Page 20 of Katabasis

T hey made camp when they were too tired to keep walking.

Alice checked the time—one in the morning, much later than they ought to have slept, but they had both kept pushing through that last stretch of their hike, forcing numb feet onward along the banks.

They were both rattled by the bone-things.

Alice was unable to shake the creeping, sticky fear from her throat, the certainty they were being watched, and the only solution seemed to be putting as much distance between them and the bone-things as they could.

This, despite Hell’s purported infinitude and unreliable topology.

Despite the fact that for all their efforts, whoever or whatever it was might still appear before them in a blink of an eye.

Alice sat, chewed through a stick of Lembas Bread, and tried not to let her despair creep.

Fortunately graduate school had prepared her for this, the constant managing of despair.

Everything was always falling apart; nothing in lab went right; you couldn’t afford groceries, your cottage had a rat problem, all your instructors hated you, you were always one step away from flushing all your life’s work down the toilet.

You shoved it to the side of your mind and went to sleep and deferred it all to tomorrow when your brain again functioned well enough to pretend.

Peter hissed as he pulled a strip of gauze from his arm.

“How’s it look?” she asked.

“Not infected, I don’t think.” He held his wrist over the fire, examining the wound. “You’d be able to tell at this point, wouldn’t you?”

“Want some more antiseptic? Just in case?”

“Yeah, all right.” She passed him her travel bottle of merbromin. He dabbed a droplet onto his arm, then held out his wrist so she could rewrap it in gauze. “Thank you.”

She settled back against her pack and closed her eyes. “Of course.”

“Hey—let’s try something.” Peter pulled a box of chalk from his bag. “I wonder if we might speed things up.”

Reluctantly Alice lifted her head. “How do you mean?”

“Well, what’s the worst thing you’ve ever seen Professor Grimes do?”

“We’ve already had this conversation.”

“I’m not saying we should find the end point,” said Peter.

“I’m just saying, maybe there’s another shortcut across the courts.

Let’s just assume we have a good guess where he is.

Suppose this—he’s done worse than greed, surely?

” Peter drew an enviously perfect little circle by his feet.

These were called test circles: pentagrams drawn in miniature for safety checks before you stepped inside the real thing.

These were best practice for spells that involved any movement or bodily alteration.

You put your hand in the test circle, and if you didn’t lose your fingers, then you could chance putting your whole self in. “Worse than wrath?”

Alice thought of flying spittle, of mugs smashed across tiled floors. Rare instances—but clear in her memory nonetheless. Professor Grimes never had patience for stupidity. “Probably, yes.”

“Then we ought to just skip ahead to cruelty, shouldn’t we?” Rapidly Peter scribbled a series of algorithms around the circle. It seemed to involve a lot of maths. Alice saw more geometry than she did Greek, and this made her head hurt. Peter set down his chalk. “What do you think, Law?”

“Hold on,” she said. “Backtrack. How does your shortcut work?”

He pulled his notebook out of his rucksack, opened it to a page in the middle, and tossed it at her. She took one glance and instantly her thinking mind shut down, as it always did when confronted with a lot of numbers. “You need to explain that to me like I’m five.”

“Gabriel’s Horn,” he said happily. “Also called Torricelli’s Trumpet. It’s a mathematical paradox that bounds a finite volume within an infinite surface area. The plane of Hell is that infinite area, and in configuring us as the volume inside the horn, we could make a finite shortcut...”

“Sorry, what?”

“It’s a bit complicated,” he admitted. “You have calculus?”

She’d taken it in college; she hardly remembered it. That was one of the joys of specializing in linguistics: the escape from pure maths. “Only the basics.”

“So then you remember curves, right?”

“Let’s pretend I do.”

“Basically, you rotate one branch of a hyperbola around its asymptote. That gives you a shape that looks a bit like a trumpet. Hence the name! Very apt, eh?” Alice was too confused to laugh.

Peter looked disappointed but continued.

“Now you can calculate the surface area with these equations here”—he tapped pertinent equations in the pentagram—“and also the volume here”—he tapped it again.

“The surface area is infinitely big, just like Hell! But the volume, miraculously, is bounded. Some people call this the Painter’s Paradox, because you can theoretically fill up this whole bounded area with an unbounded amount of paint. ”

She blinked at the page. “That’s just your twisted pseudosphere again.”

“It looks like the twisted pseudosphere,” said Peter. “Mathematically it is very different. You’re right that they both rely on some assumptions about hyperbolic space. But this spell doesn’t assume Hell is hyperbolic. Rather it produces a hyperbolic solid within the circle.”

Alice could never make up her mind on how she felt when Peter explained things. On the one hand, it was so condescending, the way he postured as if he were her tutor. On the other hand, he was very good at maths, and she really didn’t know this stuff, and competence was always attractive.

“Murdoch, this makes no sense.”

“It’s a paradox, Law! It’s all a bit abstract, I know.

But I think if I can draw the right equivalencies, we might bound the infinite space of Hell into something like—a shortcut.

A wormhole, sort of. Here: I’m going to make a shortcut ten yards to your left, just to demonstrate.

You’ll see my hand stick out of the sand.

Don’t be alarmed.” He chanted out his incantation—two steady minutes of incomprehensible geometry and calculus, and then waggled his fingers over the circle. “Voilà.”

Nothing happened.

Alice peered at the ground. All the silt inside the pentagram looked the same to her. “Well?”

“Huh.” Peter frowned at his inscriptions. “That’s funny.”

“What’s wrong?”

“It’s not sticking.” He poked at the dirt. “The ground, it’s—it’s swallowing it, almost.”

Alice slid her fingers through the sand.

It was true that chalk held better on firmer surfaces, but the good brands left smooth, solid lines no matter what the surface.

And these dirt particles didn’t seem particularly special in any way.

Perhaps they were darker, more glassy, more fine and siltlike than the sand particles you might find at a beach.

In grade school, she’d done science fair experiments with cornstarch where it held solid in your hand if you squeezed it, but melted if you let it go.

The silt felt like that. Solid to the touch, until you weren’t looking.

Discomfiting, certainly. Still, it was just sand: solid, stationary, dry.

“What grade are you using?” she asked. “You’ve got to go softer with sand.”

“I’m using 5H.” Peter clawed at the pentagram. An obsidian stream poured from his fingers; no white particles in sight. “Look! It’s not that it doesn’t draw. It just—it faded away. Like the silt absorbed it all.”

“Have you ever tried this spell yourself?”

“No, it’s new...”

“Well, could be you’ve missed something.

” Alice pulled her own chalk out and arced out a circle beside her feet.

“Small errors in the margins can make the pentagram erase itself. It’s rare, but it happens sometimes, especially in linguistics.

” She began tracing the premises of the Sorites Paradox.

Just a basic spell, something they all learned in their first year.

Sorites spells did not achieve great effects, but still no one had ever come up with a satisfactory debunking, and so they always worked at least for a little.

“A heap remains a heap,” she recited in Greek. “Remove a grain, and the heap remains...” Her voice faltered.

Her pentagram, like Peter’s, sank away into the silt.

No Barkles’-over-Shropley’s superiority, then. “I don’t understand.” She pawed at the sand. She could not see even a hint of white. All trace of the chalk was gone. “That shouldn’t—it’s never not worked before.”

Peter scooped a handful of silt into his palm and poked at it with his fingers. “I wonder if it’s a problem with Hell itself.”

“How do you mean?”

“I mean we have no understanding of Hell’s metabolism. Or Hell’s entropy.” He let the silt stream through his fingers. “Possibly its energy flows are all out of whack and it’s eating the chalk, eating its living-dead energy, instead of glitching against it—”

“But the sojourner’s accounts,” she said. “They were using magick the whole way through, and they were fine. Dante, Orpheus—”

“They weren’t drawing pentagrams, though. The magician’s accounts are all pre–Carne Abbas. They had enchanted objects, that’s not the same.”

Alice rifled through her mental catalog, trying to find a counterexample, and came up with none. “But that means...”

“That means we have no magick.”

Alice considered the implications of this, and dread congealed in her chest.