thinking today about party poison and cherri cola being literal angels from the phoenix witch
both terrible and beautiful to behold, hard to understand and communicate with but also so alluring
like poison will fix you with a cold stare if you approach her at a bar, and there’s something deeply unsettling in their eyes. she doesn’t say anything when you buy them a drink, but you feel warm to your bones and the clap the next day ends without bloodshed
or cherri will spend days alone in the wastes, smiting evil in a way too bright and grotesque to witness (why he does it so far away from human eyes). but you can’t deny the desert is safer now
Pain and dizziness and cold, terrible cold, and she’s moving him.
Then a bed. Mercy. No, not mercy, she doesn’t, she doesn’t hurt him anymore — he was good, he told her about the safehouse, she won’t hurt him now, no, that’s, no, no —
His limbs sink into the bed, and he’s sure, absolutely sure, that now that he’s been placed there he’ll never get up again. His stomach and his ribs feel like they have icicles thrust into the wounds, his hands are going numb, he shivers violently, exacerbating the pain. His body shudders with coughs, blood spilling from his lips onto the pillow. He’s patched the lung. He won’t die. Now it’s just bleeding from the trauma of using that last bit of magic, the bit that keeps his soul together.
— “c-cold,” he mumbles, desperate to let her know this, because he thinks maybe, perhaps, she doesn’t want to torture him anymore, she said no torture, and if she didn’t want to torture him she wouldn’t let him be this cold another instant—
At the sight of the scissors a pitiful whimper rises from his throat, followed by another weak, bloody cough, and then he falls back, awaiting whatever might come.
—
Cold. Yes, worryingly cold, he’s shaking - that’s a good sign?
Worse to have stopped…
“Alright,” she agrees, “Hang in there, don’t pass out…"
Cold first, or wounds first? Wounds. She thinks. She bundles the covers awkwardly around his legs and shoulders, but she knows it won’t help much if he’s not making any heat. And she doesn’t want to cover his torso while she still needs access to the wounds. They don’t own a hot water bottle - should she put him under a shower to warm him up, or would that make him worse?
"I’m going to dress these real quick,” she explains as she cuts his clothes away from the injuries. “Then I’ll – try and get you warmed up, just hang in there."
She can only pray that he’s done enough for the internal injuries as she works as fast as she can to clean the open wounds and stick them shut. It’ll sting, so she repeats herself as she goes. "Just cleaning these, you’re good, you’ve done nothing wrong, it’ll fade in a minute."
—
"T-thank y-you,” he says through chattering teeth, because if there’s one thing that she’s trained into him, it’s politeness. It costs lives, to give her a safehouse; costs nothing, to give her a thank you.
Why is he with her?
Because everything is wrong.
He flinches, winces but doesn’t make a peep other than thank you while she dresses his wounds and he shivers violently, then weaker, losing the energy to shiver. C-cold. He already told her that, he can’t tell her that again, if he’s still cold it’s because she wants him that way, or does she — between blood loss, magical exhaustion and its close companion, hypothermia, his thoughts are muddled, thudding off the frozen corners of his skull and making no sense.
—
Still moving mostly on autopilot, Ariadne scours the room for anything she can use to warm Alex up, shoving their possessions into bags as she goes. She tosses every blanket they own on top of him, but she knows it won’t make much difference. He needs a heat source, and she can’t think of anything.
There isn’t time to figure it out. She takes the bags to the car – jittery with nerves the whole while Alex is out of her sight – turns the engine on and cranks the heat up as far as it will go before she goes back for Alex.
They don’t need to go far, not tonight. They only need to not be here, not be in the building where any initial search will be focused. Thank god no one found the body yet, she thinks as she pulls away.
She has to leave Alex again while she checks them both into a nice little bed-and-breakfast place. Fear slides uncomfortably round the inside of her ribcage even though she knowshe’s as safe in a locked car as he could be anywhere. She hurries back for him.
He only protests weakly when she picks him up. And he tucks his head against her neck in a way that helps both to hide how sick he looks and to sell the illusion of a normal couple just being sweet together as she carries him into the building.
Still, she’s relying on the inherent indifference of strangers, just like Alex did when he was the one looking after her. All it’d take is one busybody looking too closely, calling the police to report something suspicious…
She sets Alex down on the bed, and unfolds the blankets from round him just long enough to check that he hasn’t bled through the dressings. “C-c-cold, ‘nterrogator,” he whimpers. “I know,” she tells him softly. His skin is still freezing to the touch. “I know, I’m sorry.”
She finds the room’s thermostat and she turns it to max. She searches the drawers and the closet, hoping for a hot water bottle, but all she finds is more blankets. She’s still not sure if putting him in a shower would be good or bad.
All she knows she has for sure is body heat.
So she shrugs off her jacket and her boots, eases the covers carefully out from under Alex, and lays herself down alongside him before pulling the covers back over them both to keep out the air.
—
Alex registers that something is warm, that there is some source of warmth for his body, and losing all other thought, he whines, presses towards the warmth.
And then he realizes it’s her and she’s holding him, and his body stiffens. He doesn’t dare pull away, doesn’t dare press closer, even though he wants both of those things simultaneously, wants to get away from the person who hurts him and surround himself with her all at once.
So he’s still.
“Th-thank you,” he mumbles again into the pillow, because never once have those words failed him.
—
It’s no surprise that he goes rigid. She’s the last person he’d want touching him. But it can’t be helped, he’s so fucking cold in her arms and he’s barely shivering any more. She’s scared. She doesn’t really understand what is wrong with him - whether it’s the effects of using up his magic, or the blood loss, or some other condition she doesn’t know enough to recognise.
But he’s far too cold, and he could still be dying.
“You’re safe,” she tells him anxiously. “No more hurting, I’m done treating those wounds. I’m not going to hurt you, you’re safe."
She lines her body up with his, pushing her warmth against his coldness all along the length of their legs, torso close against his, head against his shoulder. The practical math of maximising contact.
Only when she stops moving, satisfied with the position, does it really sink in how uncomfortable this is. Intimate, in a very different way from the familiar, guilty intimacy of watching him break down with pain and fear. He’s not just a cold bundle of limbs that she needs to warm up. He is a living, breathing, feeling person. Afraid of her touch. Guilt is a stab in the heart. She can only hope that as they lie still and nothing worse happens, the fear will slowly ease.
So she is still, breathing in the smell of him, listening to his breath and praying that it doesn’t stop. Acutely aware of the contact between their bodies, feeling the chill slowly diminish. Or at least, the perception of chill. Is he warming up, or is she just getting colder where they’re touching? But he’s not quite so stiff and his breath sounds a little better, and she starts to relax. It’s… nice, almost. Holding him close. He’s fragile. Helpless. She’s not sure if it’s a protective feeling or a possessive one, and the uncertainty makes her sick with herself. She has not done a good job of defending him, if that is what she wants.
—
The cold is — less, or as less as it’s going to get, when his very soul is cold, the blood running through his veins freezes his heart as it goes through, this cold goes deeper than the chill of being out too long on a winter’s day. The pain is less, too. Nothing he hasn’t dealt with.
In captivity, he’d fallen asleep with broken ribs, gasping shallow breaths, hoping the way his breathing made him dizzy would lure him into unconsciousness. He’d always fallen asleep knowing she could come in with a whip or a pair of brass knuckles at any time. After enough nights not sleeping, that doesn’t matter. After gutting out his magic to stop himself from bleeding all the blood in his body into his chest, that doesn’t matter either.
He’ll just sleep, while she’s letting him. If he wakes to pain — so be it.
And today she says I’m not going to hurt you, and that usually means she won’t, for at least a few hours. Perhaps after a while she’ll get bored, and see that stab wound, and wonder if she can press her fingers in and make him scream. But for now, the exhaustion goes almost as deep as the cold.
The way her arm curves over his shoulder is comforting, the way he can feel her breathe against his scarred back. Comforting. How? Because he can pretend she’s someone else. That must be it.
He whispers thank you again, for good measure, for the promise of not going to hurt you, and his eyes flutter closed.
—
Is he sleeping? Should she wake him, is it bad to fall asleep while he’s so cold? Or does he need sleep, to help him heal? She can’t bring herself to wake him, not with the certain knowledge of how his breath will catch and his eyes widen as he wonders what new pain is in store today. He deserves a little peace. He deserves more than that, so much more… but a little bit of peace is all she can offer.
She dozes too, eventually, when his breathing has been steady long enough that the terror of losing him eases its grip on her ribcage. Too shallow, thankfully, to dream. She knows how bad it will be, after this.
This is the first of two posts based on a talk given at Dmitriy Bilyk at our probability seminar. I don’t usually go to probability seminar, but have enjoyed Dmitriy’s talks in the past, and I didn’t regret it this time either :)
——
Unsatisfactory Methods
Here is how not to pick a point uniformly on the sphere: don’t pick its polar and azimuthal angles uniformly. This doesn’t work because it will incorrectly bias things toward the poles.
The problem is that this cylindrical trick only works on the 2-sphere. There is no natural notion of “cylindrical coordinates” for higher-dimensional spheres, because how many coordinates do you take linearly and how many do you take as angles?
Bilyk did not say this, but presumably no choice you can make, or at least no consistent choices that you can make for all $n$, such that you get a uniform distribution— otherwise it really would have been a short talk :P
What Bilyk did say is that there are several ways to choose points uniformly from a sphere, but “there are very few deterministic methods”. But before we can tackle these methods, we actually need to answer a more fundamental question: what does it mean to choose deterministically choose points uniformly? Generally “choosing uniformly” means sampling from a constant random variable, but that X is clearly not available to us here.
——
Riesz Energy
One way to go might be to minimize the Riesz energy. Whenever we have a collection of points $Z=\{z_1,\cdots z_N\}$, can write
We see that if the $z_k$ are very close to each other, this energy is going to be large; a set $Z$ that makes the energy small will be one whose points are generally “as far from each other as possible”. Since it’s a sphere, you can’t get too far away, and so there’s an interesting optimization to be done here.
So this seems nice, but there’s a problem. It turns out that minimizing the Riesz energy is just, like, stupendously hard. The best exact result we have was given in 2010, when Schwartz proved that $E_1$ is minimized for $N=5$ (!) by the triangular bipyramid.
To give some indication about why the problem is hard: the triangular bipyramid is not the minimizer for the Riesz energy with $N=5$ for all $s$. It is suspected that it minimizes it for all sufficiently small $s$; but one thing we know for sure that when $s$ gets large enough, the square pyramid is better.
Conjecture. There exists a critical value $s’$ such that for all $1\leq s<s’$, the triangular bipyramid minimizes $E_s$, and for all $s’<s$, the square pyramid minimizes $E_s$.
This conjecture is wide open: we don’t even know that the square pyramid minimizes $E_s$ for any value of $s$!
——
Discrepancy and a Theorem of Beck
Another, rather different way to measure the uniform-ness of a set $Z$ is by computing its discrepancy. The formal definition of discrepancy is really a lot scarier than necessary, so I won’t write it here. The idea is that if you pick any (measurable) set $S$, you can either
calculate the measure of the $S$, or
count how many of the points $z_i$ live inside $S$
Most of the time, these two numbers are going to be different, and the discrepancy $D(Z,S)$ is the difference between the numbers, divided by the number of points in $Z$. But this is not the most useful number because we have this extra data $S$ hanging around; the better idea is to let the discrepancy be the maximum of $D(Z,S)$ over all $S$ (technically the supremum).
Intuitively speaking, that if you were to have a $Z$ for which the discrepancy in the latter sense were small, then $Z$ looks “uniformly distributed”, even if $Z$ is deterministic.
However, measurable sets can look pretty wacky, and so in order to let geometry reign over set theory, it often helps to be a little more refined. Given a collection of sets $\mathcal S$, we say that the discrepancy $D(Z,\mathcal S)$ is the supremum of $D(Z,S)$ over all $S\in\mathcal S$. So it’s basically the same thing as what we did above, except that instead of doing all measurable sets, we only do the ones in the collection.
Figuring out optimal discrepancies is also not very easy, but people have over the years figured out strategies for determining asymptotic bounds. And even that tends to be pretty tough. For instance, if we’re considering things on the sphere, it may seem reasonable to look at $\mathcal S$ the collection of spherical caps:
What is the best available discrepancy in this setting? The answer, morally speaking, is that it’s “close to $1/\sqrt{N}$”, but it has a small dimensional correction:
Theorem (Beck 1984). For any positive integer $N$, there exist positive constants $C_0$ and $C_1$ such that
So you can always find a set $Z$ that does a little better, asymptotically, than $1/\sqrt{N}$; what exactly “a little” means depends on how high-dimensional your sphere is.
——
In the next post, we talk about more recent developments, including a third notion of uniform distribution which will bring us all the way to Bilyk’s work in the present day.
Isn’t the new Pixar short “The Blue Umbrella” just another re-telling of this story as shown in the classic disney short “Johnny Fedora & Alice Bluebonnet"?
…So I made a comic based on the end of this fic I wrote, but all you really need to know is that Dazai’s been pining about Chuuya’s hands since they defeated Randou, and he deals with this in a Very Normal Person way by holding Chuuya’s hand whenever he’s passed out post-Corruption use. This part takes place during that one soukoku scene in Dead Apple in the fog.
… also there are two parts to this, each with nine pages, and I’m putting most of the pages under read mores because otherwise this is going to be hellish to scroll past
