brownian motion

Euclidean spaces of more than three dimensions were first described in 1852. Four-dimensional Euclidean space is the natural generalization of everyday geometry; whereas we have three perpendicular dimensions $(x, y, z)$, 4D space has four coordinate axes $(w, x, y, z)$―all perpendicular to each other.

It was the zoning board’s fault. The Green Line Extension Project had hit a dead end halfway through the construction of the tunnel. They couldn’t dig leftwards because of a historic church, they couldn’t dig forwards because of an underground river protected by environmental law, and they couldn’t dig rightwards because the newly sprouted “Save Brighton Street Community Garden!” Facebook group was now endlessly harassing the local representatives. The representatives had caved.

The Facebook protestors were happy. Alice was not. She stared at the blueprint with the resigned fury of a civil engineer whose life had been derailed by committee indecision, constant bureaucracy, and Bostonians’ apparently innate need to protest everything in their power. Her carefully calculated vectors, her precise excavation schedule—scrapped. Every possible route through three-dimensional space had been vetoed by politics, sentiment, or sediment.

“If I can't dig around,” she muttered, “I'll dig through.”

THURSDAY TO-DO LIST.
Call Professor Chen from the local university.
Call again because she never answers the first time.
Ask HR to hire math intern. Send service request to fix coffee machine.

Perhaps the best way to think about interacting with four-dimensional space is to imagine an analogue in three dimensions. In three-dimensional space, you can step over a drawing of a line on the floor. In four-dimensional space, you can “step over” any three-dimensional obstacle in the way.

The construction was surprisingly uneventful. The tunnel-boring machines were procured. Mathematicians saw record low rates of unemployment. Salvador Dalí, hired in a posthumous consulting role via séance and AI reconstruction, insisted that the trains must melt slightly at the corners. “It is only proper,” he muttered, “that reality drip a little when pierced.” Some strange occurrences were documented, but unexpected events were to be expected. At first, people noticed small things: a coffee shop that shouldn’t be there yet, a familiar stranger reading tomorrow’s news. Some passengers began setting their watches to station time, which could run fast, slow, or occasionally in reverse.

Alice looked at the reports as they started to come in. They didn’t add up.

CHOICE COMPLAINTS FROM FIRST MONTH OF OPERATION.
“The conductor is my future son. He won’t let me ride unless I raise him right.”
“The train skipped my stop. Also, my childhood.”
“My CharlieCard is making me pay for rides that haven’t happened yet.”

A four-dimensional trip through Euclidean space should not have these side effects. The Multidimensional Bay Transportation Authority opened an investigation. Blueprints were compared; passengers were interviewed; metric spaces were measured. The cause was pinpointed. Due to some glaring oversight, the tunnel inexplicably went through time rather than space.

Minkowski space combines inertial space and time manifolds into a four-dimensional model. It is a representation of ordinary spacetime on which a time dimension has been glued on: $(t, x, y, z)$.

The architects blamed the mathematicians. The mathematicians blamed the diagrams. One intern, sobbing, admitted they’d copy-pasted the wrong tensor field from a theoretical physics subreddit. It didn’t matter. The trains were already running. The MBTA simply put up more signs.

NEW MBTA SIGNS TO SEND TO PRINT SHOP.
“Children under 12 ride free provided they haven't already boarded in future.”
“All paradoxes must be tagged and stored in the Lost & Found.”
“Commuters are advised not to make eye contact with themselves in transit.”

Now, when a rider boards at Fenwood Station, the train shifts its worldline outside of the Minkowski light cone, slipping sideways through causality by leveraging an incredible abuse of general relativity. In practice, this means the subway sometimes takes shortcuts through yesterday’s geography, or glides silently through a Tuesday that hasn’t happened yet.

Some swear the ride is instantaneous. Others insist it takes an hour and a half and ends with a gentle sense of déjà vu and the scent of cedar. A man once emerged claiming to have met his own grandson, who explained his retirement options in fluent Cantonese. Ticket prices fluctuate according to entropy, which now no longer appears to be monotonically increasing. Peak hours are notoriously unpredictable, though Wednesdays at 3:17 p.m. are generally stable.

Alice waits for the 9:15 train to take her to the office by nine o’clock. She swears that the new line is no more difficult to navigate than the rest of Boston’s subway system. The Brighton Street Community Garden remains untouched. The underground river continues to flow.

And the church bells still ring on Sundays, sometimes an hour early but always exactly on time.