Cartesian Chaos
In the town of Cartesia lived the people of Pure 3. They were a close-knit community, and a more tranquil and happy habitat would be hard to find in the whole world. It was into this idyllic harmony that the MEI headquarters decided to introduce a group of vectors from a distant galaxy. A Community Integration Unit was set up to help the newcomers to fit in. The work of the CIU consisted mainly of inspecting various cases, and if any of the immigrants did not find their conditions sufficient for absolute integration, the CIU saw to it that the estate agencies made suitable substitutions.
However, all was not well in Cartesia. There were as many variable inputs as there were inhabitants, and whilst every effort was made by the CIU to separate them into smaller, more manageable tasks, they found that the newcomers were impossible to integrate. Though no one wanted to put it explicitly, it was worming its way into many minds that a major conflict was on its way.
The story begins in Newton Park, on a brilliantly sunny day in June. Sin x and Cos x were gazing at their reflections in the x-axis, enjoying a relaxing game of Pass the Derivative. Round and round it went, until SecSquared x, who was lying under a nearby tree trying to get a Tan, felt quite dizzy. It was fixed in her mind that integration went one way and differentiation the other ('the secret of reciprocating motion', thought the reciprocal function proudly). She was about to shout a comment to the players, when CosecSquared x, her constant companion, saw her mouth open and hastily made lunch the subject.
"I'm starving," she announced, "anyone fancy a Cos pie?"
"Yeah, it must be about lunchtime," replied her friend. "Let's go to the take-away. I think I'll have power of sin with cos."
But just as they were clambering lazily to their feet, an infuriated voice screeched from a nearby tree:
"PI!!"
This was accompanied by the appearance of a large vector, leaping to the ground in indignation.
"Blasphemy! You take the name of Pi, our Plane, the Mother of All Planes, may she live forever, in vain? What in Oxyz's name is wrong with 180?"
"I really hate vectors," mumbled SecSquared x. "Always looking for another argument. As if one each wasn't enough!"
Meanwhile, Sin x, seeing one of those irksome vectors, alone, insulting radians in such an overt manner, had left his friends and bounded over to deal with the troublemaker.
"180 is the most disgusting number I ever heard!" he began. The vector shouted back,
"At least it's rational! You're off your trolley!"
Sin x fished for a comeback, but before words came to him, a young Scalar Product stepped out from nowhere and cleared her throat as if about to deliver a formal speech. Scalar Products were an enigma in Cartesia. They seemed to have come from among the vectors, but were certainly not vectors themselves, and appeared to neither agree nor disagree with either side. They generally kept to their own business, and were thought of as rather supercilious.
This particular Scalar Product certainly confirmed this view as she said, in an affected tone,
"Pi is the 16th letter of the Greek alphabet, used extensively in many branches of Mathematics: to denote a constant, transcendental and irrational, but equal approximately to 3.14159265358979323846 264338327950288, essential to trigonometry and circle theory, being the ratio of any circle's circumference to its diameter. It is also used to denote the function whose output is the number of primes up to and including the input, a symbol representing a plane, a product (finite of infinite), homotopy or a permutation. 2 pi radians is defined as the number of radians in a complete circle."
"For the respect of dx, stop being so pious!" exclaimed Sin x, but as he turned to walk haughtily off, to his terror his eyes were met by an entire gang of vectors. They had emerged from their hiding places and surrounded him during the scalar product's conveniently lengthy definition of pi.
Never had such panic seized Sin x. He was rooted to his axis, and was wavering and trembling so much that he thought he must have squared himself. After what seemed an age of the vectors' stony gazes boring into him, relief came to Sin x when he realised that his friend, Cos x, had acted quickly. He had rounded up all the radians he could find (he saw that this was no time for underestimation of the danger his friend could be in), and was leading them to face the villainous vectors.
"Let's sort out this mess once and for all!" bellowed Cos x. He spoke in a strong voice that Sin x had never heard before, but it sent a delicious tingle through his entire body, as deep as the roar of approval from the radians that followed. Every radian flung himself at the mass of vectors, and in seconds a bloodthirsty battle was raging. Blinded by visions of victory and justice, the vectors threw punches left, right and centre, sending the poor radians flying haphazardly to the ground. But the radians fought solidly, ignoring smarting grazes and throbbing bruises, and gradually began to push the vectors back and back.
Suddenly, there came an earsplitting whiz overhead, and everyone ducked. "What was that?" Yelled half a dozen voices in confusion. Another whiz, and a wail from the Integral of Cos3x. Several radians rushed to the integral's side, and to their horror they saw her beginning to break out in a rash. A thickening bulge of pus oozed from a wound in her side. She had dropped to the ground, turning deathly pale and fighting for breath. It was then that the radians realised what was happening. The vectors were hurling minus signs at them, and the young integral had been hit.
"Back, back!" called Cos x, who knew that minus signs were deadly to any radian. The terrified army began to stumble away, but before they could blink the vectors were upon them like a hailstorm, unmercilessly raining down blows. The radians were locked in a shuddering scrum, most giving up all hope of seeing the light of day again
But their brave leader Cos x was left behind, still in single combat with the chief of the vectors. It was a war of words rather than punches.
"You.you.horrid, pointy.urgh!" Screamed Cos x, unable to contain himself. "You only think in one direction - yourself! You don't deserve to live here with the respectable branches of Pure Maths! Get back to the mechanics book where you belong! Go on, get out of my sight! You."
During this tirade the vector resolved himself. Yes, he would. He felt a hint of a tremble as he prepared to hammer home the ultimate insult.
"Your mother was.was.a.a STATISTIC!!"
A shocked silence settled over the whole scene. Never had anyone heard such an enormous affront. Everyone fell back, stunned and dazed by what they had heard. The CIU Police saw their chance to act. They had surrounded the scene, and they deftly and efficiently arrested everyone in sight.
When the dust had settled, a CIU inspector came forward and demanded of all the radians present who had started it. There was some confusion over this, but eventually the general opinion emerged that it had all started at the positive x-axis, and proceeded in a counterclockwise direction.
"Right," said the inspector, "we shall put the positive x-axis on trial tomorrow morning, and the rest of you can go." With exclamations of surprise the vectors and radians left the scene.
~~**oOo**~~
The following morning, the small town courthouse was bulging at the seams. Were the vectors finally going to be removed from Cartesia? Everyone rose to their feet as His Honour the General Tangent entered and lumbered to his seat. He was enormously fat, and had a reputation for being uncommonly stupid, but on this morning there was nothing but respect for the Judge.
The Positive X-axis was ushered into the dock by two burly inspectors. He looked miserably flat and dejected, as anyone in his position might.
"Read the defendant the first four terms of the Binomial Expansion" Instructed the judge, somewhat impatiently.
"1-plus-nx-minus." began the usher's bored tone.
"Plus, not minus!" thundered the judge. "You're fired. Swear him in, then leave, and don't come back."
The usher obeyed, and slunk silently out of the room.
"Call the first witness!" bellowed His Honour. "The Negative X-axis!"
The Negative X-axis eased himself onto the witness stand, and was sworn in by the Judge himself. The prosecutor began his cross-examination in the standard form.
"How exactly did yesterday's.incident begin?" He asked, getting straight to the point.
"Well, I can't give any information on the actions of the Positive X-axis, because I was looking the other way." Drawled the axis.
"Did you not hear anything?"
"Since when did axes have ears?" Said the Positive X-axis, in mock- annoyance.
"Point taken. No further questions. You may stand down."
The Positive X-axis obediently left the witness stand, chuckling to himself.
"Call Cos2Theta!" roared the Judge.
There was a pause, and then a young woman stepped into the room and nervously approached the stand.
"Confirm your details," said the judge. "Are you Cos2Theta?"
"Well, I guess so," mumbled the girl, expressionlessly.
"What's that? Speak up, for goodness' sake! You guess so? Well, come on, either you are or you aren't. Dx, if you can't give me a straight answer to that, a fat lot of good you'll be giving evidence!"
Poor Cos2Theta was trembling and staring at the stone floor of the courtroom.
"Well, to be perfectly honest," she said, forcing herself to speak a little louder, "I'm having a bit of an identity crisis."
Immediately she said this, a murmur of sympathy rippled round the courtroom. The judge spoke to her more kindly.
"Do you swear to tell the truth?" He asked.
"I.I." Cos2Theta was in emotional shreds after having admitted her troubles to so many people. She had a lump in her throat, and all she could give the judge was such a pitifully distressed look that he had to ask her to stand down.
The prosecutor frowned. Those two were the only witnesses he had. He scanned the courtroom, and his eye fell on a strange-looking woman sitting at the origin of a graph. "She'll do," he thought to himself, beginning to edge himself towards her chair.
"WHAT are you doing?" roared the judge.
"Nothing, Your Honour," said the Prosecutor, a little too sheepishly.
"Nothing?" yelled the judge, "Leave the room and don't come back. As for you," turning to the defendant, "haven't I seen you here before?"
"Yes, your Honour," replied the axis.
"Drunk deriving, wasn't it?"
"Yes, your Honour"
"'Yes your Honour, yes your Honour,'" mimicked His Honour "I find you guilty. I sentence you to complete Exercise 3G, to perfection, by next Monday." The axis winced, as did most other people in the room. "And in terms of parameters," continued the judge, "a curfew will be imposed for the following six weeks. Case dismissed."
As soon as the defendant had been removed from the room, a chorus of disgruntled voices broke out.
"What's going on?"
"Justice for the vectors!"
"Justice for maths!"
"Justice for twos!"
"What?"
"Get rid of these scum!"
"Isn't there going to be some inquiry."
"Silence!" barked the Judge. "No, there is going to be no inquiry. Last night I met with the CIU myself." silence fell. "And we agreed that there is no particular solution to the problem of integrating this new community. The area inhabited by these vectors is no longer to be defined within Cartesia."
The chief of the vectors then rose, just in time to stem a roar of delight from the radians. "I wish to say," he said calmly, "that we vectors have decided we do not need a fixed location. We will all happily position ourselves anywhere in space, as long as there is room for our magnitudes. We are highly grateful to Cartesia for its hospitality to us over the past few months, and it is not without regret that I bid you all farewell." And with that, he led the vectors out of the room, and out of the courthouse to where a great crowd was waiting. Every vector in Cartesia had assembled for the great Exodus. Triumphantly, they followed their chief down the long road, over the hill and off into the dusty distance, leaving behind them the once more peaceful and harmonious community of Pure 3.
In the town of Cartesia lived the people of Pure 3. They were a close-knit community, and a more tranquil and happy habitat would be hard to find in the whole world. It was into this idyllic harmony that the MEI headquarters decided to introduce a group of vectors from a distant galaxy. A Community Integration Unit was set up to help the newcomers to fit in. The work of the CIU consisted mainly of inspecting various cases, and if any of the immigrants did not find their conditions sufficient for absolute integration, the CIU saw to it that the estate agencies made suitable substitutions.
However, all was not well in Cartesia. There were as many variable inputs as there were inhabitants, and whilst every effort was made by the CIU to separate them into smaller, more manageable tasks, they found that the newcomers were impossible to integrate. Though no one wanted to put it explicitly, it was worming its way into many minds that a major conflict was on its way.
The story begins in Newton Park, on a brilliantly sunny day in June. Sin x and Cos x were gazing at their reflections in the x-axis, enjoying a relaxing game of Pass the Derivative. Round and round it went, until SecSquared x, who was lying under a nearby tree trying to get a Tan, felt quite dizzy. It was fixed in her mind that integration went one way and differentiation the other ('the secret of reciprocating motion', thought the reciprocal function proudly). She was about to shout a comment to the players, when CosecSquared x, her constant companion, saw her mouth open and hastily made lunch the subject.
"I'm starving," she announced, "anyone fancy a Cos pie?"
"Yeah, it must be about lunchtime," replied her friend. "Let's go to the take-away. I think I'll have power of sin with cos."
But just as they were clambering lazily to their feet, an infuriated voice screeched from a nearby tree:
"PI!!"
This was accompanied by the appearance of a large vector, leaping to the ground in indignation.
"Blasphemy! You take the name of Pi, our Plane, the Mother of All Planes, may she live forever, in vain? What in Oxyz's name is wrong with 180?"
"I really hate vectors," mumbled SecSquared x. "Always looking for another argument. As if one each wasn't enough!"
Meanwhile, Sin x, seeing one of those irksome vectors, alone, insulting radians in such an overt manner, had left his friends and bounded over to deal with the troublemaker.
"180 is the most disgusting number I ever heard!" he began. The vector shouted back,
"At least it's rational! You're off your trolley!"
Sin x fished for a comeback, but before words came to him, a young Scalar Product stepped out from nowhere and cleared her throat as if about to deliver a formal speech. Scalar Products were an enigma in Cartesia. They seemed to have come from among the vectors, but were certainly not vectors themselves, and appeared to neither agree nor disagree with either side. They generally kept to their own business, and were thought of as rather supercilious.
This particular Scalar Product certainly confirmed this view as she said, in an affected tone,
"Pi is the 16th letter of the Greek alphabet, used extensively in many branches of Mathematics: to denote a constant, transcendental and irrational, but equal approximately to 3.14159265358979323846 264338327950288, essential to trigonometry and circle theory, being the ratio of any circle's circumference to its diameter. It is also used to denote the function whose output is the number of primes up to and including the input, a symbol representing a plane, a product (finite of infinite), homotopy or a permutation. 2 pi radians is defined as the number of radians in a complete circle."
"For the respect of dx, stop being so pious!" exclaimed Sin x, but as he turned to walk haughtily off, to his terror his eyes were met by an entire gang of vectors. They had emerged from their hiding places and surrounded him during the scalar product's conveniently lengthy definition of pi.
Never had such panic seized Sin x. He was rooted to his axis, and was wavering and trembling so much that he thought he must have squared himself. After what seemed an age of the vectors' stony gazes boring into him, relief came to Sin x when he realised that his friend, Cos x, had acted quickly. He had rounded up all the radians he could find (he saw that this was no time for underestimation of the danger his friend could be in), and was leading them to face the villainous vectors.
"Let's sort out this mess once and for all!" bellowed Cos x. He spoke in a strong voice that Sin x had never heard before, but it sent a delicious tingle through his entire body, as deep as the roar of approval from the radians that followed. Every radian flung himself at the mass of vectors, and in seconds a bloodthirsty battle was raging. Blinded by visions of victory and justice, the vectors threw punches left, right and centre, sending the poor radians flying haphazardly to the ground. But the radians fought solidly, ignoring smarting grazes and throbbing bruises, and gradually began to push the vectors back and back.
Suddenly, there came an earsplitting whiz overhead, and everyone ducked. "What was that?" Yelled half a dozen voices in confusion. Another whiz, and a wail from the Integral of Cos3x. Several radians rushed to the integral's side, and to their horror they saw her beginning to break out in a rash. A thickening bulge of pus oozed from a wound in her side. She had dropped to the ground, turning deathly pale and fighting for breath. It was then that the radians realised what was happening. The vectors were hurling minus signs at them, and the young integral had been hit.
"Back, back!" called Cos x, who knew that minus signs were deadly to any radian. The terrified army began to stumble away, but before they could blink the vectors were upon them like a hailstorm, unmercilessly raining down blows. The radians were locked in a shuddering scrum, most giving up all hope of seeing the light of day again
But their brave leader Cos x was left behind, still in single combat with the chief of the vectors. It was a war of words rather than punches.
"You.you.horrid, pointy.urgh!" Screamed Cos x, unable to contain himself. "You only think in one direction - yourself! You don't deserve to live here with the respectable branches of Pure Maths! Get back to the mechanics book where you belong! Go on, get out of my sight! You."
During this tirade the vector resolved himself. Yes, he would. He felt a hint of a tremble as he prepared to hammer home the ultimate insult.
"Your mother was.was.a.a STATISTIC!!"
A shocked silence settled over the whole scene. Never had anyone heard such an enormous affront. Everyone fell back, stunned and dazed by what they had heard. The CIU Police saw their chance to act. They had surrounded the scene, and they deftly and efficiently arrested everyone in sight.
When the dust had settled, a CIU inspector came forward and demanded of all the radians present who had started it. There was some confusion over this, but eventually the general opinion emerged that it had all started at the positive x-axis, and proceeded in a counterclockwise direction.
"Right," said the inspector, "we shall put the positive x-axis on trial tomorrow morning, and the rest of you can go." With exclamations of surprise the vectors and radians left the scene.
~~**oOo**~~
The following morning, the small town courthouse was bulging at the seams. Were the vectors finally going to be removed from Cartesia? Everyone rose to their feet as His Honour the General Tangent entered and lumbered to his seat. He was enormously fat, and had a reputation for being uncommonly stupid, but on this morning there was nothing but respect for the Judge.
The Positive X-axis was ushered into the dock by two burly inspectors. He looked miserably flat and dejected, as anyone in his position might.
"Read the defendant the first four terms of the Binomial Expansion" Instructed the judge, somewhat impatiently.
"1-plus-nx-minus." began the usher's bored tone.
"Plus, not minus!" thundered the judge. "You're fired. Swear him in, then leave, and don't come back."
The usher obeyed, and slunk silently out of the room.
"Call the first witness!" bellowed His Honour. "The Negative X-axis!"
The Negative X-axis eased himself onto the witness stand, and was sworn in by the Judge himself. The prosecutor began his cross-examination in the standard form.
"How exactly did yesterday's.incident begin?" He asked, getting straight to the point.
"Well, I can't give any information on the actions of the Positive X-axis, because I was looking the other way." Drawled the axis.
"Did you not hear anything?"
"Since when did axes have ears?" Said the Positive X-axis, in mock- annoyance.
"Point taken. No further questions. You may stand down."
The Positive X-axis obediently left the witness stand, chuckling to himself.
"Call Cos2Theta!" roared the Judge.
There was a pause, and then a young woman stepped into the room and nervously approached the stand.
"Confirm your details," said the judge. "Are you Cos2Theta?"
"Well, I guess so," mumbled the girl, expressionlessly.
"What's that? Speak up, for goodness' sake! You guess so? Well, come on, either you are or you aren't. Dx, if you can't give me a straight answer to that, a fat lot of good you'll be giving evidence!"
Poor Cos2Theta was trembling and staring at the stone floor of the courtroom.
"Well, to be perfectly honest," she said, forcing herself to speak a little louder, "I'm having a bit of an identity crisis."
Immediately she said this, a murmur of sympathy rippled round the courtroom. The judge spoke to her more kindly.
"Do you swear to tell the truth?" He asked.
"I.I." Cos2Theta was in emotional shreds after having admitted her troubles to so many people. She had a lump in her throat, and all she could give the judge was such a pitifully distressed look that he had to ask her to stand down.
The prosecutor frowned. Those two were the only witnesses he had. He scanned the courtroom, and his eye fell on a strange-looking woman sitting at the origin of a graph. "She'll do," he thought to himself, beginning to edge himself towards her chair.
"WHAT are you doing?" roared the judge.
"Nothing, Your Honour," said the Prosecutor, a little too sheepishly.
"Nothing?" yelled the judge, "Leave the room and don't come back. As for you," turning to the defendant, "haven't I seen you here before?"
"Yes, your Honour," replied the axis.
"Drunk deriving, wasn't it?"
"Yes, your Honour"
"'Yes your Honour, yes your Honour,'" mimicked His Honour "I find you guilty. I sentence you to complete Exercise 3G, to perfection, by next Monday." The axis winced, as did most other people in the room. "And in terms of parameters," continued the judge, "a curfew will be imposed for the following six weeks. Case dismissed."
As soon as the defendant had been removed from the room, a chorus of disgruntled voices broke out.
"What's going on?"
"Justice for the vectors!"
"Justice for maths!"
"Justice for twos!"
"What?"
"Get rid of these scum!"
"Isn't there going to be some inquiry."
"Silence!" barked the Judge. "No, there is going to be no inquiry. Last night I met with the CIU myself." silence fell. "And we agreed that there is no particular solution to the problem of integrating this new community. The area inhabited by these vectors is no longer to be defined within Cartesia."
The chief of the vectors then rose, just in time to stem a roar of delight from the radians. "I wish to say," he said calmly, "that we vectors have decided we do not need a fixed location. We will all happily position ourselves anywhere in space, as long as there is room for our magnitudes. We are highly grateful to Cartesia for its hospitality to us over the past few months, and it is not without regret that I bid you all farewell." And with that, he led the vectors out of the room, and out of the courthouse to where a great crowd was waiting. Every vector in Cartesia had assembled for the great Exodus. Triumphantly, they followed their chief down the long road, over the hill and off into the dusty distance, leaving behind them the once more peaceful and harmonious community of Pure 3.