Part 1 - Treasure
The warehouses lining the arrival and departure lanes of the space port are constructed out of red brick instead of the traditional glass and steel common to the colony of New Guadeloupe. They tower high above Leif, as he dashes in between them through an alleyway. Surely, he cannot keep this tempo up for much longer.
“Damn Ingfrid, that traitorous addict."
Just before another corner, Leif looks over his shoulder. The chasing squad are still on foot, but swarming above them he sees Ravens, hundreds of small, black drones, known to be relentless. Ivar was right, this steal had not been a good idea. Where is that sceptic anyway? He pushes the purple orb closer to his chest, turns the corner and stops. This is not a dead-end, luckily, but no doubt reinforcements are already circling around this way. Leif has no intention of getting trapped so easily. He needs a plan.
Yesterday Ivar and he had strolled leisurely around the docks, watching the unloading of Knarr vessels. Huge metal crates drifted out of the holds on maglev rails leading into the warehouses. They both enjoyed staring at the undulating motion of seemingly endless square snakes that held riches for trade. This was the new Gold Coast. Back in ancient history, on Earth, Tobago and Guadeloupe had been the first. Slaves, gold and diamonds. After the Danish kingdom had taken over, it was a long time before the former Viking marauders were able to rise again. Not until space opened up as the high frontier did the old Earth tribes leave in their rocket ships into a new era of exploration and colonization.
On Earth, maps were considered treasures in their own right. Possessing them meant knowledge. Conquest of foreign lands was easier, because of that. Just bring along your compass for drawing circles and a straightedge for lines. As long as you had a map and some tools, you could rule the world.
At some point the old civilizations of Earth, the Greeks and the Babylonians alike, discovered fascinating facts by pure reason alone. Such as, when you want to prove that something is impossible, you can set up a trick: You first assume it is possible and then use established knowledge to arrive at a contradiction. This result then proves the assumption was incorrect and therefore truly impossible.
Sometimes it takes a long time before some scholar conceives of a line of thinking that reveals the truth. Until that moment, it is a temptation for many a soul. Remember that squaring the circle remained elusive for millennia until finally it was found to be a total fantasy, a siren singing sweet melodies for heroes wishing to be immortalized, but fatal to have your ship track a course towards.
While Ivar concerned himself with these matters, Leif considered another view, while they were still strolling, talking, and he was not yet running for his life, as is presently the case.
Foreign conquests by nations with superior technology performed a different trick altogether by thrusting into the public conscience the idea that what seemed impossible turned out to be the opposite. Circumnavigating the globe, they leveraged the advantage of applied science and tools to overwhelm local tribes, and take advantage, for their own enrichment and sustained dominance. On New Guadeloupe and other settlements of the Danish Kingdom, this holds true to this very day.
“Surrender,” the squad leader shouts. They are close. Leif can see their blue eyes, hungry for prey. But he stays still. Closer they come. The Ravens now fill the sky above him. On the other side he sees the second squad now entering the street, just like he expected. Weapons are being raised.
Leif takes out a transparent mask from his coat and puts it over his face. Then, he dives down into the shipping lane. Obviously, there is no water here; this is a space port. He falls, fast. Their Ravens at bay, the two squads stand united on the quay, studying his accelerated descent towards certain death.
Part 2 - Map
Getting impatient after all the trouble they went through, Ivar snaps at Leif: “Place the orb here."
“And then what?"
“Then we see what it holds."
Ivar glares at him through long black hair that falls over his face irregularly. This is why they took it, so why hesitate now? Leif is still a bit disgruntled. His friend had waited a long time before he picked him out of his endless fall with the Faering light spacecraft. Wanted to be sure. Said it took a while to match his trajectory. Afraid that he would break something if it was not done well. Reasonable. And annoying as hell, at the same time. His tiny, emergency oxygen mask had been malfunctioning. Close call.
Slowly, Leif takes the purple ball out of its protective sleeve and places it on the imitation ebony reader in the middle of the hard steel table. Immediately it lights up and projects an intricate three-dimensional web of dots and lines. They lean forward and start to take in the information.
The Magistrate had kept this hidden for a long time. Only through thorough research, supported by stealthy reconnaissance, did Leif and Ivar figure out how to acquire this little gem. Resident ruler of the capital, the Magistrate is a perfect example of the upper class: long, white-blonde hair contrasting with a light-brown skin teint. Emerald eyes with a hint of blue. This is how they all look, men and women. They do not mingle with the rest of society.
When they were younger, Leif and Ivar had separately been recruited into one of the most infamous gangs in the city: Skugga. Selected from hundreds of eager candidates who every year wish to join from the poor lower classes. Natural aptitude is a baseline requirement, but the final cut is made by choosing those who derive real pleasure from the thrill of it.
Their leader had paired the two of them up for a job in their first month being officially part of the gang. Ivar was the planner, the schemer, while Leif was the man of action. The assignment went so well that ever since they were a set team.
Now, with a map in their possession, they scrutinize its intricate details. What does every dot mean to the other, how do they connect? Ivar has crawled through numerous science textbooks since he was able to read, figuring out mechanisms, deciphering math, appreciating electronics and the laws that govern them. There are two special numbers, indicated by the letters "e" and "π" that come up all the time.
Whenever a colonizing force's governor, chief, warden or keeper fancies calculating the latest projections of wealth accumulation, he realizes he needs a formula in which "e" features prominently. It is a strange number that this letter represents. The formulas with "e" are called exponential. They are fed with regular numbers when you apply it to a typical growth scenario, but never is the result a regular number. You have to round it off to arrive at a humanly usable one. It seems as if nature wants to deceive us—the patterns and formulas that govern its inner workings are never regular like something as simple as "1 and 1 is 2" or "the square root of 4 is 2."
Leif knows that Ivar understands this intimately, even though he does not to the same degree. Exponential functions. Normal, regular numbers, like whole numbers, or fractions, or even square roots—these are all familiar to man, since the time of Pythagoras. The ones that nature enjoys revealing like clockwork, though, like "e," are labeled transcendental.
Part 3 - Ship
Leif and Ivar are back on the quays, tall red brick buildings everywhere. Some of them have ornate plaques attached to the front, clearly owned by some prosperous cooperative. All of them have hooks and large entry doors for loading cargo. For a moment Leif and Ivar stand admiring the architecture. It reminds them dimly of something familiar. Perhaps an ancestral place, who knows.
After they had memorized the key points of the projected map, they hid the inconspicuous ball behind one of the secret panels they had installed in their quarters a long time ago. When you start out in any profession, you make so many mistakes that it can seem that you will never master it. This pair, though, has been at it for quite a while.
In fact, too long, if you ask them. Usually, after about five years in the gang, members will start to make mistakes. No one has survived past the eight-year mark. Jobs are always dangerous and slipping up does not end well. Ivar and Leif have decided they want out.
They need to find something on their leader, to make sure he does not just put a contract out on them. This is the moment, they are still competent, at the peak of their abilities. But they have to act in secret. If someone finds out they are gone too long, it will be suspicious. The clock is ticking. Quickly they make their way into the city again.
On a street corner sits a smaller building with a copper roof. They walk inside, greet the barman and settle down in a corner. Time to find a ship.
When any of the ancient Earth people started to explore their environment by sail or oar, they sketched maps on which they worked with lines from one place to another, and they traced circles to determine how far a vessel could travel. The Greeks in particular made fundamental discoveries about hidden rules that the universe had bestowed upon these drawings and calculations. For instance, the surface area and circumference of a ring is proportional to the diameter by a strange number they could not get a proper grasp on. They denoted it "π" for the first letter in the word "periphery."
In his profound studies Ivar marks one moment as the one of the highest significance. He encountered a formula called Euler's Identity. It electrified his brain. Other people may experience something similar when they hear beautiful music or read evocative poetry or observe a particularly splendid painting. For Ivar it was seeing this identity, which combines the worlds of growth calculations and sketching circles with regular human numbers, in just one single, simple equation.
"It is amazing." Ivar tried to convince Leif afterwards. "How this can even exist."
Seeing the incomprehension, he continued with: "The result of a formula with "e" filled with "π" turns up a normal number. In fact, the result is "-1."
Ivar ignored it: "So, if an exponential function always produces a transcendental number if you give it a normal number, how come you get a regular number when you feed it with π?"
That merely produced a confused, almost resentful look from his companion.
"Because it is not a regular number. It is also transcendental."
"Ok, I get it. And this is useful how?"
"We will get to that."
Part 4 - Journey
Magnus has the exact name for how he looks. However, the vessel he commands is the opposite. It is about twice as big as a Faering, which holds just about four people, tightly packed together. But, he is a capable captain and does not talk about his voyages, even when he drinks a lot. He just gets even more quiet than he already is. A low grunt escapes his throat presently. Leif shakes his hand and transfers the cash by holding a plastic rectangle over Magnus’s payment cube. A short synthetic sound confirms the transaction. They are on their way.
All aboard the Astrid, the crew settles in their bunks, while the navigation application of the board computer plots a course. Traditional tools like the compass and straightedge are not needed anymore. Computers are the tools to replace all tools.
Curiously so, though, numbers like "π" and "e" escape the otherwise exact ability of computers to determine a solution. Running a program to ascertain the digits behind the comma for any of these two transcendental numbers will have the machine endlessly churning without any repeating pattern emerging from the chaos. The old tools face no such issues.
Barely thirty minutes later, they can see the port of New Guadeloupe receding into the distance. Here and there a Raven flits over the top of a building, looking for trouble.
“What exactly are we getting ourselves into?” Ivar ponders, half aloud. He has not devised any strategy beyond this point, did not research further. It scares him.
“Our bloody freedom,” Leif stresses in a surprisingly steady tone of voice, but his hands fidget around in his pockets.
Their stop will be at Ghana II, another place where the Swedish-African Corporation had set up their new trading posts and accompanying homes. The locale was more mixed, from dark-skinned voodoo practitioners to battle-ready Viking traders; and many variations in between. This was one of the first settlements. All the main commerce houses, insurance companies and banks were founded and are still headquartered there. Many a rich man or woman finds his or her origins in the tales of the first days of Ghana II.
Nowadays, the place is still considered dangerous. Those that know their way around have profited tremendously from engaging in more and more unethical enterprises. The three passengers on the Astrid are not from around here, even if they are only too familiar with the stories.
The Astrid floats into one of the old docks. Here, there are fewer amenities but also, crucially, there is far less security. Magnus confidently berths at the pier. Ivar limps off onto a gangway that the gravity of this planet pulls him towards. Big change. He stumbles and falls. Just in time. A projectile grazes his hair and digs itself into the hull of the Astrid with a sharp thwack.
Part 5 - Pirates
Haarfager raiders! Leif dives behind a bulwark to the side fast, taking care not to leave anything vulnerable exposed. Magnus is inside. He immediately understands the situation and starts to check the ship’s cameras for signs of the attackers. Ivar lies spread out on the ground initially, slightly shaken and unnerved, but now he rolls behind a tower of metal crates, fluently like a pro. Nothing happens for a very long time.
On the other side of the dock, a cloud of Ravens surges up into the sky and swirls towards the opening hatch of the Astrid. Individually they are beyond what you can hope to deal with without losing limbs or vital bodily functions, but in groups they are absolute killing machines. On New Guadeloupe Leif had been well prepared, with multiple back-up options at his disposal. And even then, he had felt the panic rising at the moment he saw the swarm of Ravens.
It is much worse now. Stygian Blood Ravens, as they are called, are not ethically bound by government rules. They are programmed to follow pirate command, without any restraints. The vanguard few are already circling Ivar’s hideout. Panicking, he kicks the crates, so that they tumble over and on top of him, creating a provisional hiding place. Magnus decisively closes the ship’s hatch with two button presses and disconnects from the dock. The Astrid starts to float away, as the Ravens spread out to find Leif and Ivar. The Haarfager troupe has not even shown itself.
Beneath the crates, it is really dark. It comforts Ivar, because that tells him there are no holes for bloody Ravens to get through to him. He checks his communications bracelet. Coverage is also out. All is dark. The longer he is able to remain where he is, undetected, the more Ivar feels encouraged that they may let it pass. The ship must be out of reach already. What would you need one of the little thieves for?
Metal scraping. A distant boom. Ivar pays attention. The scraping comes closer, like steel pincers working on a tin can, squeezing and bending it out of shape. A tearing sound, loud and abrupt. And then.
Ivar has to shield his eyes that immediately water up. The sudden light is unforgiving. He feels relentless hands grab him and prop him up. A rough throat chuckles disdainfully.
The brightness fades to normal levels and Ivar looks around. He is surrounded by a gang of pirates of the most dreadful kind. High-tech outfits, onyx breastplates, desert camouflage capes and barely hidden weaponry, mostly guns but an occasional serrated knife. Perhaps for effect. Nowhere does he see Leif. A few of the men are scouring around in the vicinity of the bulwark. Ravens crowd the sky.
Part 6 - Reversal
His captors have also covered his eyes, of course, so Ivar has no idea where they are taking him. It is of no real consequence, because the outcome must manifest itself as expected in these situations. The only question is whether they will torture him or not, but death is inevitable nonetheless. He prefers the path without torture, obviously, but then again, it is supremely likely to happen anyway. Leif was not found. This settles it. Ivar mentally prepares for severe pain in uncomfortable places.
His blindfold is removed and still it is dark. Someone stomps away; slams a door shut. The air is musty and moist. He smells intoxicating liquor mixed with other heady fragrances that no doubt belong to a concoction of drugs that needs to be inhaled for consummate effect. This is not looking too well, Ivar thinks. Carousing next door is a troupe of drunken, doped-up pirates with a penchant for violence, holding a captive unwilling to rat on his friend. Blood has been spilt in less tenuously menacing circumstances. What is truly disconcerting, however, is that nobody seems to be coming for him. Ivar sits in this dark room, tied up, while listening to the raucous party that is in full progress somewhere else, not too far away.
He hears scratching. It is coming from one of the corners. Is it a rat? No, it is bigger than a rat. Now it comes from the other corner. Is there more than one? Maybe the debauchery of wanton Viking pirates is not the worst that can happen to him. It is too much to bear. Ivar shifts around, pulls on his restraints. He has to get out of here, he has to, no matter what. No way is he going to be eaten by bloodthirsty rodents that are too big for comfort.
He — “Ivar, damn it, here you are.”
He recognizes this whisper. This voice that never misses a chance to interject a curse or two. Leif!
His buddy frees him quietly, expertly one might say. They look at each other, both relieved to see the other. Explanations will be given later. Tiptoeing out of the room through the hole that Leif made to get in, they enter a hidden space in the wall. Carefully, very carefully, they move along. On the other side a celebration seems to be going into an exuberant stage of inebriation.
Near the back of the Haarfager compound, they exit the building onto a small inner square. Three technicians are walking around with tablet devices in their hands, tipping the surface with their fingers in indistinguishable patterns. On the ground rest thousands of Ravens. Quickly Ivar and Leif duck behind the nearest large object that casts a shadow to hide in. Not a crate this time, but a control system with a power source connected to it. They watch how the flying menaces are being inspected for possible physical damage and checked with the tablets for system errors or anomalies.
Never before have they been as close and still able to observe how it all works. They now notice that the technicians have button-shaped transponders attached to their coats. Very interesting. A bit farther away a black plastic box holds more of them. Without a word between them, they decide to steal a few and figure out how to make them work.
Leif glances at Ivar. Yes, transponders involve wave equations, which means π hides just around the corner. Knowing the math brings a world of engineering into Ivar's grasp like a map guides the captain of a vessel, at sea or in space. Still, he would need to work together with a skilled cryptographer to make any modifications without setting off the failsafe alarms.
Part 7 - Journey
The two professional thieves have left the place of their captivity behind and are now sneaking through the alleyways of Ghana II. Ivar is still curious: “How did you escape at the docks?"
“Jävla skit (holy shit)! I put on my oxygen mask and let myself drop below the quay. Then I kept going underneath until I came to the other end."
Leif looks at him as if this is nothing special. Always more the action type than Ivar who is the worrier, which does make him the better one at preparations. They come upon a bar and go in. Settling into a corner table in the shadows, they wave the waitress over, order food and drinks, look around to see if they are being noticed and finally ease up a bit.
“I slipped the Drotts’s ID card from him,” Ivar says.
Leif grins and leans forward: “What does it say?"
“Now that sounds fricking familiar."
They lock eyes. The Magistrate. They must be related. Possibly nephews, even brothers. At the table next to theirs a large, dark figure starts to snicker.
“Found something interesting, boys?"
His face is hidden. Several figurines are spread out on the table in front of him. Their limbs are crooked. Needles are stuck in at nasty spots. Must be one of those Voodoo priests. The dark shape lights up a cigar, one of those Habana Secunda ones, big, fat, juicy and definitely illegal. Briefly the priest’s face is illuminated: Blackest of skins, green eyes, blonde mustache. High-born.
“Nothing special,” Leif says.
“I see,” the priest drawls. He grabs a figure and makes a few adjustments. With studied nonchalance he takes out a pin from the shoulder and twists it into the forehead. The room shrinks. Peripheral vision disappears. They are alone with the black priest. The world has faded away to insignificance.
“You know,” he goes on, “these frontier planets were settled by my ancestors. Their descendants rule to this day. In government as well as in the underworld."
“What is your name, Houngan?"
“Victor Scylfing. My family has been opposed to the Scaedugengas for centuries."
Leif and Ivar breathe out. Victor seems satisfied and removes the pin. The world returns.
“My enemy’s enemy and so forth."
“Will you help us?"
“One condition: square the Orb."
He hands them a small cube.
The door of the bar bursts open.
“Come with me, now."
Victor leads them into an antechamber, opens a hatch in the floor and takes them through a tunnel, urging silence and speed.
It is a long passageway, prompting Ivar to consider momentarily how they ended up here. Obtaining the map had elevated their spirits. Intelligence had appeared within their grasp, which now fast-tracked its way into uselessness. They are still trapped in the Skugga gang until the early retirement waiting for all members. This Houngan could shift the balance, with his gift, a cube no less. What's next? A pyramid? Probably not.
Ivar's thoughts go to the Drott and the Magistrate. That is what you'd call a striking coincidence. The Haarfager raiders nurture their fame across several of the known worlds. Even though the name refers to the founder's fairness of hair, many of the members come from all sorts of backgrounds. It is a mixed group, much like Skugga, their own gang. Where Skugga operates in the shadows, the domain of thieves and extortion, the Haarfager are a more militant troupe with a typical violent intent.
Part 8 - Ship
The Houngan leaves them as soon as they come outside close to the docks. They search for a suitable ship to stowaway on. When they turn around to the priest to thank him, he is gone.
Acting as normal as they can, Leif and Ivar walk into the lights of the docks. They know a few ways to hide on a space vessel. A Raven flies overhead. It halts. Damn. They run.
As they approach the other end of the quay, a group of Haarfager raiders is now chasing them full-on. Luck has run out.
Suddenly there she is rising from below, the Astrid, cargo doors opening. They jump in and Magnus pushes down on the throttle. The ship accelerates into space. They kind of expect explosions. Or Ghana II collapsing behind them, like in those adventure stories. But nothing of the sort. Just a terrible view of pirates boarding their boats to pursue.
Tense hours play out as their ship burns fuel to keep the pirates at the exact same distance. Luckily, they are outside of firing range. Ivar mentally notes this is obviously a section of a ring surrounding the enemy ship and that its span is determined by "π.”
How long this can go on they do not know. Magnus does not take the direct route home, of course. That would be leading the wolf to the meat locker and opening the door.
There are no asteroid fields to heroically lose themselves in or interstellar plasma clouds that might deregulate the enemy’s navigational systems. None of the usual saviors of protagonists in a tight pinch. They have to come up with something. The situation is like a metaphor: sitting comfortably in their own protective bubble, deathly danger is chasing them. It cannot get close unless they slow down, but in the end, energy will run out. A confrontation is inevitable.
Part 9 - Map
“Hail, Ingfrid,” Ivar says.
“What? No fucking way. That crazy dopehead almost got me killed last time.”
Leif stands up, jittery with anger. He slaps the desk with his flat hand to make his point.
“She could do it,” Magnus adds, placing his big hands on Leif’s shoulders, easing him back down.
Another hour passes. The status quo remains. No new ideas were voiced.
In unison: “Ok. Let’s do it then."
Besides possessing an extremely unreliable personality, made worse by the chemicals she lets her body consume, Ingfrid is one of the best hackers anyone knows. Ivar persuades her via an encrypted connection to come to the rescue. She owes Leif at least this, he points out. She agrees.
It still takes quite a long time, a few hours, before Ingfrid has sabotaged all the pirate ships without them detecting any intrusions. One by one they come to a halt and become distant specks for the crew on the Astrid. Magnus takes her in a complex evasive pattern back to New Guadeloupe.
They all gather in the thieves’ hideout front parlor. Ivar has produced the orb they stole, while Leif presents the cube from the priest. Magnus and Ingfrid look at the two fiercely glowing objects with unveiled admiration.
“Was the Houngan being esoteric or literal?” Leif says, as he leans towards his friend.
“Perhaps there is some kind of procedure we can find for this,” Ivar replies but gives back a look that says he does not know it, all the same.
So much time has passed on Ghana II. They will have to report in with Skugga, their gang. Even though their hands are not empty, it feels like they are. Still clueless, Ivar holds the orb next to the object in Leif’s hand, hoping to see something in their appearance, a similarity, a glimpse of a solution.
This is the moment of revelation. It somehow makes Ivar remember a peculiarity that was proven only quite late in history by a German mathematician.
It goes like this: The surface of a square is the length of the sides multiplied, so if you know the surface, the square root is the side, still a normal number in human terms. If it was a circle, you have to multiply by "π," so it always is a strange number, a transcendental one.
It's almost as if nature wants to emphasize something— what men want and construct just does not exist, not really, not fundamentally.
The very first time the two objects come close they pull each other together like magnets. Immediately they merge. Fine dots of light appear and start to group together. They fade away and become tiny black amoeba shapes. They undulate like a swarm of birds does across the sky, moving in a kind of meta-pattern that follows an unknown set of rules. Chaotic, yet ordered. The surface area is wobbly and shifts continuously between a cube and a globe.
And then they see it.
Part 10 - Treasure
There is a red brick storage building on the pier of New Guadeloupe. Countless Ravens flock around it like black amoeba, meta-pattern and all. One group swirls straight through the other, without any of them touching or losing their trajectory. Leif and Ivar are sneaking around looking for places to hide. Following Ivar's explanation, Ingfrid has reprogrammed the transponders they stole from the pirates. She and Magnus wait for the boys to return. Or not.
Several transponders are placed at the opposite side of the pier, intended to attract Ravens as a diversion. Both thieves are wearing a transponder that should clear them, just like the guards. The decoys are activated. The black swarms move in new ways. A few of them make their way to the far end. Leif and Ivar walk into the street, backs straight, making small agitated gestures, eyes alert, hearts in their throats.
Unhindered they enter the building. Quiet.
A library of sorts.
Back in the hideout, Magnus is reading the same chapter of a novel again and again. Normally he relaxes by engrossing himself in a story, but now he can’t keep his attention. Ingfrid is sleeping, passed out after she took some pills of dubious origin from a dirty plastic container that suddenly appeared from one of her jacket pockets.
The door opens quietly. Two men slip inside.
Old, parched paper documents are placed on the table. Four pairs of eyes: eager, excited, unsure and disappointed.
What the priest wanted them to find turns out to be proof of how society in the New Colonies is purposefully designed. Often old tales describe how the lower classes consist of degenerates and the luckless, while the beautiful and wealthy rule in pleasant isolation. Sometimes the story has a twist and reveals how depraved the upper class can become when there is no real life, no juice, in their daily existence.
This could be expected to be the case. There is a blood link between the Magistrate and the Drott, after all. The reality is different, however. All the worlds appear to be ruled by the underworld chiefs. They have placed relatives at senior positions in official government and each pretends to hate the other, executing fake campaigns to eradicate the powers on either side. A game of balance is being played. And it has been played for a very long time.
What would happen if these documents were made public?
The priest hopes that his family can take over, most likely.
Leif, Ivar, Magnus and possibly Ingfrid fear they will end up imprisoned.
How long would this news stay news and hold its influence over the population?
There is no way they can use this to bribe their way out of the gang. Everything that happened, straight from getting their hands on the purple orb to finally finding the true map of society, hidden in a warehouse, still amounts to a definite but miserable fact: They are back to where they started, without a way out. Nothing has changed.
Perhaps it ends up being just a story, both linear and strangely circular, written down by one of the four protagonists. For posterity.
The ancient Greeks originally stated the problem of squaring the circle in terms that can be paraphrased as “constructing with compass and straightedge a circle from a square, so that both have the same surface.”
The impossibility of squaring the circle can be found from a proof published in 1882 by German mathematician Carl Louis Ferdinand von Lindemann. The publication itself is about the transcendence of the number "π," which means that it cannot be written as a power equation that takes either whole numbers, fractions or roots like the square root of "2."
In short, the surface of a circle is related to "π," while that of a square is a power equation. It turns out they belong to completely different number universes.
The “circle squared” has been used as a metaphor in literature from as far back as 414 B.C. and is referred to by many authors of note, including Dante Alighieri, Alexander Pope, James Joyce and Thomas Mann.
Ironically, practically speaking, you can get very close to constructing the square of a circle, just like Leif and Ivar almost managed something extraordinary. The solution they found was a map. It unfortunately turned out to be impossible to navigate.