we never even see the same river again? More disturbing than the question, What if we never get from A to B? is the question, What if we continue past B but never return to it? In practical terms, Zeno’s paradox can be disproven on a small scale, when we look downward at the radius of a circle. Yet its frightening opposite can’t be disproven in practical terms on the large scale, when we look upward, where lines radiate out into the universe, and, perhaps, into an infinite number of universes.

She takes a train northward, into Missiones Province, with its Jesuit schools and all the good intentions in the world. From there she travels up to the cataracts of Iguazú, with their indescribable beauty.

From there she travels on foot and by float-plane into the vast stretches of the Amazon.  

This woman will never come back. Even though Mateo never saw her face, he still remembers the sharp line of her shoulder blade and the tightness of her jeans, moving slowly ahead of him in the murky hours before midnight.

She continues northward in her tight blue jeans. She takes a series of planes, skipping lightly from stone to stone on the arc of the Lesser Antilles till she reaches the great rock of Puerto Rico, and the firm footholds of Hispaniola and Cuba. From there she jumps to Cape Canaveral, where with running step she plants her foot one last time and leaps into the Great Beyond, shuttling outward, past her life on this planet and into the stars. Past the Big Dipper and the Little Horse, she follows a trajectory more disturbing than the earthly paradox of Zeno.

For Zeno can be disproven, both practically and logically. Our reality in space is not chopped up into discreet temporal packets that operate apart from one another. The tortoise inches along while Achilles flies past him (unless of course you are Aesop, but that’s another matter). Yet no one can prove that the universe — or the universes — aren’t infinite, and that our beautiful Argentinian’s journey must ever come to an end. No matter what system or theory we devise to show that space is finite, we can never be sure that there isn’t space outside that system. Practically, there always is. As a result, we can’t ever really understand our universe. For how can we understand something if we don’t know its dimensions, and if we don’t even know where it is? Perhaps other universes have other systems, and perhaps their systems are changing ours all the time. Perhaps other dimensions cut through ours and link it to an infinity of dimensions. Or perhaps there’s no system that links universes or dimensions. Perhaps there’s no fourth dimension at all. Perhaps this escape from meaning goes on forever.

Mateo thinks that there should be a religion called Perhaps. For what bothers him more than two lines that in theory never meet is a single line that in all likelihood will never end.

In his treatise, The Infinite Line, Granlineas the Mystic argues that if there’s one single line, then there are an infinite number of single lines. And each of these lines moves in two directions. Mateo applies this to the trajectory of our beautiful Argentinian, who at this point might be called Andromeda, or Anunitum, the Babylonian antecedent of Andromeda, or Anuradha, the ‘sweet and delicate’ constellation in Indian astronomy. Every step she takes can be matched by a step in the opposite direction. Which means that the further she goes in any direction, and in understanding the places she finds there, there will be an equal and opposite number of things that she will neither encounter nor understand. Every step she takes is one step away from understanding not only what she left behind, but also from understanding what was behind what she left behind. Multiply those two trajectories by infinity, because if there’s one single line, then there are an infinite number of single lines.

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Andromeda left the circle at dead north, where 360 meets Zero, where fullness meets emptiness. Yet where in her northern trajectory is Shakespeare’s pole star, that “ever-fixed mark, / whose worth’s unknown although his height be taken”?

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Next: 🔭 Alas, Poor Yorick

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