A Subway Named Mobius
1950
In a complex and ingenious pattern, the subway had spread out from a focus at Park Street. A shunt connected the Lochmere line with the Ashmont for trains southbound, and with the Forest Hills line for those northbound. Harvard and Brookline had been linked with a tunnel that passed through Kenmore Under, and during rush hours every other train was switched through the Kenmore Branch back to Egleston. The Kenmore Branch joined the Maverick Tunnel near Fields Corner. It climbed a hundred feet in two blocks to connect Copley Over with Scollay Square; then it dipped down again to join the Cambridge line at Boylston. The Boylston shuttle had finally tied together the seven principal lines on four different levels. It went into service, you remember, on March 3rd. After that, a train could travel from any one station to any other station in the whole system.
There were two hundred twenty-seven trains running the subways every weekday, and they carried about a million and a half passengers. The Cambridge-Dorchester train that disappeared on March 4th was Number 86. Nobody missed it at first. During the evening rush, the traffic was a little heavier than usual on that line. But a crowd is a crowd. The ad posters at the Forest Hills yards looked for 86 about 7:30, but neither of them mentioned its absence until three days later. The controller at the Milk Street Cross-Over called the Harvard checker for an extra train after the hockey game that night, and the Harvard checker relayed the call to the yards. The dispatcher there sent out 87, which had been put to bed at ten o’clock, as usual. He didn’t notice that 86 was missing.
It was near the peak of the rush the next morning that Jack O’Brien, at the Park Street Control, called Warren Sweeney at the Forest Hills yards and told him to put another train on the Cambridge run. Sweeney was short, so he went to the board and scanned it for a spare train and crew. Then, for the first time, he noticed that Gallagher had not checked out the night before. He put the tag up and left a note. Gallagher was due on at ten. At ten-thirty, Sweeney was down looking at the board again, and he noticed Gallagher’s tag still up, and the note where he had left it. He groused to the checker and asked if Gallagher had come in late. The checker said he hadn’t seen Gallagher at all that morning. Then Sweeney wanted to know who was running 86? A few minutes later he found that Dorkin’s card was still up, although it was Dorkin’s day off. It was 11:30 before he finally realized that he had lost a train.
Sweeney spent the next hour and a half on the phone, and he quizzed every dispatcher, controller, and checker on the whole system. When he finished his lunch at 1:30, he covered the whole net again. At 4:40, just before he left for the day, he reported the matter, with some indignation, to Central Traffic. The phones buzzed through the tunnels and shops until nearly midnight before the general manager was finally notified at his home.
It was the engineer on the main switchbank who, late in the morning of the 6th, first associated the missing train with the newspaper stories about the sudden rash of missing persons. He tipped off the Transcript, and by the end of the lunch hour three papers had Extras on the streets. That was the way the story got out.
Kelvin Whyte, the General Manager, spent a good part of that afternoon with the police. They checked Gallagher’s wife, and Dorkin’s. The motorman and the conductor had not been home since the morning of the 4th. By mid-afternoon, it was clear to the police that three hundred and fifty Bostonians, more or less, had been lost with the train. The System buzzed, and Whyte nearly expired with simple exasperation. But the train was not found.
Roger Tupelo, the Harvard mathematician, stepped into the picture the evening of the 6th. He reached Whyte by phone, late, at his home, and told him he had some ideas about the missing train. Then he taxied to Whyte’s home in Newton and had the first of many talks with Whyte about Number 86.
Whyte was an intelligent man, a good organizer, and not without imagination. “But I don’t know what you’re talking about!” he expostulated.
Tupelo was resolved to be patient. “This is a very hard thing for anybody to understand, Mr. Whyte,” he said. “I can see why you are puzzled. But it’s the only explanation. The train has vanished, and the people on it. But the System is closed. Trains are conserved. It’s somewhere on the System!”
Whyte’s voice grew louder again. “And I tell you, Dr. Tupelo, that train is not on the System! It is not! You can’t overlook a seven-car train carrying four hundred passengers. The System has been combed. Do you think I’m trying to hide the train?”
“Of course not. Now look, let’s be reasonable. We know the train was en route to Cambridge at 8:40 A.M. on the 4th. At least twenty of the missing people probably boarded the train a few minutes earlier at Washington, and forty more at Park Street Under. A few got off at both stations. And that’s the last. The ones who were going to Kendall, to Central, to Harvard — they never got there. The train did not get to Cambridge.”
“I know that, Dr. Tupelo,” Whyte said savagely. “In the tunnel under the River, the train turned into a boat. It left the tunnel and sailed for Africa.”
“No, Mr. Whyte. I’m trying to tell you. It hit a node.”
Whyte was livid. “What is a node!” he exploded. “The System keeps the tracks clear. Nothing on the tracks but trains, no nodes left lying around—”
“You still don’t understand, A node is not an obstruction. It’s a singularity. A pole of high order.”
Tupelo’s explanations that night did not greatly clarify the situation for Kelvin Whyte. But at two in the morning, the general manager conceded to Tupelo the privilege of examining the master maps of the System. He put in a call first to the police, who could not assist him with his first attempt to master topology, and then, finally, to Central Traffic. Tupelo taxied down there alone, and pored over the maps till morning. He had coffee and a snack, and then went to Whyte’s office.
He found the general manager on the telephone. There was a conversation having to do with another, more elaborate inspection of the Dorchester-Cambridge tunnel under the Charles River. When the conversation ended, Whyte slammed the telephone into its cradle and glared at Tupelo. The mathematician spoke first.
“I think probably it’s the new shuttle that did this,” he said.
Whyte gripped the edge of his desk and prowled silently through his vocabulary until he had located some civil words. “Dr. Tupelo,” he said, “I have been awake all night going over your theory. I don’t understand it all. I don’t know what the Boylston shuttle has to do with this.”
“Remember what I was saying last night about the connective properties of networks?” Tupelo asked quietly. “Remember the Mobius band we made — the surface with one face and one edge? Remember this—?” and he removed a little glass Klein bottle from his pocket and placed it on the desk.
Whyte sat back in his chair and stared wordlessly at the mathematician. Three emotions marched across his face in quick succession — anger, bewilderment, and utter dejection. Tupelo went on.
“Mr. Whyte, the System is a network of amazing topological complexity. It was already complex before the Boylston shuttle was installed, and of a high order of connectivity. But this shuttle makes the network absolutely unique. I don’t fully understand it, but the situation seems to be something like this: the shuttle has made the connectivity of the whole System of an order so high that I don’t know how to calculate it. I suspect the connectivity has become infinite.”
The general manager listened as though in a daze. He kept his eyes glued to the little Klein bottle.
“The Mobius band,” Tupelo said, “has unusual properties because it has a singularity. The Klein bottle, with two singularities, manages to be inside of itself. The topologists know surfaces with as many as a thousand singularities, and they have properties that make the Mobius band and the Klein bottle both look simple. But a network with infinite connectivity must have an infinite number of singularities. Can you imagine what the properties of that network could be?”