Hieda no Akyuu hummed softly as she set out the tea things on the low table. The ninth Child of Miare wondered if she ought to wash the ink stains from her fingertips before her guest arrived, but decided that it probably wouldn't matter to Kozusu. The bookshop owner would probably appreciate the point, come to think of it.

"Hard at work again, are you?"

Akyuu yelped, dropping the teapot. It plunged towards the bare wood floor, only to plunge through a hole that suddenly appeared in midair below it. An eye seemed to wink at her from out of the gap before it closed again.

"I apologize for startling you."

Akyuu turned to the source of the familiar voice, which she'd already placed from memory as belonging to Yukari Yakumo. Sure enough, the tall, blonde youkai was already seated on the other side of the tea table. Last time she'd visited, she'd been wearing a violet Western-style dress with long white gloves; today, she had on a white one with deep purple insets and carried an ornate fan. With her other hand she extended the fallen teapot to Akyuu.

"I suppose that I should have announced myself properly at the front door, but the last time I paid a call that poor maid of yours nearly had a fit of nerves."

"Well, isn't that the appropriate response when face-to-face with a youkai?" Akyuu poured tea into one of the cups.

Yukari chuckled softly and accepted the offered tea. "Perhaps so. But you don't seem to be subject to such reactions."

"Well...I did want to see you, as a matter of fact, so I'm rather glad you called." She wondered for a moment how, exactly, Yukari had known that Akyuu wanted to see her, given that she'd only mentioned it to Kozusu and not even to the Hakurei shrine maiden, but gave it up as a fruitless thought.

To say nothing of how she dropped in just in time for tea.

"Indeed?"

"It's about the Chronicle..." Akyuu began, then glanced down at her fingertips.

"A brush isn't quite as efficient as modern methods of printing, but don't you find it has an elegance all of its own?"

"Oh, yes! I didn't know that you were a writer, though, Miss Yukari."

The youkai sage offered a shrug.

"I dabble. Or did you mean, 'she rarely does things herself, leaving practically all of her work to a beast she controls'?"

Akyuu probably would have blushed at the unflattering reference to Yukari's laziness from the Chronicle, had it not been for the fact that it was one of the edits Yukari had inserted to the volume.

"...Either one?"

Yukari smiled at her and took a sip of tea.

"In all seriousness, though, what would you like to know?"

"Well, it's a bit hard to ask, but it's about something that I wrote about you that made me curious. It's about mathematics."

"Oh?"

"I understand that shikigami would have a high degree of mathematical knowledge, but..." She trailed off again, finding it hard to say things like, but what does a gap youkai use it for? It was easy enough to understand that Yukari was considerably more intelligent than a human, but she didn't quite follow why it was in that particular area of study.

"It seems like an unusual hobby for a youkai?" Yukari suggested after letting her dangle for a moment.

"It rather does, if you don't mind?"

"No."

Akyuu blinked.

"Wait, did you mean 'no, it's not an unusual hobby,' or 'no, I don't mind you asking'?"

"Yes," Yukari said, and drank more tea. "It's good for that as well."

"What is?" Akyuu said, completely lost by now.

"But the main point is, without an advanced knowledge of mathematics and the ability to make high-speed calculations, I wouldn't be able to properly exercise control over boundaries," Yukari took pity on her.

"Really? It seems so unusual. After all, youkai are creatures that are inherently fantastical. Even Gensokyo itself exists because the boundary between reality and fantasy is embodied in the Great Hakurei Barrier, doesn't it? But math is, well, it's so grounded in specifics and rigorous rules, that it...doesn't seem very youkai-like."

Yukari chuckled softly.

"You're greatly underestimating mathematics if you think that, Miss Hieda. On the contrary, math is just as capable of creating entire constructs of rules and systems that exist to describe what does not and, indeed, cannot exist, simply because it happens to be useful to do so. After all, if you think it is complex to make calculations concerning what exists, then how much more complex does it have to be when the world outside is telling you that the subject of your thoughts does not exist?"

Akyuu wasn't entirely sure if she understood that, but it reminded her of something Ran Yakumo had once said, that her mistress would be capable of calculating the depth of the Sanzu River, even though the Sanzu River was bottomless.

"Consider even something as simple as one of my gaps," Yukari elaborated. "Imagine for a moment the calculations required of how, exactly, two discrete points in time and space are connected to one another, and where the boundary lies between here and there, to do something like reaching out and fetching fresh tea from your kitchen to replace what I'm drinking here." The air in front of her shimmered, and opened into another "hole" in space, into which she leaned, her head and shoulders vanishing.

~X X X~

Reimu Hakurei sighed and leaned back into the steaming water. Youkai, she decided, should be courteous enough to harass people by night. Four hours of cleaning up the byproducts of two rival teams of kappa engineers fighting over the best way of finishing Kanako's latest scheme to bring science to the masses (and why wasn't Sanae exterminating problems in her own backyard, anyway?) left a shrine maiden tired to the bone, and it wasn't even seven at night yet.

Still, the problem was dealt with, and Reimu now had a hot bath, to say nothing of a full bottle of equally hot sake to warm her insides to match. There was a lot to be said for the simple comforts of home.

She wondered what it said about her life that she'd become so familiar with the sensation of the fabric of reality being ripped apart in her immediate proximity that she could recognize it immediately when it happened.

"Dammit, Yukari!"

~X X X~

The youkai sage's upper body jerked backwards out of the gap and she fell over, the wooden washtub that had apparently been the cause of her sudden repulsion (and her bleeding nose) rattling off the floor before the gap snapped shut.

"Miss Yukari, are you all right? What happened?" Akyuu gasped.

"...Forgot to carry the two."

~X X X~

A/N: Yukari's line, "It's good for that as well," is a reference to the "Mathematician's Answer": to answer either/or questions with yes/no answers (so-named because in math, "is it A or B?" would be answered "yes" if the answer was either A or B and "no" it was none of the above), which is what she was giving Akyuu.