Twilight of the Triangles

[Tritopea]

Twilight of the Triangles
Winner of the Hugo Award

The right hand of Tri-056 shakes with anticipation, intricate calculations flowing from the pointed tip of the pen it clutches. Finally, the nearly incomprehensible sea of symbols is complete, punctuated with a single number triumphantly enclosed in a furiously etched triangle of ink. The pen is cast aside, revealing the deep, red imprints on the hand formed from the pressure of its sharp, tetrahedral edges.

A look of self-satisfied triumph quickly dissolves into one of horror, weeks of delicate manouevring through new frontiers of stochastic analysis yielding a result at once mathematically beautiful and personally terrifying.

If only he were not sworn to the truth. They will surely have his head for this, another blasphemous sphere to be purged from this world.


Three quiet knocks, followed by a barely audible fourth, signal the moment he has been dreading. In reflection, their clandestine protocol was an unnecessary risk, but such matters were undoubtedly irrelevant now. With a heavy sigh, Tri-045 rises from his tripodic chair and opens the door.

A haggard man, head held low, sheaf of papers clutched in one hand, enters the room. He quickly glances behind him into the darkness outside and closes the door gently. Tri-056 has not slept well in weeks, at first because of the torments of his research and more recently the implications of its results. He places the papers onto a nearby table and slouches into the nearest chair.

Tri-045 takes a seat at an adjacent face, perusing the notes without understanding much. There isn't any need to look, since the conclusion is obvious enough from 56's state.

"I take it that you've completed the analysis?" Tri-045 stares into 56's balding head, his face now buried into the table.

"Yes - the projection is another 100 years at most. The least sustainable population is at least 1000 - not even close," 56 mumbles gloomily.

There is a long pause as Tri-045 considers this dire news. "So there is no way to rescue our people gradually?"

"No. We would have to abandon our principles entirely for... generations."

"And then, " their voices in unison now, in a mocking drone, "we would become impure, no better than the sideless circles."

Tri-045 shakes his head, placing the papers neatly on the table. He'd like to cast them theatrically, but disrespect to the mathematics wasn't worth it. "There must be something that I can suggest, perhaps an increase in the tolerances, or-"

56 looks up suddenly, glaringly hollowly in the direction of his colleague's face. "I said, no. We have proceeded to far, and now we are to be consumed by our own traditions." With that burst of desperate anger fading, he once again lays down his head and continues, now in a leaden whisper. "Ironic that our trajectory is to a point... ever triangular, except for the eternity of nothingness that awaits."

Tri-045 sits in silence for several minutes, reluctant to say the inevitable. "I have to report this."

"I know. I'm sorry. I wish I could will the mathematics to say otherwise."

"They won't remove you as they will me. Your role in this is only technical."

56 slouches deeper in his chair. "And if I survive, what can I do? Now that I have seen the end, my existence is pointless. I don't want to watch our civilization die."

"There are others." Consoling 56 is an impossible task, but 45 understands it to be important. A plan begins to form in his mind... "If we cannot be incremental, there is one other option."

This gets the attention of 56, who stands and looks directly at 45 for the first time. "Revolution is impossible. That much the Elders have right."

"No, not that." 45 shakes his head, anticipating the response. "Not in the sense of our glorious war against the circular oppressors," an ironic smile is forced by 45 to acknowledge that he too, finds such tales no more than myth. "But if we look to the rest of the world... perhaps with the help of others, we can reform."

"Do they exist?" A skeptical raise of an eyebrow indicates that 56 is now curious, a brief respite from the deep depression of the previous moment.

"I believe so. We have some evidence," 45 replies, emphasizing the conspiratorial "we."

Probabilities materialize in 56's mind, a reflexive and futile attempt to determine what the odds might be of success. But there is only a clutter of algebraic gibberish. He sweeps these thoughts from his mind, repeating to himself that they are irrelevant. "I do not share your optimism. But I respect your conviction - I will sleep. Tomorrow you will have your official report."

With that, 56 collects his papers and leaves. Could this path to reform actually work, wonders 45 as the door closes. His plan is only a single, preposterous idea, but perhaps there could be some way? Whatever it may be, one thing was certain: Once the Elders read his dreaded report, there would be no turning back, and so little time to act.

[Tritopea]

One Hundred and Five, Briefly One Hundred And Twenty

Synchronization approaches. In only two weeks, the entire population of Tritopea will gather at its ceremonial grounds for its most holy of rituals. Each Tritopean takes a place in the grand triangular figure drawn there, arranged according to the most ancient of traditions. Each of three Elders is positioned at a vertex, giving an elaborate speech on the (always positive) state of the nation. It is this annual event that enforces Tritopean's strict population constraints. The population must, with no exceptions, form a triangle. Only through the purest of forms can the population be infused with its energy and continue to function. At one time, Synchronization was a great feat of organization and discipline, but declining population has made the ritual a painful reminder of Tritopea's failures. For most of the Tritopeans, however, size is not a concern, so long as the triangular form is preserved.


Trees are one of a multitude of theological conundra that plague Tritopea - reconciling necessities and natural phenomena with their triangular vision. Though relentless topiary aggression, the flora of Taijitu could be molded into compliance. But branches and flowers and stalks cannot be trimmed so easily. Pruning trees to conform to the fractal branching of triangular degree, revered only by the Tritopeans and perhaps the occasional mad graph theorist, is both a science and a holy art perfected by Tritopean groundskeepers over many generations.

A lattice of pyramidal evergreens, hollowed in the center, forms the border of one of Tritopea's well-maintained parks. A path winds through the copse, suitable for an afternoon stroll. Arranged precisely in the center are benches (unsurprisingly, three in number).

On one of the benches, incongruously situated are a mere two Tritopeans - numbered 67 and 83 - engaged in a whispered conversation. There is no one else nearby this morning, as most are either busy at work or meditation. Perhaps most importantly, there are so few Tritopeans that its parks are attended predominately by those assigned to maintain them.

67 turns to his companion with a puzzled expression. "It's curious that no one remember the visit," he remarks. Long ago, a delegation of sorts from a foreign nation arrived in Tritopea, but it was turned away. The drastic implications of the visit - that the populated world extended far beyond their island - were of enough concern to warrant secrecy. After this moment the world was convinced of Tritopea's irrelevance (and insanity), and future explorers made certain to avoid the very conspicuously triangular island. Records of the visit were kept, however, circulating occasionally as rumor and finally surfacing in the inquisitive Tri-083.

83 has become somewhat of an expert on the subject over the last month. His position in one of Tritopea's great Libraries (as the Tritopeans find acquisition of knowledge quite important, their libraries are surprisingly large compared to their population) has enabled furtive access to those secret records without much trouble.

He replies matter of factly, with a slightly bemused shrug. "The ones who welcomed him are dead, and it wasn't deemed information worth passing on. We should be thankful that our Most Venerated Elders at least at had the foresight to obtain a map."

"I can't imagine how they must have felt, traveling so far only to be turned away. It is so inhospitable." Of course, Tritopeans are not known for either their imagination or hospitality, so the rejection and 67's lack of cross-cultural empathy were only to be expected. I would have done the same, reflects 67 to himself.

87 nods. "Perhaps they will return with more appropriate ships." Then, adding with irony, "We were clear enough on that point."

Sensing the degeneration of the conversation into existential bitterness, 67 smirks briefly and shifts the subject. "Have you seen these maps? I assume they show much more than our own."

"No, but I've heard about them - the originals have been burned, I think, but copies with the appropriate coordinate system were made. It is the only sign our proposal might be taken seriously."

"Privately at least." 67 reminds himself to remain optimistic. "Still, I suppose it is a good sign."

"Indeed. That our languages coincided was intriguing enough..."

"Yes, that does not seem to make sense. The Universe arranges itself in most curious ways."

"It's almost as if there wasn't any thought put into it."

Their conversation wandering into idle philosophical speculation (the most dangerous kind) indicates that it has come to an end. Tri-067 smiles and walks away, their brief respite from the demands of Tritopean society at an end as the sun, defiant in its sphericality, rises into place to signal the morning proper.


An unusual liveliness suffuses the Tritopean island, lifting spirits from depressed and hopeless to slightly-less-depressed. Hope spreads throughout and dares the populace to believe their dull existence might mean something. For the first time in many years, a new layer can be added to the population, marking a reversal of the devastating trend of decline that has brought Tritopea near extinction. If the ceremonies can be arranged on time, then one hundred and twenty Tritopeans will stand at this year's Synchronization, a most numerologically auspicious occasion. 

But for a few Tritopeans, there are only the same doubts, their well-cultivated cynicism withering away any accommodation for that foolishness of 'hope'. Thus it is a time of tentative celebration, elaborate planning of rituals and foremost, of silent despair. In that sense it is just another Tritopean day.


Tri-048 surveys the 15 children with pride. They are between 15 and 28 years old, some not children in the conventional sense (but nevertheless children to the Tritopeans, who would not consider anyone mature until being inducted into the society). The children have gathered here in the assembly hall of a Tritopean school,the only such school currently in operation. For Tritopean children, education takes the form of general knowledge and preparation for their integration into society, should it be possible. Years of careful education have groomed these children well, and should they become true Tritopeans, their education will continue as they learn an assigned trade. Detailed records of performance on various tests have already determined the career path of each child, though this may not yet be known to them.   

It is time for a farewell speech of sorts. 48, having taught the children most of what they know, feels it is appropriate. It is a sacred duty to teach children of Tritopean ways, but the great of reward of their acceptance into society is only available to a few. 48's predecessors have had the unpleasant misfortune of their hard work amounting to naught, their students left (eventually) to expire.

"Soon, you will become Tritopeans." 48 immediately regrets being so unprepared for this speech, concerned that emotions might render it an embarrassment. "You have been equipped with the knowledge to succeed in whatever you are assigned, so that you may become productive members of our society." The students sit in rapt attention, understandably nervous about their passage into maturity but also trained to listen to their elders without question. "You understand accept our sacred truths, and I am certain that you will respect our traditions and keep our society alive. You are our hope for renewal, a new layer that may herald our return to glory. Let that be your inspiration, and let our teachings be your guide."

It was hardly a memorable speech, but it would have to do for now. After all, there is paperwork to do, and ceremonies to arrange. For once, 48 looks forward to it.

[Tritopea]

Tritopea is now fifteen greater in number, and congratulations and introductions are formally exchanged as its new citizens take their place. For the most part, children are safeguarded by their teachers and monitors, and so these nascent Tritopeans are only now escaping their sheltered childhood to find a slightly less sheltered existence on the island at large. Apprenticeships commence, homes are assigned, and everyone is in good cheer.

Everyone, save one brooding Breeding Monitor, who has languished too long in a tortured state of indecision over the dreaded report. But today a pyrrhic spark of hope promises closure. Tri-056 has never seen the executioner's prism, but nightmares of its mercilessly geometric decapitation are more than enough to instill him with dread. Yet even Tri-045 has principles, and to fabricate the report would be a lie of unbearable magnitude.

A letter, half-written, lies on his desk, by a shaking hand, while the report now rests askew in a tetrahedral waste bin. With a heavy sigh, Tri-056 lights a match and lights the report, watching its pages blacken and curl. Turning away from the smoking bin, he continues a final letter of a very different sort.

Cowardice is the only compromise I can bear, but if it is any consolation some good may come of it. The excess layer you've sought now exists. Albeit at my expense, but it is all my weary soul can manage - I cannot bear the burden of martyrdom. I am sorry you could not convince me otherwise, but this end was preordained by our cursed society - blame the Elders if you must channel your sorrow anywhere.

A glance at the clock indicates the time is near - his invited guests are soon to arrive, and preparations must be completed. With a few rambling words, Tri-056 completes his last words, loathing the clichéd tearstained pages and running ink. He folds the letter delicately and addresses it to Tri-045, who (ever the punctual mathematician) undoubtedly has already left for this meeting. With a whispered goodbye, he imbibes his last drink, and shortly collapses to the floor, limbs incongruously arranged against the tesselated tiles of the floor.


Tri-055 has never seen a corpse before despite the drastic Tritopean need for senseless murder, and looks away to the distraction of the letter and then to the ashes of the bin. Making sure the door is closed and he is alone, he unfolds the letter and reads, confirming his suspicions.

"I will make sure some good comes of this. Don't worry about it," whispers Tri-045 to the room at large, unwilling to meet the eerie, bloodshot gaze of his departed colleague. A nervous laugh escapes irreverently from his throat, the absurdity of the plan becoming clear.

[Tritopea]

One-Hundred and Nineteen

"This is a great disappointment," remarks Tri-002, second Elder of Tritopea, with fitting coldness.

"If only the necromantic arts could be perfected," says the third Elder, smiling slightly.

"We could never welcome the coward back into our ranks, anyway. It would be fruitless." The eldest Elder, Tri-001, gazes sharply at the third, not amused by the offhand comment.

Elder the Third rebounds quickly enough - "It would be enough to have a proper execution," is his dry reply, which elicits a chuckle from the First (quite a relief for the Third, whose wit has been in question ever since a disastrous fractal pun).

"We will have executions aplenty, anyway, unless we arrive at some more prudent solution." There is a sense of urgency in Tri-002's voice, as the thought that Elder the First might hold to tradition and remove the windfall layer is a legitimate possibility despite its patent stupidity. Dictatorial intoxication has made the First a bitter old man. and not one to humor practical suggestions. For the Elders must follow the logic of the Triangles, and never resort to base justifications of a pragmatic nature.

Says the first: "Do you refer to that proposal we have received in the report of our Triangular inferiors?" Through the surprisingly elaborate bureaucratic channels of a people numbered only 119, a suggestion had been made to send ambassadors from Tritopea to other (currently unknown) nations by repurposing exiles (who would otherwise be terminated). This was likely a breach of protocol or tradition somewhere in the convoluted chain of formalities and reports, and the Elders weren't oblivious to the dissidents involved. But now was certainly not the time for prosecution of citizens, and after all, Tritopea was a nation ostensibly committed to progress, albeit of a psychotic sort.

Elder the First's permanent sneer curls further, imagining some less than subtle social manipulation to inconvenience these these tradition-flouting dissidents. But he willing at least to consider that they are right, if not to accept their arguments.

"Yes, I do, although I did not wish to speak of it directly. I know you find it distasteful." Elder the Third remains quite as the Second speaks, watching the conversation unfold intently.

"Indeed. But I know you wish to convince me. Do so, then, and with none of their vaunted pragmatism."

"It is simple. We seek to purify our imperfect world, so that it may be triangulated. Beyond the borders of our equilateral domain lie heathen nations (at least one of which has briefly visited us, as you must be aware). Their ships are degenerate in form [i.e. having circular shapes], so one can only imagine what geometrical abominations are in their homelands. This asymmetry cannot be left forever, and now that we know it exists we can no longer feign ignorance. To reform the foreigners, we must know more of them."

"Is that all?" Elder the First raises an eyebrow doubtfully, more to make the Second uncomfortable than anything else.

"Yes. No further argument is needed for the imperative of exploration. Do you agree?"

Elder the First pauses, feigning thought (a popular pastime of Tritopeans). "I accept your premise. By ignoring the foreigners, we tacitly acknowledge asymmetry. But this is irrelevant if we have no means of action. Tritopeans must stay in Tritopea or else the Triangulation could not be satisfied, and the terminated are no longer of us. Why send the terminated layer away - why grant them a task we would not bestow upon our own?"

"True, it would be nonsense to appoint ambassadors of our citizens. But the exiled, to use the term, have been Tritopeans in the past, and they have learned our culture. They are representative of our ways even if they are no longer of us. Moreover, if we stipulate that they cannot return, then their impurity cannot infect us - it exists only in the context of the already impure foreigners. If anyone, it is only the exiled who can explore at all."

"Sending the impure to discover the impure. I suppose. But we must remain in contact, and that is unacceptable. We do not consort with the exiled, as you call them."

The Second nods. "But there is not much precedent, anyway, for the exiled have hitherto been dead. Their information, though admittedly tainted by their perceptions, is still in essence true, and therefore of importance to us. We do not deny logic."

"We also do not listen to heretics."

"Fair enough. But think of the ambassadors as another caste, as we designate for prisoners and children. There is no reason to suggest the terminated must be killed outright, nor that we should ostracize them fully. If used cautiously, their information would be worthwhile."

"Your avoidance of the word "useful" is amusing. But I appreciate your tact. Elder the Third, what are your thoughts?"

"Hmm?" Elder the Third collects his thoughts through the extended utterance. "I believe we should establish ambassadors as suggested. If foreign affairs are handled strictly by certain appointed citizens here in Tritopea, there should be no trouble. For as the Triangles conquered the Squares, so the Tritopeans must - eventually - conquer the world."

"I do not fancy compromise, but I find sufficient argument of an acceptable sort to make a decision. We will send the ambassadors. They will not be of us, but will have been of us. They shall be in contact with foreign ministers to be appointed. Let this be an extension of our holy search for knowledge, and our prerogative of purification. And ensure that our citizens do not take too much interest."

Elder the Second grins wickedly. "How practical of you." An irritated glare from Elder the First causes the grin to retract, as the snide remark could quite possibly have changed his mind.

Thankfully, this is not the case. "I will begin preparations tomorrow, and inform those fit to be involved. There may be a few terminations of the traditional kind anyway." This announcement ends the meeting, and they exit the meeting hall solemnly.

[Tritopea]

One Hundred and Five, 15 Exiles and Three Corpses

TO BE CONTINUED