The Last Rectangle Read online

  The lender stood against the wall of someone’s house and watched the empty square. Soon he heard someone walking a long way behind him up that street. It was the borrower coming towards the square and it was too late for him to retreat so he continued walking. As soon as he reached his lender, they walked together in silence and towards the middle of the empty square.

  In the middle of the square, they stood further apart than people usually stand when conversing - - - but not further than needed for reasonable social discourse. No one said anything.

  Soon, the borrower reached into his pocket and handed the lender a stack of notes. The lender did not need to count it to know that it was short of what was lent. He was grateful that something was being settled after all. He just suggested that they meet in the square next Thursday. The borrower nodded in agreement but did not leave. They stood apart as they had and for around half an hour. Then they left without saying much. Something of the emptiness of the village square had ensured that the transaction was willingly completed and in a gentlemanly manner. Virtual eyes in silent houses watching lenders and borrowers could have been the reason they behaved humanely towards one another.

  It must have been the lender or the borrower, presumably the lender and out of gratitude, who let out the word that this was a suitable place to settle debts. On the coming Thursday, they both came using different routes and found other persons in the square. Transactions were completed at almost the same time, as one sole cricket can launch the dawn chorus.

  In time, lenders and borrowers would gather in the square and soon separate into two groups as oil separates from water. Each group would eye the other in silence. They all felt they should wear dark clothes. The scene must have looked quite theatrical if watched from one of the windows overlooking the square. Conveniently, none of the villagers peeked through their curtains or even walked around during that period.

  Very soon and not in unison, hands reached into their pockets. Someone would start by handing over a bundle. Once delivered to its lender, both lender and borrower stayed within their congregations. No sound was made. No talk was needed. Short settlements were implicitly rescheduled for the coming Thursday. There must have been a natural countdown period that caused everyone to stay for half an hour after the handing over and then leave.

  On one particular Thursday, a situation arose that became an issue. There were persons who had borrowed from someone and lent to another at the same time or at least for settlement on that Thursday. It was difficult for them to know with which group to stand. Some migrated towards the lender group but were shunned back to the borrowers. It was soon implicitly agreed that such persons had to settle their debts first then join the lenders. This would cause a commotion to take place so that along with seeing the dark colors of the suits, the villagers could now hear the scratchy sound of shuffling shoes. When the villagers cleaned the gravel from the square this only resulted in shoes emitting deeper bass sounds.

  Gradually, groups of five or six started to form containing both borrowers and lenders. That broke up the initial segregation. These individuals were mostly borrowing and lending within that group and found it convenient to stay together. But soon this would create a confusion as there was no agreed upon location for these groups.

  Gone was the segregation into two groups where lenders and borrowers stood. These new groups started floating around and it was inevitable they would collide with one another. Boundaries were contravened. Conflicts arose. Little groups created their own ethical codes and considered it offensive when others did not respect them. This broke the boundaries further and soon, individuals started ceding from their groups. The slow return to individual processing of debts meant that the square became empty again and maltreatment of lenders and borrowers by one another resumed.

  The Last Rectangle

  The Archway

  Throughout the years, the painting stood in the archway of the main Madrasa5 in Tripoli. It was a monument to what would one day be seen as artistic treason. The fine wood of the easel would rot and fray and be replaced by skillful craftsmen, but the painting remained solid and resolute. The Madrasa had a rectangular court in its middle. It was sometimes used for lectures when the air was dry and conducive to learning. Around the court there was an archway made up of 8 arches on the long side of the court and 5 on the short. The inside of the archway led to the doors of the lecture rooms and the studios. As was usual with such Madrasas, the building was made up of three floors most of which were used as lecture rooms or studios while those on the top floor were the quarters of tutors, generally those who had come from other cities.

  Although Islamic in its architecture and design, the design motif of this particular Madrasa was based on Phi, the golden ratio or the divine proportion associated with the Greek sculptor Phidias. The long side of the court was 1.618 times the length of the short side. Other relations in the Madrasa were supposed to be based on this same ratio. However, a fault crept into the design of the Madrasa at the very beginning of its life. It remained an innocent crack in its heart, treacherous but not fatal. This was not unusual. Many designs or artistic works contain a fault that seems so naturally correct to the eye of the designer. Some contain more than one. Others contain faults not caused by the designer but part of an accepted view. Many contain faults caused by eager designers bent on correcting prior faults. Few questions are asked at design time. The rest of the work grows basing itself on the main idea but grafting the fault into itself in such a natural way that the principle of the fault becomes part of the work, sucking its logic inwards like division by zero but not quite able to overcome the original subversion.

  A bit more about the golden ratio or what renaissance artists called the Divine Proportion. Interestingly, Phidias did not call it the golden ratio nor was it known as Phi in his time. He started observing the many passionate traits of this wondrous ratio all around him. He used it in many sculptures. It was some mathematician early in the 20th century who, in honor of Phidias, gave the name Phi to the Divine Proportion or the Golden Ratio. That it deserves to be called golden or divine is beyond doubt. Phi is an irrational number which means that it cannot be expressed as the ratio of two integers like 2 / 3 or 5 / 6. Usually, irrational numbers have digits to the right of the decimal which are not known to end.

  To get the golden ratio, draw a line AB that is one unit long. Find a point C on the line so that the ratio of the longer to the shorter segment would be the same as the ratio of the whole line to the longer segment. Such a point can be found only if the ratio is 1.618. Sometimes, 0.618 is referred to as the golden ratio because dividing 1 by 1.618 gives 0.618. This also means that Phi is the only number whose reciprocal is 1 less than it is!

  Golden rectangles are even more interesting. Say you had a rectangle with 1 unit for its width and 1.618 for its length. If you measure 1 on the long side and mark off a square, you will be left with a smaller rectangle with the measurements 1 x 0.618. This would also be golden since the division of 1 by 0.618 gives Phi. Measuring off another square in this new smaller rectangle will leave you with another golden rectangle with measurements 0.382 by 0.618 which also gives Phi when divided by one another. And so on. If arcs of circles are drawn in each square, a helix will result, a curve that is commonly found in nature. And so Phi intimately lines our geometric thought and biologic realities from pentagrams to sunflowers.

  And so it was not a small reason that helped the architects decide that the whole Madrasa will have Phi as its basic design motif. As the architects started the design, they defined the longer side of the court to be the equivalent of today’s 60 meters which meant that the shorter side had to be 60 / 1.618 = 37.08 meters. That was their starting reference, a perfect golden rectangle. The number of arches on the long side was 8 while that on the short side was 5. This was not a strictly Islamic convention but the Architects used this number since the ratio of 8 / 5 = 1.6 was close enough to the
Golden Ratio but not exactly it. Since the long side of the court had a length of 60, the width of each arch was 7.5 meters. On the short side, the court had a width of 37.08. Dividing this by 5 gave the arch a length of 7.416, a little less than the width of the arches on the long side. A fault of 8.4 cm but one that few would notice.

  To proceed, the architects then sketched a strip around the court which was to be the archway. Since they meant this strip to have a constant width of 3 meters, they overlooked the fact that adding 3 meters to the length and 3 meters to the width of the rectangle of the court would result in a new rectangle with 60+3 = 63 and 37.08+3 = 40.08 as its dimensions. Dividing 63 by 40.08 results in 1.571 which was not Phi, but a smaller proportion. What they should have done was to add 3 meters to the width and 3 x 1.618 meters to the length. The outer rectangle would have remained golden. In effect, the archway’s outside rectangle became a rectangle that was a bit more squarish than the golden rectangle. Again, as the architects measured off another strip of 10 meters to place the rooms around the archway, the same error crept into the design of the outer edges of the building. This resulted in a further decrease of the ratio to (60+3+10) / (37.08+3+10) = 1.457 making the outer perimeter of the Madrasa an even more squarish rectangle. Of course, the gardens around the Madrasa would be fenced off by a further rectangle which was offset by the width of the garden so that the ratio was even smaller, reaching around 1.208.

  As the ratio of successive rectangles tended towards 1 and away from Phi, the rectangles tended towards being squares. This steady transformation was completely missed by the architects. It was not till years later when a similar transformation but one that was opposite in direction took place that the inherent design fault was recognized.

  Year 1618 - The Teacher

  The painting was started in the early years of the 17th century, a few years after the Madrasa was built, 1618 to be precise. The painter was a tutor in the Madrasa, well liked by his students for his passionate classes and his involvement in his own art unlike the other dry theoreticians. In the middle of many of his classes, he would illustrate points by taking time to sketch, draw or paint elaborate segments that later on wove themselves into his paintings. On this particular occasion, the painter in him overpowered the tutor and as he started lecturing on the golden ratio, he brought in an easel and while talking, busily cut and hammered small bars of wood to make a golden rectangle which he covered with canvas. He set it up on the easel. His plan was to paint the Madrasa from the viewpoint of someone standing in the archway in the middle of the south wing and looking towards the north wing across the court. There it was, the boundary of the painting was a golden rectangle and it housed the court which was also a golden rectangle.

  With the first few strokes on the lower edge of the painting, he sketched the steps leading from the archway to the court. He now painted the steps. He then continued his sketching on both sides of the painting never touching the middle area, painting over the sketching when he felt satisfied with the result. The two vertical sides started taking shape as he painted parts of the archway on either side. Above the archway, he painted a few floors. He finally reached the top edge of the painting where he figured in some clouds. The middle remained blank, unusual as a practice in his time. The class finished without any conclusion. The students dragged themselves out and reappeared promptly the next day to find a note on the door saying that on such a lovely day in May, the teacher wished to conduct the class out in the open, on the steps of the south wing. The students went down in a hurry. The teacher was bracing the easel in the middle of the archway. Knowing the architecture of the Madrasa intimately, he oriented the easel so that it faced north, allowing him to get the best possible lighting on an afternoon where the air could be tasted in the mouth.

  As the students settled down, some in the court and some on the steps leading to it, he began to touch up the painting in silence so that by now, the strip surrounding the edges was completely painted. The first bead in our story fell out of its Masbaha6 when the teacher suddenly clutched his chest in pain flinging the brushes and charcoals in his hand far and wide. He fell in one bundle onto the floor at the foot of the easel and must have died instantaneously. His students rushed to carry him out leaving the easel and the painting in the archway.

  Days passed and in a communal gesture of admiration and respect for their teacher, the students never even discussed the possibility of completing the painting. Instead and as a commemoration, they agreed to fix the easel and painting just as they were left by the teacher. A few of them dug three little holes in the floor and placed the legs of the easel one in each, cementing them firmly so that the easel became a permanent fixture of the archway. The painting was also permanently fixed to the easel. Mostly empty, its presence in the archway was monumental and all because the teacher had intended to steal the perfect northern light on that dry May afternoon. The painting remained there, an idea in the mind of a painter no longer there to complete it.

  Years passed and the students who were present on that afternoon graduated and left the Madrasa giving way to others who got to know about the painting by hearsay. Gradually, the original students grew older and one by one died till there was no one around who had a direct experience of that afternoon. Fewer and fewer people mentioned the painting and its sanctity dwindled, but it stood there.

  Year 1762 - The Foreign Painter

  The second bead of the Masbaha fell in a completely different age, 144 years after the death of the teacher. In the latter decades of the 18th century, in 1762 to be precise, on a cold winter’s day yet sunlit and clear, a man with a long ragged hard coat and a graying beard, obviously coming from a colder northern climate, walked into the town with determined steps and aimed at the Madrasa without asking any questions. He knew where everything was. He entered the front gate in the west wing, walked across the grass court and settled his bags right at the foot of the easel. He reached into one of them with his thick arm and pulled out a broken loaf of bread. As he chewed on it, he measured the painting with a fixed stare, as if he were confirming what he already knew. He placed the half finished loaf back in the bag and wiped his hands on his coat. He eyed the edges carefully and pulled out a sketching charcoal. He measured a distance from the top edge inwards using two fingers placed together. With the charcoal, he ran a line around the four edges marking off a smaller rectangle inwardly offset from the edge of the painting. Already, the travesty was in action. Here was a painter who had made the same mistake as the architects but in the opposite direction. By reducing both length and width by two fingers on each side, the new inside rectangle was not golden and for the same reason. The canvas was constructed as a golden rectangle. The drawn rectangle was slightly more elongated than a golden rectangle. The strip that he created with the charcoal ran around the edge of the painting like the archway around the court, realistic but not golden. It contained the initial material of our first artist, the only part that he had completed. In the middle, the painting was mostly empty. The foreign artist pulled out his brushes and paints, mixed them up himself without assistance and started to paint the inner rectangle leaving the outer strip as painted by the first artist. His application was manic. His pace was that of a silent hunting lion who knew where his prey was. No one thought of questioning his intrusion or of even stopping him. Each afternoon as the light faded, he would clean up around the easel, put his tools in his bag, pull out his loaf, chew a few bites as he looked hard at the painting and leave the court the way he came.

  This went on for a few afternoons until he finished his own painting within the inner rectangle. In his mind, he had closed the painting, repaired the fault, but unknown to him at this stage, he had launched a subversion that opened up the initial rectangle. He left the town with the same determined steps as when he came in. Again, a handful of students witnessed the whole process and they in turn grew older and died away until fewer and fewer people mentioned the painting.


  Year 1851 - The Soldier

  In the early parts of the seventeenth century, 89 years after the foreign painter, sometime in 1851 to be precise, the third bead of the Masbaha fell. One day, an army from the north moved into the outskirts of Tripoli. No one knew what the army was doing there and there was no real cause for alarm for had there been anything to worry about, it would have been over by the first day and the army would have occupied Tripoli in no time. It did not. It stayed like a large beast camped on the northern bank of the river encircling the northern periphery of the town, but the army had something to worry about as it feared attacks in the night. The solders slept through most of the day and kept watch in the dark hours. The town was totally ignored except for some army supply personnel and some higher ranking officers. These latter met with the notables of the town mostly as a matter of courtesy and generally to reassure them that there was no need to worry about occupation or any other interference.

  In the night you could hear commotion and movement by the river. Groups of soldiers guarded the edges of the camp while others polished swords, sewed uniforms, tents and repaired saddles. On one of those nights where the stars shone with strength, a young soldier with a reddish beard and a round face stood up, opened his arms to the sky and started shrieking, much like a mouse at first, then he toned down and simply churned out a long groan, a mixture of a shriek and a call. He bounced up and down on his heels and danced round and round and before anyone could stop him dashed into the river, still bouncing up and down, crossing it fast. He raced into the town and straight to the Madrasa. There, he ran amok in the fields and woke everyone up with his strange moan. He did this all night long until the west gate opened in the early morning. He rushed straight through the gate, across the court and stood in front of the painting, still bobbing up and down and still shrieking. He paused a while then dashed to the classes opening one door, peeking in and then closing it until he came across a studio where there were some brushes and paints. He grabbed what he could and rushed back to the painting. In an almost foreordained manner, he sketched a second smaller rectangle inwardly offset from the last one by two fingers, perpetuating the error made by the foreign artist. As the golden ratio was betrayed a second time, the new rectangle was becoming even more elongated. This time, the shrieking artist could not close his rectangle around blank space. This second rectangle covered the inner part of the painting already completed by the foreign artist. The soldier soon began painting over the foreign artist’s work. He painted throughout the day in a frenzy worthy of a soldier in battle. He did not stop with the failing light but continued painting in total darkness. Those watching him fell back and slept on the grass or even went home for a troubled sleep as they could still hear his shrieks till the early hours of the morning. Those that stayed and those that returned found the painting completed on top of the previous painting but within the second rectangle which meant you could now see two strips from two different paintings as well as the new painting. He finished his work and left the town only to find that the army had left as silently as it came. He ran after it shrieking and waving his arms. The students who saw the third artist kept their memories until they died and no one talked about the third painting any more.