It was not until later, when Ethan was trapped, alone in another world, that he would reflect on what might have happened had the coin landed the other way up. Perhaps it was always going to land that way, its fate, like his, a slave to the laws of causality. Perhaps everything that had happened since was not his fault after all.
He recalled how the coin had spun in the weak glow of the ceiling lamp, casting a fleeting shadow across the windowless hall. The shadow passed over a mirrored sphere, as tall as he was, resting on a low plinth. The coin’s reflection glinted in the sphere’s surface, as if the sphere itself was winking at him.
Ethan reached out to catch the coin. Heads.
He glanced at the reflection in the sphere. Tails.
***
It was Professor Taraskin who had set Ethan on this path, several weeks earlier. Ethan had knocked on his advisor’s door that day and paced about nervously until an abrupt ‘Come,’ broke the silence.
‘Mr Brice,’ Taraskin said, as Ethan stepped tentatively into the professor’s office. ‘You have something to show me this time?’
‘I’m afraid I’ve reached an impasse,’ said Ethan.
Ignoring Taraskin’s look of feigned surprise, Ethan turned to the blackboard. He picked up a stub of chalk and transcribed the steps of his deadlocked calculation. At first, the algebra flowed naturally from one line to the next. Then, as with his previous attempts, one term of the equations refused to simplify. The taint in his formulae took hold, and within a few lines he reached a familiar dead end.
He stared helplessly at his work, as if this time the offending terms might resolve themselves. It was like reading a novel only to find its final pages missing. The computation, which described the interactions of subatomic hexon particles, might only have been a modest contribution to theoretical physics, but it would have been his contribution, the first step on his path to emulating his father and becoming a professor. That dream, the only one he had ever known, now seemed impossibly distant.
‘I see aetherics is proving more difficult than you anticipated,’ Taraskin said.
Ethan turned reluctantly from the blackboard.
‘There’s another problem too.’ He fixed his gaze on the patch of carpet at Taraskin’s feet. ‘I managed to secure the final night slot at the Freeman Field Manipulator next week. I have to finish this calculation by then so I can compute the corresponding Manipulator settings.’
He stole himself for Taraskin’s inevitable rebuke. Since their first meeting a year ago, during which Taraskin had made it clear he believed Ethan had only been accepted onto the doctoral program because of his father’s standing, the professor had never missed an opportunity to belittle him. He must be relishing seeing Ethan flounder like this.
‘If you waste the slot, they won’t grant you another,’ Taraskin said. ‘I’ll see to that.’ He studied the board for a moment, then added, ‘Unfortunately, since it is not just your reputation at stake, I will have to help you.’
Taraskin reeled off the names of several researchers whose journal articles he suggested might provide a route through the obstruction in Ethan’s calculation. Ethan scribed the references frantically onto his Babbage screen.
‘Come back the morning before your slot,’ said the professor. He smirked, then added, ‘if you still haven’t figured it out, it may be time to consider an alternative career.’
Ethan spent the following days locked in his office in an endless cycle of hope and frustration, pulling paper after paper from the hub, and working through each one in the vain expectation of finding an idea that would unblock his calculation. As the days passed, he followed a trail of increasingly obscure references, until he was finally confronted with two papers too old to be found on the hub.
With his meeting with Taraskin looming the following morning, Ethan cycled straight to the University Library in search of physical copies of the papers.
The first piece, a slim nineteenth-century textbook entitled On the philosophy of discrete aethers by Goodwin, contained a series of meandering monologues on how to interpret the physics of the time. There was nothing within it of use to Ethan though.
After fruitlessly searching the bookstacks for the second item, he approached a librarian and unrolled his Babbage screen, on which he had scribbled the reference, Journal de Mathématiques Pures et Appliquées, January 1864, Jean Gaston Darboux. The librarian directed Ethan towards a room in the basement he had not previously known existed.
Ethan padded along a bookcase in the rayed light of the basement window, following the receding dates embossed onto the spines of the journals, until he found himself staring up at two copies of the January 1864 edition. Why there was a duplicate of this edition only did not concern him; there were more important matters at hand. He eased one of the books off the shelf and laid it carefully on a nearby desk.
The cover was charred, its faded letters all but illegible. As he lifted it, an acrid smell displaced the familiar mustiness of the library. The first few pages clung to each other, as if unwilling to yield their secrets. Ethan rifled through it, wincing as pages came loose from their brittle binding.
The volume contained a compilation of research papers from numerous authors. Darboux’s paper, when Ethan finally found it, comprised a series of lengthy derivations, interspersed with French prose. It was indecipherable but for the occasional familiar-looking word. A part of him felt reluctant to study it further. There was nowhere else to look after this, but as long as it remained unread, there was hope of salvaging his calculation. And his career.
Nevertheless, as the sunbeams traced an arc across the dusty floor, Ethan worked through the paper, reproducing the formulae for himself, filling in missing steps and recasting the archaic notation in modern form. Normally, he would have derived a deep satisfaction from the task, savouring the manner in which terms cancelled as he manipulated them, his real-world troubles forgotten. Today, however, the symbols seemed restless, as if aware of the importance of the calculation.
When he reached the end, he leant back, closed his eyes and mentally worked through the trail of papers he had been following. His uneasiness increased with each paper, and by the time he had reconsidered Taraskin’s original suggestion, it had grown into a panic that smothered all other thoughts. The techniques he had spent the last week developing would not bypass the blockage in his work.
He slammed the journal shut in disgust. A yellowed page slid out and floated to the floor. He sighed and picked it up. At first glance it appeared blank, but when he held it towards the window faded marks caught the light. In one corner were the initials MG and a date, 1st July 1877. The remainder of the page was covered with barely legible handwritten equations. The crossed-out formulae and smudged symbols suggested somebody in frenzied excitement, rushing to commit an idea to paper before it escaped their grasp. Quite what the idea was remained unclear, but there was a familiarity to the work, and Ethan had the sense of someone, another struggling student perhaps, following a similar line of enquiry to his own. But to expect this scrap of paper to help him was, he realised, nothing more than a mark of his own desperation.
He flipped the sheet over to reveal more equations and, close to one blackened edge, the words, “What is Cedric’s orb?” scrawled in cursive hand.
A polite cough jolted his attention from the page and he turned to see a librarian standing in the doorway.
‘I’m afraid we’re closing now,’ the librarian said, with an apologetic smile.
Without thinking, Ethan slipped the sheet between the pages of his notepad. He replaced the volume on the shelf, and hurried out of the library.
***
Ethan cycled across the twilit city towards the Freeman Field Manipulator, whose great hemispherical dome dominated the horizon. Nestled in the Manipulator’s shadow was a cluster of buildings forming the Cambridge Fundamental Physics Institute, and there, in the outlying Anderson building, was his office, a windowless box off a basement corridor.
There was room for little more than a desk and chair in the office. But for Ethan it was enough that it formed part of the Institute, and in spite of Taraskin’s threats he could not conceive of working elsewhere. Nor, for that matter, could he imagine working with any other professor, for few could match Taraskin’s brilliance. Many times during the past year, as he struggled with his work, he had imagined himself making a discovery of his own, earning not just Taraskin’s respect, but his envy too.
He sat at his desk and wrote up some notes to show Taraskin tomorrow. Before long he heard a series of muffled cries from beyond the door. He stepped out of his office toward the source of the noise.
The corridor receded into dimness in either direction, but before Ethan had gone far, another door was jerked open. A suited man strode out shouting, ‘… the space, and the decision is final!’ He slammed the door behind him and, ignoring Ethan, marched off towards the exit. Ethan glanced at the nameplate on the door – Clarence Kettle, Emeritus Professor of Emergent Physics. He caught himself before a smile could form. Kettle, a lone, aged scientist, whose backwater research was of little interest, must just have been visited by one of the Institute’s administrators and told he no longer even merited an office in this forgotten corridor.
The door opened again and a diminutive figure rushed out, head bowed. Before Ethan could move his bulky frame aside, she had collided with him. Her Babbage machine, which had been clutched to her chest, clattered to the floor.
Ethan loitered awkwardly, one arm half extended, as she picked up the machine. Only then did he recognise her as Olivia Windell, an algorithmist working on the colossal Babbage engines that controlled the Manipulator. He had often noticed her around the Institute, but had never before mustered the courage to speak with her.
‘What were you doing in Kettle’s office?’ he asked.
‘Well, nice to meet you too,’ Olivia replied, hinging open her machine to check for damage. ‘I’m fine, by the way.’
Ethan’s face began to burn. This was not how he had imagined their first meeting might go. ‘I only mean I didn’t think you did emergent physics,’ he stammered. ‘You’re just an algorithm writer aren’t you?’
Olivia’s eyes widened. ‘I’m sorry, I didn’t realise only important people were allowed down here. And who are you again?’ She folded the display over the keyboard and stalked off down the corridor.
***
After finishing his notes, Ethan made his way to his room, in a featureless accommodation block on the far side of the city. For a time he moped on his bed, reliving his excruciating encounter with Olivia, while rolling a coin over his knuckles, a nervous habit he had had for as long as he could remember.
As the evening light faded, he dragged himself over to his desk for a final attempt at his calculation, only to encounter the same impenetrable obstruction that had plagued him for so long, like a glass wall blocking his way against which he could find no purchase. He leaned back and tossed his pencil aside. Tomorrow morning he would have to confess to Taraskin that he had made no progress.
Only then did he remember the sheet of paper he had found in the library. Carefully, he removed it from his notebook.
But for a single word “Suppose” part way down one page, and the question, “What is Cedric’s orb?” on the other side, it contained nothing but erratically written equations in old-fashioned notation. He flipped the paper over, wondering which side came first. It likely contained nothing more than the indecipherable musings of a nineteenth century researcher, doggedly pursuing one of the many incorrect theories of the time. Certainly there was nothing here that would help him overcome his own problems.
Nonetheless, Ethan appreciated the elegance and flow of the work. He studied the column of equals signs on the left, where the author had fought line by line to simplify the terms in the calculation. This was clearly a fragment of a much longer derivation, whose beginning and end were both missing.
Several times, as he followed the algebra, he was convinced he had found a mistake, only to recall an obscure theorem, applied here in an ingenious manner, which justified the step. Each innovation sent the calculation in an unexpected direction. Although Ethan had no idea what the symbols represented, he could not help but sense the excitement MG must have felt.
The word “Suppose” preceded the introduction of a new formula, of unknown origin. He felt a flicker of recognition; he could not recall seeing the equation before, yet its shape was somehow familiar. The equations halted, unfinished at the end of the page.
Sensing the direction in which the algebra was heading, Ethan ripped a fresh sheet from his notepad and continued the calculation. It seemed to proceed of its own accord, Ethan merely a spectator as his pencil flew across the page in a scramble to record each line of equations. Perhaps MG had experienced the same fervour in 1877.
After several more pages, it was clear he had reached a terminus; there was a finality about the last expression, with no further simplifications available. But what the expression meant remained unknown.
Ethan stretched his arms over his head and gazed through the crack in the curtains. The stars were fading into the brightening sky and a lone warbling blackbird heralded the beginning of the dawn chorus. He had wasted the whole night.
Yet he did not regret studying the note. Never before had he experienced such purpose in a calculation, nor such a feeling of resolution upon finishing it. He looked at the question, “What is Cedric’s orb?” on the note, and wondered what it meant.
With his mind abuzz and sleep a distant prospect, he thought he may as well return to his office. He would spend the remaining time before meeting Taraskin working out how best to explain that the final night slot would go to waste, before the Manipulator’s immense negatonic magnets were deactivated for the winter.
The city seemed indifferent to Ethan’s troubles as he cycled towards the Anderson Building; the air was still and clear, marred only by his cloudy breaths, and a light frost lay upon the ground. He had just finished fumbling with his frozen bicycle lock when Olivia Windell strode past. He looked up eagerly, then cringed and averted his gaze as he recalled their previous meeting.
‘Morning, then.’
She had stopped and was staring at him. Tousled hair fell about her pale face and her eyes, which yesterday had shone with zeal, were bleary.
‘Just finished the night shift?’ he asked, glancing at the concrete mountain behind her, whose upper slopes glowed in the sunrise.
‘My penultimate one until the spring,’ she said. ‘You look like you’ve been up all night, too.’
Ethan smoothed down his hair self-consciously. ‘And now I’m off to get yelled at by Taraskin for failing to finish my sums,’ he said.
For a moment the terms in his failed calculation hovered before him, as if he might grasp them and force them into the required form. When they faded from view, Olivia had already begun to walk away. He lingered for a time, savouring this brief connection in their otherwise disjoint existences.
***
Ethan entered Taraskin’s office in the Newton Building to find the professor hunched over a pad of paper, pencil in hand, as if to show he had far more important theories to consider. After an interminably long time finishing whatever he was writing, he looked up. Ethan passed on his news with a simple shake of his head.
Taraskin seemed to tower over him, despite remaining seated. ‘You are overconfident in your abilities,’ he began. ‘That might be forgivable, given your father’s standing. But to abuse my name to secure your slot at the Manipulator, while overstating the importance of your work …’
Having already decided not to engage in this part of the conversation, Ethan let Taraskin’s admonishment wash over him and resigned himself to his situation with surprising calmness. After a year of repeated failures, it would almost be a relief if Taraskin refused to continue working with him. He gazed out of the window behind the professor, where a murmuration of starlings rippled hypnotically across the sky. A bird of prey, which had been hovering ominously above, tucked in its wings and plunged into the murmuration, which flowed apart into two distinct flocks.
The professor’s rising voice dragged Ethan’s attention back to the room, though only as far as a small blackboard beside the window. On it was a single formula, its shape familiar … if the theta and sigma were replaced with –
‘What’s that equation?’ Ethan said.
Taraskin seemed momentarily taken aback by the temerity of the interruption, but soon recovered his stride. ‘I have not finished,’ he barked. ‘You –’
‘I’ve got something!’
Taraskin glared dangerously at Ethan. ‘This had better be good.’ He paused, then added, ‘I obtained the expression recently. It’s a reformulation of the equation describing the coupling field.’
And in that instant Ethan knew what he had derived the previous night.
Ignoring Taraskin’s protests, he stepped up to the larger blackboard on the wall, swept the board rubber across it and wrote down the standard equations of discretised aetherics. Within a few lines he had transitioned into MG’s algebra. Where MG had introduced his new formula, Ethan instead substituted in Taraskin’s equation. Such was the strength of the narrative in the derivation that he remembered each line with ease.
He filled the blackboard with the remainder of the computation then stepped back to contemplate his work. It appeared to provide a connection between the established physics of discretised aetherics and the coupling field, an additional fundamental force discovered at the Manipulator only weeks ago. But how could MG have known this in 1877?
He turned from the blackboard to find Taraskin standing beside him. His anger appeared to have dissipated.
‘Show me how you obtain these lines.’ The professor pointed in turn to several lines of algebra where Ethan had applied MG’s novel techniques.
When Ethan had described the techniques, the two of them turned their attention to the last line. ‘What happens if you take the limit as we approach the Noetherian length scale?’ said Taraskin. There was an eagerness in his voice now. He passed the chalk to Ethan, though there were plenty of sticks on the shelf below the blackboard.
Ethan squeezed another line of algebra onto the board, letting the variables approach the tiny length scale suggested by his advisor. ‘I don’t understand,’ he said. ‘It seems to … separate.’
‘A bifurcation,’ Taraskin said, ‘though what it means physically, I don’t know.’ He pushed his hair behind his ears, depositing chalk marks onto his cheeks, then paced before the board. Finally he stopped and turned towards Ethan. ‘I propose we discard your work on hexon states for now,’ he said. ‘Let us instead ascertain whether you can use your Manipulator slot tonight to investigate this bifurcation formula.’
Ethan let Taraskin guide him through the steps required to derive the Manipulator settings from the formula. His lack of sleep soon began to tell and his contributions became increasingly sparse. Taraskin, meanwhile, seemed to relish the opportunity to demonstrate the mathematical abilities that had earned him such renown. By the time the frost had melted from the grounds of the Institute beyond the window, they had computed an unusual configuration of fields, at the limit of the Manipulator’s capabilities, that would allow them to explore the physical implications of the bifurcation formula. For the first time in days, Ethan felt a faint hope that his career might be salvaged after all.
As Ethan opened the door to leave, Taraskin called out, ‘Wait!’
He paused, one hand on the handle.
‘How did you come up with this?’
Ethan shrugged.
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