She couldn't believe him. That he sounded sane made none of this more plausible. “And you have no idea how this gate works?” she challenged.

His gaze flashed. “Of course I do. It's a branch. From here to your mountains.”

“A tree, you mean?”

“No. A branch cut to another page. Your universe is one sheet, mine is another.”

She gaped at him. “Do you mean a Riemann sheet? A branch cut from one Riemann sheet to another?”

“That's right.” He hesitated. “You know these words?”

She laughed unsteadily. “It's nonsense. Not the sheets, I mean, but they're just mathematical constructs! They don't actually exist. You can't physically go through a branch cut any more than you could step into a square root sign.”

He was watching her with an expression that mirrored how she had felt when he told her about his prophecy. “I have no idea what you're talking about.”

“It's glorious,” Janelle said. “Is this the Hall of Arches?”

“No. The Fourier Hall.”

“Fourier?” She blinked. “Like the mathematician?”

The woman gave a sharp wave of her hand. “It has always been called this. That is all I know.”

Janelle didn't push. 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. She had discovered early on that if she wasn't certain how her words would be received, it was often better to say nothing.

She couldn't stop staring at the arches, though. 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. Such a strange thought, that she could capture in mathematics the essence of a dream palace that couldn't exist.

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.

Think about the graph of the function f(x)=e^x. 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: if the point (a,b) is on the graph, that means that when f(a)=b. We can invert the function by reflecting its graph through the line y=x. Then we get the graph of the function y=ln(x), its inverse. This also passes a vertical line test and now has the property that (b,a) is on the graph if (a,b) was on the graph of y=f(x).

All of that is fine and normal precalculus stuff. But, things get messier when we try to include trig functions in the discussion. The graph of the sine function (of course) satisfies the vertical line test. However, if we try the same trick of reflecting it to get an inverse, we run into trouble. If the sine graph runs up the y-axis, then it does not satisfy a vertical line test. Technically, this means that the sine function does not have an inverse. That's not a surprise, it is a consequence of the fact that the sine function takes the same value at many different points. So, whereas it makes sense to ask for the value of x at which e^x takes the value 1 (at x=0), one cannot do the same for sine because it takes the value 1 at infinitely many different x-values. Still, we want to have something almost like an inverse for it. So, the standard thing one does is take just a little piece of that flipped sine graph, a piece which does satisfy the vertical line test, and call that the inverse. This works fine as long as one stays in the middle of that piece, but things seem unpleasant when you get to the end and "fall off the end of the world". That's when you realize that there is not just one choice of such a "piece" of the graph, there are many of these "branches", and when you leave one, you can just jump to another "branch" and that makes things better. Those places where you switch from one to another are the "branch cuts".

Even though everything I've said above was in the context of real numbers, I'm not aware that anyone worked out the mathematical details in this context. Instead, this was really investigated in earnest for the first time in the context of the logarithm function on the complex plane. The thing is, unlike f(x)=e^x for real numbers, for complex numbers this function takes the same value at different input values. (For instance, f(2πni)=1 for any integer n and so there is no longer a single input value that produces the output value 1.) It was in this context that the logarithm function began to be thought of as having a graph that looks like a parking garage, where each floor of the garage is a "branch", a full collection of inverse values for the exponential function, but which goes up and up covering each of the different possible choices that would make the inverse "function" single-valued.

The next leap the field takes is to begin to think of these "Riemann surfaces" as geometric objects in their own right. In other words, forget about having to start with a function and find an inverse, just think about the ways in which one can join branches together with cuts to make surfaces. This not only becomes part of modern geometry, but has applications in quantum physics (where, for instance, Feynman integrals are computed as sums over the set of Riemann surfaces) and in my own field of soliton theory (where even ocean waves are studied in terms of "spectral" Riemann surfaces).