The Spacetime Pool

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Onto some more specific mathematical details. The following exchange takes place shortly after a mysterious man has appeared and apparently taken Janelle through a portal to another universe: That he sounded sane made none of this more plausible. From here to your mountains. A branch cut to another page. Your universe is one sheet, mine is another. A branch cut from one Riemann sheet to another? That is all I know. Having lived as the child of a diplomat for so many years had taught her a great deal about dealing with cultures other than her own, and she could tell her interactions here were on shaky ground.

What an exquisite challenge, to portray those graceful repeating patterns as a periodic function. Their Fourier transform would be a work of art. An unsteady urge to laugh hit her, followed by the desire to sit down and put her head in her hands. Contributed by Alex Kasman I wanted to say just a bit more about branch cuts and Riemann surfaces. I'm putting it in this box so that it can be more easily ignored by those who are not interested in the technical details. You can tell it is a function because it passes "the vertical line test", which is to say that no two points on the graph have the same x -coordinate.

In terms of input and output, we think of it this way: All of that is fine and normal precalculus stuff. But, things get messier when we try to include trig functions in the discussion. People came and went. Silk wrapped them from neck to ankle, glistening in the smoky torchlight, crimson and saffron, shot through with gold threads. Their shimmering dark hair fell to their waists. The trio stopped in front of Janelle. The oldest woman, a matron with silver hair, spoke in melodic phrases that almost sounded like English, but that went by too fast to catch.

As Janelle set off with them, accompanied by her guards, she glanced back at Dominick. The older woman spoke curtly. It was dizzying, all that geometrical beauty gleaming in the torchlight. The older woman was watching her face. That is all I know. Having lived as the child of a diplomat for so many years had taught her a great deal about dealing with cultures other than her own, and she could tell her interactions here were on shaky ground. What an exquisite challenge, to portray those graceful repeating patterns as a periodic function.

Their Fourier transform would be a work of art. An unsteady urge to laugh hit her, followed by the desire to sit down and put her head in her hands. At the back of the hall, they passed under a huge arch built from gold-veined marble rather than the wood used in the Fourier Hall. A true corridor lay beyond, with stone walls tiled in star mosaics.

Its size dwarfed their party, and other halls intersected it at oddly sharp angles. The pillars at corners where the halls met were carved to portray men with great broadswords or women in elegantly draped robes holding long-stemmed flowers. Janelle tried to keep track of their route through the maze of halls, but exhaustion dulled her mind.

She was lost by the time they stopped at an oaken door.

Publication: The Spacetime Pool

The guards stayed outside while the women took her into a small room. Plush rugs covered the floor, and mosaics with pink tulips and swirling green stems graced the lower half of the walls. Something odd about the stems tugged at her mind, but she was too tired to puzzle it out. In one corner, a white table supported a blue vase with real flowers.

Blue velvet bedcovers lay in another corner, on a thicker pile of rugs, with pillows heaped there like a tumble of rose and jade clouds. They led her across the room and under an archway. In the chamber beyond, a small, sunken pool steamed, and a lamp glowed dimly in a seashell claw on the wall. The older woman finally spoke.

But I can manage. The trio bowed and gracefully exited the chamber. A moment later, the outer door creaked on its hinges. Unsure what she would find, she returned to the bedroom. An oil lamp hung on a scrolled hook by the entrance. To her relief, the door had a lock on this side and opened when she tried it.

One of her guards stood a short distance down the hall, severe in his leather armor. Light from a wall sconce glinted on the hilt of the broadsword strapped across on his back. He turned with a start.

He seemed bemused by her attention. She could think of many reasons Dominick might post a guard: For all its extraordinary beauty, his world had a starkness that kept her off balance. Ill at ease, she explored her suite. In the bathing room, an elegantly carved bench stood against one wall, with a jade-green towel, a silver brush inlaid with mother-of-pearl from abalone, two soaps carved like tulips, and a crimson silk robe.

It was all gorgeous, everything handmade. The suite, however, had only the one exit. They had closed her in well. More than anything, she wanted to clean up. She carried the soaps to the pool, an oval filled with scented water, but then she hesitated. The idea of undressing made her feel vulnerable. The grimy scrapes on her arms and legs decided her; she quickly peeled off her clothes, shivering as the cold air chilled her bare skin.

Then she slid into the heated pool. Warmth seeped blissfully into her body as she lay back. Silence filled the room, a contrast to the muted city roar she had lived with these last years, at MIT. No sirens or engines interrupted the quiet, none of the constant hum that rumbled even in the deepest hours of an urban night. She was immersed in a great ocean of quietude. They came from complex numbers. Just as 12 was the same at the start and finish, so 0 and 2p were the same.

To avoid that contradiction, z slipped through a branch cut to a second sheet for the second cycle F. Just as 3 am and 3 pm were different times, so F on each sheet was considered different. Her twelve-hour model was an only analogy; she had no idea how long would she have to wait before the actual gate reopened.

Nor was that her only problem. Suppose she divided F by 3. Divide F by 4, and she needed four. Many sheets could exist. Her head hurt, and the water had cooled. Putting away her thoughts, she soaped her body and washed her hair. Then she climbed out and dried off with the luxuriant towel. She reached for her wrinkled sundress, but then paused. The robe was far nicer and scented with perfume, certainly more pleasant than her gritty clothes.

She slipped on the robe, and the sensuous glide of silk against her bare skin stirred her thoughts of Dominick. She tried to smile at her reflection in the pool. She padded barefoot into the other room. She was so tired she could barely stand, but she felt too exposed to sleep. The bed consisted of no more than layers of rugs covered by velvet. She sat on it in the corner, with the wall at her back, facing the door as she drew pillows around her.

Her eyelids drooped, and she forced them up.

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The lamp swung on its hook, moving shadows on the walls, back and forth, back and forth The scrape of wood against stone roused Janelle. She lifted her head, disoriented. She had slid down and was lying amid the pillows. The lamp had burned low, leaving the room swathed in velvety shadows. The scrape came again. She thought she said, Who is it? The door swung inward, moving slowly. Dominick stood in the archway, filling it with his height and his presence.

The dim light turned his shirt a darker blue and glinted on the hilt of his sheathed dagger. The way he loomed, his face harsh and starkly intense, evoked the specter of conquerors who swept across continents, laying waste to their enemies. Such a quiet greeting for so dramatic a man. She appreciated that he asked, given that he could have done whatever he wanted.

He entered, and the room seemed to shrink. He closed the door, then came over and knelt on the other side of the bed. His shirt was open at the neck, revealing a tuft of chest hair, black and curly. He watched her watching him, and his lips curved upward. The shadows eased the hard edges of his face. Sitting on the bed, he tugged off one of his boots. Now he was taking off the other boot. He set it next to the first and started to undo his shirt. He took up the entire length of the bed. She could see why he might like sleeping on the floor; his legs were too long for a mattress. Maybe she should ask him to leave.

But she dreaded being alone. He continued to watch her, his head tilted to the side as if she were a puzzle. His knuckles brushed her inner thigh. He traced his finger along her cheek. But public ceremonies are traditional and expected, especially for the royal family. Is that all right? He kissed her deeply, and she tensed, wanting him both to stop and to keep going.

Her only experience with seduction was on the level of sending out for pizza and Cokes; she was so far out of her depth here, she was drowning. Then he drew back, his face unexpectedly tender. His palm stretched up her torso and his fingers closed around her breast. Of course she looked at his chest where he had unfastened his shirt.

A mat of hair curled over his muscles. She laid her palm against his abdomen, feeling the springy hair, the hard muscles. But very intimidating, too. Laying her on her back, he stretched out on top of her, easing his hips between her thighs. Then he reached for the waistband of his trousers. He lifted his head, his eyes glossy with arousal. He brushed his lips across hers. She lay still, meeting his gaze. Dominick groaned and rolled off her, onto his back.

Then he threw his arm over his eyes and inhaled deeply. He stayed there, silent and still, except for the rise and fall of his chest. Gradually his breathing slowed. Finally he lowered his arm and turned his head to her. Better than Make up your damn mind. She could also, she realized, end up pregnant. Dominick studied her with that close focus of his. Then he slid down the velvet cover and drew it over them both. Settling on his back, he pulled her into his arms.

She closed her eyes, relieved, letting her head rest in the hollow where his arm met his shoulder. As she drifted into slumber, she wondered if he dreamed of the towns and countryside that would someday fall to his army. He could be gentle with her, but she had no doubt he was capable of conquering a continent.

Would he wrack his world with the ambition that led men to create empires—at immense human cost? She lay in a pleasant haze, listening to the dawn. Her eyes snapped open. She was still in the palace. The room otherwise looked as she remembered, beautiful and spare. She rubbed her eyes. Yesterday she had been a new graduate with good prospects; today she had nothing but the unknown. She thought of Rupert Quarterstaff, the lawyer who dealt with her inheritance. Two years ago, when she had been paralyzed by grief, Rupert had stepped her through the estate settlement with a solicitude that went beyond his professional duties.

He expected to see her in a few days.

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It would be a mess. Janelle sat up, rubbing her eyes. She needed a library. Pushing off the covers, she shivered in the cold air. She went into the other room and bathed, then dried off with a towel someone had left while she slept. Her clothes from yesterday were gone. As Janelle searched for something to wear, she kept noticing the walls. Stepping closer, she peered at the mosaics. Wavelike curves intertwined in the tulip designs. They were too accurate for coincidence; someone had understood them well enough to reproduce the curves.

It was another piece of the puzzle, along with the Fourier Hall and Riemann gate. Deep in thought, she returned to the bedroom. Someone had come in while she bathed; her robe were gone, and the bed had been remade, with fresh rugs and a jade-green bedspread. As she toweled her hair, she surveyed the empty room. When the doorknob turned, she jumped. She barely had time to wrap herself in the towel before the door opened. The three women from last night stood there, each holding a large box decorated with abalone and opals.

Her greeting seemed to be the signal they expected. They bowed and entered the room. The older woman took an ornate key off a hook under the lamp and handed it to a soldier outside. He closed the door, and a loud click came from the lock. Janelle watched them uneasily. Janelle was becoming accustomed to the dialect and understood better this morning.

It reminded her of times she had spent with the families of dignitaries who visited her father, how she had striven to learn their language. To her, such new words were gems strung together to create sparkling necklaces of meaning. Danae offered her box. Prince Dominick-Michael and his men must leave today to discover what Maximillian plans. It would give her time to adjust, though, and to learn about the gate. You consider yourself above them?

Farimah jerked up her hand as if to strike her. Then she took a deep breath and lowered her arm. Her voice was ice. He told me to call him Dominick. She glanced at Farimah with sympathy. With so much unrest in the provinces, a woman needs more protection than in normal times. Running her fingers over the necklace, she realized it was a delicate version of the heavy chain Dominick wore.

The bracelet had the same pattern as the abalone in his shirt cuffs. Janelle blinked at her. Farimah sighed as she rose to her feet. The skirt fit low on her pelvis, showing too much of her abdomen. The hem almost reached her knees, but a slit went up the left side to her hip. She knelt by her box and withdrew a girdle designed from beaten coins, with a border of little gold bells.

Janelle squinted while they fastened it around her hips. Heavy and snug, the girdle fit over the skirt and sparkled with sapphires and mother-of-pearl. It jangled when she moved. Then Silvia brought out a bra made from silver coins, with loops of abalone and opal beads. Enough is enough, Janelle thought. It is too small. Silvia went to the door and knocked.

As the guard outside opened it a sliver, Silvia blocked his view of the room. A child squeezed past her, a girl of about three with black curls and a sweet face. Silvia glanced back at Janelle, her gaze malicious, then slipped outside and closed the door. The child ran to Farimah. Then she froze, her gaze darting to Janelle. Panic surged over her face. Puzzled, Janelle gave the child a friendly smile. Farimah lifted the child into her arms, her attention riveted on Janelle. Both Farimah and Danae had gone deathly pale.

Her mother died in childbirth. The girl was watching her with big, dark eyes that somehow looked familiar. Janelle stared at Farimah. Selena hardly sounded like a concubine, if Dominick had lived monogamously with her for so many years, raising a family. Had some stupid prophecy kept them from marrying? No wonder Farimah resented her. But her frozen look thawed a bit. She took the girl to the door and gave her into the keeping of someone outside.

Silvia returned then, watching them with an avid gaze. Janelle wanted to sock her. Silvia could have kept the girl outside and protected Farimah from that heart-stopping moment when the grandmother realized she would have to tell Janelle about the children. What had Silvia hoped to achieve? It created a dynamic foreign to Janelle, an unstated enmity and maneuvering for sexual power. Silvia was a beauty, with glossy black hair and a voluptuous figure. Maybe she believed discord between his new wife and the mother of his former favorite could work to her advantage.

Janelle had no interest in such machinations. Compared to this place, her world was so enlightened it glowed in the dark. Given the halter Silvia was holding, they would have to melt the damn things. It held her breasts in a scanty gold mesh with a few jewels in strategic places and more of those bells fringing the bottom. Her groom would certainly have no trouble finding her, given all the noise she would make in this outfit. Her companions regarded her politely. You obviously have the first two. Maybe a few of the others. Janelle barely managed to hold back her retort.

They ignored her protests and inflicted make-up on her next. Silvia brushed her hair, working until she had dried and fluffed up the curls. Then they took her into the bathing chamber, where a long mirror hung on the wall. Her reflection stopped her cold. She glistened in gold and sea colors. Her eyes looked larger and greener than normal, and her hair floated around her shoulders like a gold cloud. Even her bangs curled in traitorous perfection.

She had to admit, the effect was impressive—and in that it became seductive. They turned her into a woman of mystery and beauty, and it tempted her to believe it increased her worth. Farimah threw up her hands. She had no clue what that entailed, but she suspected she was supposed to think of ways to entice the groom. Maybe she should entertain herself by deriving equations for the sinusoids on the walls. She stepped up on the bench in the bathroom to look out the window—at a spectacular panorama. Mountains towered on both sides, east and west. In the south, before her, they dropped to a mesa several miles distant, where mounted riders moved in chess-like patterns.

It had thousands of men. Then again, maybe Maximillian was a saint and Dominick just coveted his throne, as disenfranchised brothers had since time immemorial. Wood grated in the other room. Janelle returned to the bedroom and found a group of strangers waiting for her. Six older women stood in the front, their carriage and jewels surely marking them as noblewomen.

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Blue silk wraps covered them from neck to ankle, making Janelle even more self-conscious about her skimpy attire. Behind them, an array of servants carried platters of food. They offered her the feast and waited while she ate. Everyone declined her invitation to join in, but no one seemed offended by the thought. The meal was delicious, though odd, with Janelle standing up, surrounded by silent people, sampling foods and wine. Right now, a few shots of whiskey would have done nicely.

When she finished, they took her outside. Twelve warriors waited in the corridor, hulking in armor, with what looked like ceremonial broadswords on their backs, the gilded hilts inlaid with jewels. While the servants took off with the platters, the noblewomen and soldiers escorted Janelle the other way.

She went in a daze. She wanted to believe this was a delirium; maybe a car had hit her and she was lying in a hospital. But it felt all too real. Up ahead, shouts echoed in the halls. It seemed out of place with the reserve of the people here. Crashes sounded in the distance. More shouts came, and the halls vibrated with a great pounding. The guards split their group into two, half of the warriors taking the noblewomen one way and the others hurrying Janelle into a side corridor. They ran hard, with drilled precision, while all around them the rumble intensified.

The Spacetime Pool

A rangy soldier kept pace with Janelle. The rumble surged into a roar—and raiders thundered out of a cross-hall, all astride biaquines. The man in front brought his mount to an abrupt halt, and it reared, its hooves smashing the pillar of an arch that framed the corridor. Biaquine screams rent the air, and metal rang as swords flashed. The rangy soldier pulled her into a side hall, and they ran hard down the corridor. The bells on her clothes chimed as if announcing their location. Only a few lamps lit the area. Despite the dim light, her guard took the turns with confidence, always choosing hallways too narrow for a biaquine.

Until they hit a dead end. I saw no symbols I recognized on those men. Taking a lamp off a hook in the recess, he motioned her forward. She entered the passage. Even if he were willing to commit such an atrocity, too much chance exists that in the heat of the attack, you would be killed despite his orders. His answer had an obvious corollary: They followed an ancient tunnel. Cracks cut through the walls, and lichen encrusted them in eerie patterns.

The damp air smelled musty, and the stone chilled her bare feet. She shivered, wishing she had more clothes. Then it hit Janelle: Wave functions oscillated down here, too, engraved in the stone. She indicated the patterns. The levels above are in better repair.

A murmur of flowing water came from ahead. The path widened into an open area, and a crude rail blocked the way, with walkways curving to either side. She went to the rail and looked down into a well about ten feet across. It plunged into darkness. She toed a pebble over the edge, and a good five seconds passed before she heard a faint splash. Janelle had already stepped forward when the lamplight revealed the ground had collapsed into the well. She jerked back and stumbled into the guard. Grasping her shoulder with a steadying hand, he held her until she caught her balance.

She stared bleakly at the fissure. It was too large to jump, and the rail that bordered the well was broken. Although two sheets of wood lay across the gap, neither looked solid. Whatever bridge they had once belonged to had fallen into neglect.

Her guard squinted at the boards. The tunnel contained nothing they could use to repair the bridge, and the rail around the well consisted of sections too short to bridge the gap. The chill seeped into Janelle, and the clink of her clothes seemed muted in the damp air. She pried off the bracelets and anklets and hid them in a crack to retrieve later—if she survived to tell anyone. The guard knelt to examine the boards. Or captured, which could be worse. His gaze never wavered. In battle, death is always possible. She knelt next to him. People know of these tunnels.

You must not be caught. The rest is secondary. But it is my honor to serve Prince Dominick-Michael. She managed a smile for him. She walked forward, her hand clenched on the lamp, and the span bent under her weight. Halfway over, one of the boards snapped. Janelle flailed, dropping the lamp, and it plummeted into the well.

As she fell to her knees on the remaining board, darkness closed around her. A splash took away the last hint of light. Are you all right? She moved another inch and her knuckles hit the jagged, broken edge of the path. Even as relief surged over her, the remaining board creaked. In the same instant that she threw herself forward, the board snapped and dropped out from under her.

Her torso landed flat on the path, but her legs hung into the fissure.

She scrabbled at the ground, frantic as rocks fragmented under her and clattered away. With a heave, Janelle hauled herself onto the path and sprawled on her stomach. She groaned as the girdle jabbed her skin. Instead, you treat us with grace. I am just a soldier. I have no great knowledge of other places. But it seems to me that you are a gift to His Highness. She valued the chance to learn other cultures. But according to their ways, Dominick and his people had treated her well. The last passage will let you out near there. She went with care, probing each step with her foot before she put down her weight, lest she stumble into another chasm.

Her palm hit stone. Alarm surged through her, but she pushed it down and searched the surface. She did indeed find tiles, as Kadar had described, and she pushed them in the sequence he had given her.

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