Having a day free, Kit, Yashmeen, and Günther decided to make a farewell visit to the little-known but rewarding Museum der Monstrositäten, a sort of nocturnal equivalent of Professor Klein’s huge collection of mathematical models on the third floor of the Auditorienhaus. They traveled in a motor-diligence out toward the Brocken. The brushland grew hilly and witchlike, clouds came from directions indeterminate and covered the sun. “An older sort of Germany,” commented Günther, with a less-than-reassuring smile. “Deeper.”

It was not so much a conventional museum as a strange underground temple, or counter-temple, dedicated to the current “Crisis” in European mathematics . . . whether intended for exhibition, worship, study, or initiation could not be read from any exterior, because there was none, beyond an entranceway framing a flight of coal-black steps sloping downward in a fathomless tunnel to crypts unknown. As if to express the “imaginary” (or, as Clifford had termed it, “invisible”) realm of numbers, the black substance from which it was constructed seemed not so much a known mineral as the residue of a nameless one, after light, through some undisclosed process, had been removed. Now and then a load-bearing statue was visible in the form of an angel, wings, faces, and garments streamlined almost to pure geometry, and brandishing weapons somehow not yet decipherable, featuring electrodes and cooling fins and so forth.

They found the inside strangely deserted, lit only by a few whispering gas-sconces which receded down the corridors leading away from the shadows of the entrance-hall. Yet there was the smell of German tidiness constantly exercised, of Sapoleum and floorwax, of massive applications of formalin gas still pungently lingering. The corridors seemed swept by generations of sighing, which occasionally had reached wind force—a sadness, a wild exclusion from the primly orthogonal floor-plans of academic endeavor. . . .

“Somebody must be in here, at work?” it seemed to Kit. “Guards, staff?”

“Perhaps they hide from visitors they do not know,” Günther shrugged. “How could anyone’s nerves here remain unafflicted?”

From time to time, where there was light, it was possible to make out spacious murals, almost photographically precise, their colors unmodified by the daily interior purifications, depicting events in the recent history of mathematics, such as Knipfel’s Discovery of the Weierstrass Functions and the recently installed Professor Frege at Jena upon Receiving Russell’s Letter Concerning the Set of All Sets That Are Not Members of Themselves of von Imbiss, which exhibited parallax effects as one walked past, with such background figures as Sofia Kovalevskaia or a mischievously hydrophobic Bertrand Russell actually entering and departing the scene, depending on the viewer’s position and velocity. “Poor Frege,” said Günther, “just about to publish his book on arithmetic, and this happens —here he is basically saying ‘Kot!’ which is German for ‘How much will it cost me to revise these pages?’ You see the way he appears to be striking himself on the forehead, which the artist has cleverly indicated with the little radiating streaks of green and magenta...”

Signs directed them down into a corridor vaulted with iron trusses, which led to a series of panoramas of quite stupefying clarity known to convince even the most skeptical of visitors, who might find themselves surrounded 360 degrees by a view of ancient Crotona in Magna Grecia, beneath the precipitously darkening sky of an approaching storm, with robed and barefoot Pythagorean disciples, in some spiritual transport whose illumination was mimicked here by the fluorescence of gas-mantles soaked in certain radioactive salts . . . or seeming to have entered the very lecture hall at the Sorbonne where Hilbert on that historic August morning in 1900 was presenting the International Congress with his list of the celebrated “Paris problems” he hoped to see solved in the new century—yes, here was Hilbert beyond a doubt, Panama hat on his head, somehow optically presented as three-dimensional and even more lifelike than a figure in a wax museum, even to the thousands of drops of sweat running down everybody’s face. . . .

According to the design philosophy of the day, between the observer at the center of a panorama and the cylindrical wall on which the scene was projected, lay a zone of dual nature, wherein must be correctly arranged a number of “real objects” appropriate to the setting—chairs and desks, Doric columns whole and damaged—though these could not strictly be termed entirely real, rather part “real” and part “pictorial,” or let us say “fictional,” this assortment of hybrid objects being designed to “gradually blend in” with distance until the curving wall and a final condition of pure image. “So,” Günther declared, “one is thrust into the Cantorian paradise of the Mengenlehre, with one rather sizable set of points in space being continuously replaced by another, smoothly losing their ‘reality’ as a function of radius. The observer curious enough to cross this space—were it not, it appears, forbidden—would be slowly removed from his four-dimensional environs and taken out into a timeless region. . . .”

“You will want to go that way, Kit,” Yashmeen said, indicating a sign that read ZU DEN QUATERNIONEN.

Of course, of course, no business of Kit’s, they obviously needed some time together, departure in the air, things to say. . . . Released, Kit descended dark stairways uncomfortably steep even for the moderately fit—as if modeled after some ancient gathering-place, such as the Colosseum in Rome, stained with Imperial intention, promises of struggle, punishment, blood sacrifice— and stood at last before a rubber curtain, waiting, until it was mysteriously drawn aside and he found himself injected into overamplified Nernst light at the verge of white explosion, and there he was, undeniably at the canalside in Dublin sixty years ago as Hamilton received the Quaternions from an extrapersonal source nearly embodied in this very light, the Brougham Bridge receding away in perfect perspective, the figure of Mrs. Hamilton gazing on in gentle consternation, Hamilton himself in the act of carving into the bridge his renowned formulae with a pocketknife part real and part imaginary, a “complex” knife one might say, though a “real” reproduction of it was on view in a nearby gallery dedicated to famous “props” in the general mathematical drama, pieces of chalk, half-finished cups of coffee, even a thoroughly crumpled handkerchief, said to have belonged to Sofia Kovalevskaia and dating from Weierstrass’s time in Berlin, an example of Lebesgue’s notorious “surface devoid of tangent planes,” an eccentric distant cousin of the family of functions, everywhere continuous and nowhere differentiable, with which Weierstrass, in 1872, had inaugurated the great Crisis that continued to preoccupy mathematics even to the present—there in its own freestanding display case, under a hemisphere of glass, illuminated from somewhere below, preserved in a constantly renewed atmosphere of pure nitrogen. How did this handkerchief get into its tangentless condition? Repeatedly wadded up in a tightened fist? Opened, cried and noseblown into, squeezed again back into a tight ball? Was it a record, a chemical memoir, of some extraordinary passage between the kindly professor and the student with the eloquent eyes? From wherever she had been, Yashmeen had reappeared to take Kit’s arm and gaze for a while at the forlorn relic.

“She was always my inspiration, you know.”

“All jake with you and the Teutonic god there?”

“He’s very sad. He said he will miss you. He wants to tell you himself, I think.” She wandered off as Günther, his eyes gleaming in the shadow of his hatbrim, approached Kit with a look of deep, though not fathomless, discontent. He was enroute to Mexico to manage one of the family coffee plantations. His father had been adamant, his uncles were looking forward to his arrival.

“Practically my neck of the woods,” Kit said. “If you get up to Denver—”

“It is our strange German vertigo, everything in motion, like water draining down the sink, this unacknowledged tropism of the German spirit toward all manifestations of the Mexican, wherever they may occur. The Kaiser now seeks in Mexico the same opportunities for mischief toward the U.S. as Napoleon III before him . . . no doubt I have some blind pathetic little part to play.”

“Günni, you seem kind of, I don’t know, short on that old selfconfidence today—”

“You were right, you know. That day of our duel. I have been only another Rosinenkacker on holiday, lost in his banal illusions. I must now bid farewell to the life I might have had, and take up again the stony road, a pilgrim on a penitential journey. No more mathematics for von Quassel. It is a worldline I shall, after all, never travel.”

“Günni, I was a little harsh, I think.”

“You will be kind to her.” With a, you might say, Germanic emphasis on “will” that Kit didn’t know how seriously to take.

“I’m her road partner for a week or so, that’s really all it is. Then, so they tell me, other forces come into play.”

“Ach, das Schicksal. From chloral to coffee,” Günther brooded. “The antipodal journey from one end of human consciousness to its opposite.”

“Fate is trying to tell you something,” Kit speculated.

“Fate does not speak. She carries a Mauser and from time to time indicates our proper path.”

They moved along in regret and reluctance, feeling through the ponderous stone envelope the afternoon as it deepened. Back in town waited another evening in its coercive penultimacy, and yet none could quite suggest detaching from these corridors commemorative of the persons they had once imagined themselves to be . . . who, each of them, had chosen to submit to the possibility of reaching that terrible ecstasy known to result from unmediated observation of the beautiful. Were their impending departures not only from programs of mathematical study but indeed from further hope of finding someday that headlong embrace?

“Children.” The voice could not be located, it was everywhere in the corridors. “The Museum is closing now. The next time you visit, it might not be exactly where it stands today.”

“Why not?” Yashmeen could not refrain from asking, though she knew.

“Because the cornerstone of the building is not a cube but its four-dimensional analogy, a tesseract. Certain of these corridors lead to other times, times, moreover, you might wish too strongly to reclaim, and become lost in the perplexity of the attempt.”

“How do you know?” said Günther.

“Who are you?”

“You know who I am.”

from Against the Day by Thomas Pynchon (2006)