#MATHS IS BETTER

[d4nny-z0la]

Good Evening.

I HATE English language and literature with my WHOLE SOUL.

Any English defenders can kindly piss off, thank you.

Maths is better.

[inkbybambi]

chemistry teacher soap catches gym teacher simon fucking you in the bathroom during a school dance, tattooed hand over your mouth to muffle the pretty noises you make. he doesn’t think either of you notice, until those dark eyes turn on him, a slight tilt to his head in invitation.

soap decides he has a better way of keeping you quiet, stuffing his cock down your throat, enjoying the way you choke on it and the way simon tugs him forward to lick into his mouth, a hum of approval low in his throat.

[visenyaism]

it’s been six years so i’m sure we’ve discussed this without me but the same cold snap that caused the terror to start terroring was also the one that fucked the donner party down in the sierras because 1846 weather simply had that dog in it. crazy

[cyfffff]

you’d think since dick was a mathlete hed b the number one guy to ask for help w math homework but actually he knows so much advanced math that the first time tim asked him for help with a simple geometry problem he pulled out his dusty ti-84 and started solving it in the most convoluted way that took an hour bc he was also just trying to relearn the math he hadn’t done in like 5 years. cue him screaming in frustration when he realized the actual way to do it would’ve taken him 2 minutes (tim asked him at the beginning if that way was right and he waved him off saying no) none of the batsibs can ask him for math help until they get to pre-calculus at least

[inkskinned]

love when men cry about body hair bc "it's hygiene" and yet 15% of cis men leave the bathroom without washing their hands at all and an additional 35% only just wet their hands without using soap. that is nearly half of all men. that means statistically you have probably shaken hands with or been in direct contact with one of these people.

love when men say that women "only want money" when it turns out that even in equal-earning homes, women are actually adding caregiver burdens and housework from previous years, whereas men have been expanding leisure time and hobbies. in equal-earning households, men spend an average of 3.5 hours extra in leisure time per week, which is 182 hours per year - a little over a week of paid vacation time that the other partner does not receive. kinda sounds like he wants her money.

love that men have decided women are frail and weak and annoying when we scream in surprise but it turns out it's actually women who are more reliable in an emergency because men need to be convinced to actually take action and respond to the threat. like, actually, for-real: men experience such a strong sense of pride about their pre-supposed abilities that it gets them and their families killed. they are so used to dismissing women that it literally kills them.

love it. told my father this and he said there's lies, damned lies, and statistics. a year ago i tried to get him to evacuate the house during a flash flood. he ignored me and got injured. he has told me, laughing, that he never washes his hands. he has said in the last week that women are just happier when we're cooking or cleaning.

maybe i'm overly nostalgic. but it didn't used to feel so fucking bleak. it used to feel like at least a little shameful to consider women to be sheep. it just feels like the earth is round and we are still having conversations about it being flat - except these conversations are about the most obvious forms of patriarchy. like, we know about this stuff. we've known since well before the 50's.

recently andrew tate tried to justify cheating on his partner as being the "male prerogative." i don't know what the prerogative for the rest of us would be. just sitting at home, watching the slow erosion of our humanity.

[kenjakusbraincum]

Heey, I LOVE your writings on soft sukuna, you write so beautifully🩷 please can you do one where he is jealous (fluff)😭🩷

Thank you sm for the kind words!!! Here's my best attempt at doing your idea justice <3

Sukuna has no real reason to be jealous. He practically owns you, controls every aspect of your life, who or what could he possibly be jealous of? Every servant who dared approach you in an inappropriate way would be dealt with swiftly. And you're a good pet, who has eyes for no one other than your master. You really don't give him a reason.

But there's this one thing... Since you've been so good and obedient, Sukuna has allowed you many liberties. You're permitted to skip around the mansion, watch Uraume cook, even enjoy little hobbies. You've tried many before you found that crocheting particularly piqued your interest. Ever since you've learned the basics, you've been spending hours working on perfecting your skills. At first it was cute, watching you squint in concentration as you move the hook. But then the math became really simple - having this hobby to keep you busy meant you approached Sukuna out of boredom a lot less. And he noticed it. It irked him, but you're not technically doing anything wrong. You were still as happy to serve him as ever, he just had to ask. But why would he have to ask? You should be all over him on your own. He should have to push you away, not beg you to give him attention. He didn't like this disturbance in your master and pet balance that this little hobby of yours caused.

He stands at the door now. You're crocheting again. You and your favorite servant laugh at your failed creation so sweetly, you don't even notice he's waiting. He clicks his tongue to establish his presence, and your servant falls to her knees immediately. You however, are not held to that high of a standard anymore.

"Master!", you call him, and hop up to greet him with a deep bow. Before he can say anything, you've picked up the piece of fabric you've been working on and ran into his arms to show him.

He looks at the ugly form and scoffs. "This is what I'm sponsoring?", he says and pulls a loose piece of yarn, making your little creation fall apart. He always was a bully, but you note his bad mood.

"I'm only a beginner...", you sulk.

"That much is obvious.", he flicks the yarn away and it falls onto the floor. Before you can bend to pick it up, he seizes your wrist and pulls you back. "Aren't you a little young to waste time with hobbies for the elderly?", he asks. You look at him with your cutest, practiced doe eyes, but it doesn't work.

"Come, pet. I know an activity more suitable for your age.", he says when you don't respond, and steps out of the room. You hop after him, unaffected by his condescending comments. You know that they're just for show. If he really thought you were a hag, you would've been gone a long time ago.

"Sitting at your throne all day?", you tease innocently and join him at his side, sliding your arm underneath one of his. You hope your playfulness will distract him from whatever is bothering him. "Or in a bath?" His lower set of eyes peeks at you and smirks, noticing that you're feeling particularly daring today. He's not sure how he feels about that. "Or in your bed." He rolls his eyes gently and opens the door to his chambers.

"At least then you'd be serving your purpose and actually spending time with your master.", he comments and shuts the door. His comment catches you a bit off guard and you stop in front of his bed. He makes his way towards you, and you look up at him with an insulted expression.

"Master, are you jealous of a ball of yarn?", you ask playfully, and squeal when he suddenly pushes you down to sit on the bed. Now you're at eye level... with his crotch.

"You've got quite a big mouth today. Put it to good use for a change, will you?", he runs his hand from the crown of your head to the back of your neck. You seem to have struck a nerve, so it really is the ball of yarn. Is it possible that Sukuna is this clingy?

"Will you?", he repeats and tugs on your hair and narrows his eyes. You smile obediently and reach behind him to untie his obi.

"Yes Master."

-

You try your best to manage the time you spend crocheting from then on, working on productivity in the hours that you dedicate to developing this skill. And it helps that you have a specific goal in mind now: helping Sukuna realize that this hobby is a friend, not an enemy. He still catches you engaging in it sometimes, and gives you a dirty look, but you're as quick as ever to drop what you're doing and join him. That seems to satisfy him.

When you're finally happy with the result of your creation, you look for Sukuna around the mansion. It's not really that hard to find him, as he frequents three places most of all: the dining room, his bedroom and his throne room. This time, he's sitting on his throne, and a small line of people wait for their turn to be gifted his attention. You on the other hand, don't have to wait in line to get it. His lower set of eyes spots you the moment you enter the chamber. You're allowed to roam the mansion, but barging in unannounced is not standard even for you.

Still, Sukuna has learned that you usually only feel daring enough to cross boundaries when you're sure he'll like what you have in mind. So for now, he will let this slide. He's bored as hell anyways. The people are dismissed and you pass by them on your way to his throne, nestled on a pile of bones. You stop in front of it and greet him with a bow.

"Master, I come to you with a humble offering.", you say with your hands on your thighs and your eyes fixated on the ground.

"Show me.", he says simply, but you recognize entertainment in his voice. You climb up the bones and feel his stare scan you from head to toe, before you sit on his knee.

"May I ask you to close your eyes?", you ask and flutter your lashes. Oh the way you seduce him. Who else could ask Sukuna to do something as dangerous as close his eyes? Give his opponent valuable time to land an attack. Who else could dare? And who else would he ever listen to and really close his eyes? Really do as he's told? Oh how safe he feels with you.

You take one of his large hands into yours, and gently pry his long fingers away to open his palm. He has beautiful hands. The only ones you've ever known, but you're sure they're the most beautiful hands in the world. So dangerous, so elegant. You want to press a kiss to his palm, but you hope your gift will have the same, maybe even more profound effect.

Something soft touches his skin, and then you speak, as politely as before. "You may look.", in your softest voice. And when he opens his eyes, he finds himself looking at you first. You're an offering on your own.

Then he looks at his hand. Two crocheted plush figures resembling him and yourself lay flat on his palm, connected through their holding hands. At first glance, it looks like they're two separate creations. In a sense, they are, but... He tries to part them.

"We're sewn together.", you explain. He hums in amusement and inspects your gift more closely. His plush is bigger, recognizable by the pink hair and four buttons for eyes. It's even wearing his favorite kimono. Yours is smaller and less detailed. You look like any other human when placed next to him, insignificant. But in a sea of pets, entertainers and lovers he's had in the past, he would never fail to recognize it as you.

He's spent so long looking at it with that face of his that you just can't read. You're starting to grow restless in his lap, and he feels your eyes dwell into his soul. When he looks back at you with one pair of eyes, your brows are furrowed in worry and you're fiddling your hands in your lap. He pats you on the head and pulls you closer, so you have no choice but to lean on his frame.

"It's beautiful, darling.", his fingers run through your hair, scraping your scalp softly. "No loose threads either.", he looks at you with all four eyes now, and you feel so small in his arms. You're not used to receiving this many compliments from Sukuna at once. Not ones that weren't directed at your body or performance. Especially not when he's looking at you so tenderly, when every word sounds so loving and genuine. "You've improved so much.", his hand is on your face now, and you catch him glancing at your lips. You part them to start thanking him, but you already know how much he hates listening to that.

You stay quiet instead, and lean closer, letting him take you. And he kisses you so softly, fingertips light against your heated skin. You feel like you're floating, like a lily pad in a warm pond. The littlest gesture of his affection has you melting in his embrace. The power he has over you... and how wonderful it is to surrender yourself to it.

None of the liberties and privileges you've been awarded with compare to this. You know that many pets have walked these halls before you. Many warmed his bed and claimed the title of his favorite. But how many loved him like this? Enough to dedicate time of their day to making intricate gifts. How many could say Sukuna kissed them lovingly, for no other reason than to show gratitude and affection?

You're flushed completely red by the time his lips leave yours. You can't hold the intensity of his gaze, as he stares at you in adoration. "I'm happ.. I'm glad you l-like it...", you stumble through the words and win a giggle out of him. You are just so cute. Like a pet should be. He rubs your head again and pushes you away lightly.

"Go now, the people await me.", he says with a benevolent smile gracing his face. "I'll see you tonight."

You bow to him and leave.

And when you visit him that night, he is as gentle as he was when he kissed you earlier, still in a good mood after your gift. Caressing your hair, shoulders and back, as you lay comfortably with your head on his chest. Keeping you warm in his embrace. You're trying your best to follow the conversation, but sleep is slowly taking over you. Sukuna notices and plants a kiss to your forehead, wishing you goodnight. The last thing you see before your eyes close, is your handcrafted plushies sitting on his nightstand.

[i-may-be-an-emu]

Shout out to people who can not tell the time

Shout out to people who need a little longer to figure out the time

Shout out to people who can't do "quick" maths in their head

Shout out to people who need to use a calculator for even "simple" maths

Shout out to people who need others to read number a for them

Shout out to people who cry over maths and numbers

Shout out to people who say the wrong numbers when reading or talking about prices and the time of day

Shout out to people who can't read charts and graphs

Shout out to people who get confused with mathematical concepts

Shout out to people who can't read music because it seems mathematical to them

Shout out to people who's maths struggles limit them

Shout out to people with dyscalculia or math struggles, basically. I am with you. I am here for you. I see you. It sucks, and most of the world doesn't even know about dyscalculia. But it'll be ok. It won't go away but people will help you, you can adapt. I love you. Screw math ❤

[theluckiestlb]

girl...WHEN??? WHERE????

what are you talking about???? when you kept him isolated for 14 years??? when you robbed him of bodily autonomy???? when you exposed him to his mother's corpse to akumatize him, TWICE????

wait, wait—OH you mean when he was terrified for his life, literally begging you to stop after you beat him up. my bad.

no? OH, you mean when you forced him to leave the country and the love of his life.

Huh.

[themandolinscrimblo]

Here's a little.... "warm-up" sketch that turned out so good that I wanted to post it.

Yes, I forgot Eddie's belt.

And yes, the blur, dizzy lineart on Frank is unintentional (but in a good way).

[yuurionviktor]

Gonna finish Gideon one soon, but for now you can have this Harrow

[locusfandomtime]

Doing the maths: Grian's failure at getting a mending book

lots of talk about maths and probabilities below the cut! but there's a graph and simple explanation at the end if you want to get the gist of it and are bad at maths.

(I am still young and learning maths, critique/advice always welcomed)

What are the odds of getting a mending book in Minecraft?

(I am assuming Grian has been doing all his fishing with Luck of the Sea 3)

The probability of a mending book is actually a bit annoying to estimate. The Minecraft Wiki lists fishing up an enchanted book as 1.9% chance. This is for ANY enchanted book. The Minecraft wiki talks about how the chance of an enchantment being selected is calculated. Mending has a weight of 2. Using the table, mending has a probability of 2/135.

However, Grian is looking for any book with mending, not just a pure mending book. Additional enchantments are calculated in a different way, involving RNG, which means it won't be as easy to model. Due to this reason, I'll just be using the odds for a pure mending book throughout.

TLDR: a mending book has a 0.028..% chance (2/135*0.019*100)

Grian's Data

According to this screenshot, Grian has used a fishing rod 5679 times. This number may not be fully accurate, as it includes the times he's fished other players, rather than just fished for items, but it is a good estimate.

To help visualise this data, with a median waiting time between catches of 17.5 seconds, Grian has spent over 20 hours fishing so far! He may have a problem.

Is this statistically significant?

Hypothesis testing (p-value approach):

H0: p = 19/67500 (the null hypothesis - he has no mending books because of chance)

H1: p < 19/67500 (the alternate hypothesis - he has no mending books due to different odds)

5679 trials, 0 mending books

X ~ B(5679, 19/67500) (binomial distribution, 5679 tries with a probability of a mending book being 19/67500, where X is the number of mending books)

p(X=0) (what is the probability the number of mending books being 0)

p = 0.2021473392

Now, the point at which data becomes significant is subjective. For instance, you *could* get a million heads in a row flipping a coin, it's not impossible, but at a certain point, you can begin to say "okay there's something not normal about this". For this approach, the closer the p-value is to 0, the more evidence there is against the null hypothesis . The p-value here is far above a significance level of 0.01, or 0.05, or 0.1. There isn't a clear line between significant/non-significant, but this is answer is quite a bit far from 0

With this, I cannot reject the null hypothesis.

Personal conclusion: this is not statistically significant, Grian is just unlucky.

Are other values statistically significant?

Gem's proposed 9000: results in a p-value of 0.079... more significant than Grian's number but I don't imagine Mojang would be too concerned. As said though, it's all subjective.

I am bad at maths, what does all this mean?

Here is a graph, showing what number of mending books you might have after 5679 tries. The height of the bar represents the probability of getting that amount. The numbers at the top are the (rounded) numbers I used in my calculation

The pink column is 0 mending books - like what Grian has! As you can see, it is less likely than getting 1 or 2 books, but not too uncommon to happen.

End conclusion: Grian has bad luck. Like, not as hilariously bad as he thinks, but still bad. If he keeps going, chances are he will get a mending book, but I think he should probably stop fishing because at this point he has a problem.

[lotus-pear]

i think you guys are onto smth..

i unironically got invested in this HELP

[wallflowers-in-the-wind]

Whenever I see people discussing Feyre in ACOMAF it’s always about Feyre’s trauma and how Tamlin ‘wasn’t a good partner’ or ‘didn’t help her with her nightmares’. But no one seems to be asking what did Feyre do for Tamlin.

Didn’t Tamlin also have nightmares? He suffered too and did everything he could to save her. It’s literally stated in the book that he slept at the end of the bed in his beast form to protect them from intruders. He threw himself into his job to ensure their safety as much as possible. Feyre and her tendency to put herself in harms way regardless of what it may do to everyone who cares about her didn’t help.

Trauma can cause people to close up and focus on what happened to them. Both Feyre and Tamlin did this. But only Tamlin gets called out for not doing enough to help Feyre. He’s painted as a horrible monster who doesn’t care while she can ignore everyone else’s feelings. And no it doesn’t count to reach out once them back off when things get too difficult.

It’s a ridiculous double standard and Feyre needs to be held to the same standard of scrutiny for her actions. Because what she did was a lot worse than what he did to her.

[jackwhiteprophetic]

The fact that Eddie's parents watched him tell his son "I love you no matter what" and "I know you're mad but I love you" and didnt leave without even sparing a second to give their own son a fucking hug is fucking evil

[physicallyimprobable]

what's the 3-dimensional number thing?

Well I'm glad you asked! For those confused, this is referring to my claim that "my favorite multiplication equation is 3 × 5 = 15 because it's the reason you can't make a three-dimensional number system" from back in this post. Now, this is gonna be a bit of a journey, so buckle up.

Part One: Numbers in Space

First of all, what do I mean by a three-dimensional number system? We say that the complex numbers are two-dimensional, and that the quaternions are four-dimensional, but what do we mean by these things? There's a few potential answers to this question, but for our purposes we'll take the following narrative:

Complex numbers can be written in the form (a+bi), where a and b are real numbers. For the variable-averse, this just means we have things like (3+6i) and (5-2i) and (-8+3i). Some amount of "units" (that is, ones), and some amount of i's.

Most people are happy to stop here and say "well, there's two numbers that you're using, so that's two dimensions, ho hum". I think that's underselling it, though, since there's something nontrivial and super cool happening here. See, each complex number has an "absolute value", which is its distance from zero. If you imagine "3+6i" to mean "three meters East and six meters North", then the distance to that point will be 6.708 meters. We say the absolute value of (3+6i), which is written like |3+6i|, is equal to 6.708. Similarly, interpreting "5-2i" to mean "five meters East and two meters South" we get that |5-2i| = 5.385.

The neat thing about this is that absolute values multiply really nicely. For example, the two numbers above multiply to give (3+6i) × (5-2i) = (27+24i) which has a length of 36.124. What's impressive is that this length is the product of our original lengths: 36.124 = 6.708 × 5.385. (Okay technically this is not true due to rounding but for the full values it is true.)

This is what we're going to say is necessary to for a number system to accurately represent a space. You need the numbers to have lengths corresponding to actual lengths in space, and you need those lengths to be "multiplicative", which just means it does the thing we just saw. (That is, when you multiply two numbers, their lengths are multiplied as well.)

There's still of course the question of what "actual lengths in space" means, but we can just use the usual Euclidean method of measurement. So, |3+6i| = √(3²+6²) and |5-2i| = √(5²+2²). This extends directly to the quaternions, which are written as (a+bi+cj+dk) for real numbers a, b, c, d. (Don't worry about what j and k mean if you don't know; it turns out not to really matter here.) The length of the quaternion 4+3i-7j+4k can be calculated like |4+3i-7j+4k| = √(4²+3²+7²+4²) = 9.486 and similarly for other points in "four-dimensional space". These are the kinds of number systems we're looking for.

[To be explicit, for those who know the words: What we are looking for is a vector algebra over the real numbers with a prescribed basis under which the Euclidean norm is multiplicative and the integer lattice forms a subring.]

Part Two: Sums of Squares

Now for something completely different. Have you ever thought about which numbers are the sum of two perfect squares? Thirteen works, for example, since 13 = 3² + 2². So does thirty-two, since 32 = 4² + 4². The squares themselves also work, since zero exists: 49 = 7² + 0². But there are some numbers, like three and six, which can't be written as a sum of two squares no matter how hard you try. (It's pretty easy to check this yourself; there aren't too many possibilities.)

Are there any patterns to which numbers are a sum of two squares and which are not? Yeah, loads. We're going to look at a particularly interesting one: Let's say a number is "S2" if it's a sum of two squares. (This thing where you just kinda invent new terminology for your situation is common in math. "S2" should be thought of as an adjective, like "orange" or "alphabetical".) Then here's the neat thing: If two numbers are S2 then their product is S2 as well.

Let's see a few small examples. We have 2 = 1² + 1², so we say that 2 is S2. Similarly 4 = 2² + 0² is S2. Then 2 × 4, that is to say, 8, should be S2 as well. Indeed, 8 = 2² + 2².

Another, slightly less trivial example. We've seen that 13 and 32 are both S2. Then their product, 416, should also be S2. Lo and behold, 416 = 20² + 4², so indeed it is S2.

How do we know this will always work? The simplest way, as long as you've already internalized the bit from Part 1 about absolute values, is to think about the norms of complex numbers. A norm is, quite simply, the square of the corresponding distance. (Okay yes it can also mean different things in other contexts, but for our purposes that's what a norm is.) The norm is written with double bars, so ‖3+6i‖ = 45 and ‖5-2i‖ = 29 and ‖4+3i-7j+4k‖ = 90.

One thing to notice is that if your starting numbers are whole numbers then the norm will also be a whole number. In fact, because of how we've defined lengths, the norm is just the sum of the squares of the real-number bits. So, any S2 number can be turned into a norm of a complex number: 13 can be written as ‖3+2i‖, 32 can be written as ‖4+4i‖, and 49 can be written as ‖7+0i‖.

The other thing to notice is that, since the absolute value is multiplicative, the norm is also multiplicative. That is to say, for example, ‖(3+6i) × (5-2i)‖ = ‖3+6i‖ × ‖5-2i‖. It's pretty simple to prove that this will work with any numbers you choose.

But lo, gaze upon what happens when we combine these two facts together! Consider the two S2 values 13 and 32 from before. Because of the first fact, we can write the product 13 × 32 in terms of norms: 13 × 32 = ‖3+2i‖ × ‖4+4i‖. So far so good. Then, using the second fact, we can pull the product into the norms: ‖3+2i‖ × ‖4+4i‖ = ‖(3+2i) × (4+4i)‖. Huzzah! Now, if we write out the multiplication as (3+2i) × (4+4i) = (4+20i), we can get a more natural looking norm equation: ‖3+2i‖ × ‖4+4i‖ = ‖4+20i‖ and finally, all we need to do is evaluate the norms to get our product! (3² + 2²) × (4² + 4²) = (4² + 20²)

The cool thing is that this works no matter what your starting numbers are. 218 = 13² + 7² and 292 = 16² + 6², so we can follow the chain to get 218 × 292 = ‖13+7i‖ × ‖16+6i‖ = ‖(13+7i) × (16+6i)‖ = ‖166+190i‖ = 166² + 190² and indeed you can check that both extremes are equal to 63,656. No matter which two S2 numbers you start with, if you know the squares that make them up, you can use this process to find squares that add to their product. That is to say, the product of two S2 numbers is S2.

Part Four: Why do we skip three?

Now we have all the ingredients we need for our cute little proof soup! First, let's hop to the quaternions and their norm. As you should hopefully remember, quaternions have four terms (some number of units, some number of i's, some number of j's, and some number of k's), so a quaternion norm will be a sum of four squares. For example, ‖4+3i-7j+4k‖ = 90 means 90 = 4² + 3² + 7² + 4².

Since we referred to sums of two squares as S2, let's say the sums of four squares are S4. 90 is S4 because it can be written as we did above. Similarly, 7 is S4 because 7 = 2² + 1² + 1² + 1², and 22 is S4 because 22 = 4² + 2² + 1² + 1². We are of course still allowed to use zeros; 6 = 2² + 1² + 1² + 0² is S4, as is our friend 13 = 3² + 2² + 0² + 0².

The same fact from the S2 numbers still applies here: since 7 is S4 and 6 is S4, we know that 42 (the product of 7 and 6) is S4. Indeed, after a bit of fiddling I've found that 42 = 6² + 4² + 1² + 1². I don't need to do that fiddling, however, if I happen to be able to calculate quaternions! All I need to do is follow the chain, just like before: 7 × 6 = ‖2+i+j+k‖ × ‖2+i+j‖ = ‖(2+i+j+k) × (2+i+j)‖ = ‖2+3i+5j+2k‖ = 2² + 3² + 5² + 2². This is a different solution than the one I found earlier, but that's fine! As long as there's even one solution, 42 will be S4. Using the same logic, it should be clear that the product of any two S4 numbers is an S4 number.

Now, what goes wrong with three dimensions? Well, as you might have guessed, it has to do with S3 numbers, that is, numbers which can be written as a sum of three squares. If we had any three-dimensional number system, we'd be able to use the strategy we're now familiar with to prove that any product of S3 numbers is an S3 number. This would be fine, except, well…

3 × 5 = 15.

Why is this bad? See, 3 = 1² + 1² + 1² and 5 = 2² + 1² + 0², so both 3 and 5 are S3. However, you can check without too much trouble that 15 is not S3; no matter how hard you try, you can't write 15 as a sum of three squares.

And, well, that's it. The bucket has been kicked, the nails are in the coffin. You cannot make a three-dimensional number system with the kind of nice norm that the complex numbers and quaternions have. Even if someone comes to you excitedly, claiming to have figured it out, you can just toss them through these steps: • First, ask what the basis is. Complex numbers use 1 and i; quaternions use 1, i, j, and k. Let's say they answer with p, q, and r. • Second, ask them to multiply (p+q+r) by (2p+q). • Finally, well. If their system works, the resulting number should give you three numbers whose squares add to 15. Since that can't happen, you've shown that the norm is not actually multiplicative; their system doesn't capture the geometry of three dimensions.

[inkskinned]

you were raised in comparison.

it wasn't always obvious (well. except for the times that it was), but you internalized it young. you had to eat what you didn't like, other people are going hungry, and you should be grateful. you had to suck it up and walk on the twisted ankle, it wasn't broken, you were just being a baby. you were never actually suffering, people obviously had it worse than you did.

you had a roof over your head - imagine! with the way you behaved, with how you talked back to your parents? you're lucky they didn't kick you out on your ass. they had friends who had to deal with that. hell, you have friends who had to deal with that. and how dare you imply your father isn't there for you - just because he doesn't ever actually talk to you and just because he's completely emotionally checked out of your life doesn't mean you're not fucking lucky. think about your cousins, who don't even get to speak to their dad. so what if yours has a mean streak; is aggressive and rude. at least you have a father to be rude to you.

you really think you're hurting? you were raised in a home! you had access to clean water! you never so much as came close to experiencing a real problem. sure, okay. you have this "mental illness" thing, but teenagers are always depressed, right. it's a phase, you'll move on with your life.

what do you mean you feel burnt out at work. what do you mean you mean you never "formed healthy coping mechanisms?" we raised you better than that. you were supposed to just shoulder through things. to hold yourself to high expectations. "burning out" is for people with real jobs and real stress. burnout is for people who have sick kids and people who have high-paying jobs and people who are actually experiencing something difficult. recently you almost cried because you couldn't find your fucking car keys. you just have lost your sense of gratitude, and honestly, we're kind of hurt. we tell you we love you, isn't that enough? if you want us to stick around, you need to be better about proving it. you need to shut up about how your mental health is ruined.

it could be worse! what if you were actually experiencing executive dysfunction. if you were really actually sick, would you even be able to look at things on the internet about it? you just spend too much time on webMD. you just like to freak yourself out and feel like you belong to something. you just like playing the victim. this is always how you have been - you've always been so fucking dramatic. you have no idea how good you have it - you're too fucking sensitive.

you were like, maybe too good of a kid. unwilling to make a real fuss. and the whole time - the little points, the little validations - they went unnoticed. it isn't that you were looking for love, specifically - more like you'd just wanted any one person to actually listen. that was all you'd really need. you just needed to be witnessed. it wasn't that you couldn't withstand the burden, but you did want to know that anyone was watching. these days, you are so accustomed to the idea of comparison - you don't even think you belong in your own communities. someone always fits better than you do. you're always the outlier. they made these places safe, and then you go in, and you are just not... quite the same way that would actually-fit.

you watch the little white ocean of your numbness lap at your ankles. the tide has been coming in for a while, you need to do something about it. what you want to do is take a nap. what you want to do is develop some kind of time machine - it's not like you want your life to stop, not completely, but it would really nice if you could just get everything to freeze, just for a little while, just until you're finished resting. but at least you're not the worst you've been. at least you have anything. you're so fucking lucky. do you have any concept of the amount of global suffering?

a little ant dies at the side of your kitchen sink. you look at its strange chitinous body and think - if you could just somehow convince yourself it is enough, it will finally be enough and you can be happy. no changes will have to be made. you just need to remember what you could lose. what is still precious to you.

you can't stop staring at the ant. you could be an ant instead of a person, that is how lucky you are. it's just - you didn't know the name of the ant, did you. it's just - ants spend their whole life working, and never complain. never pull the car over to weep.

it's just - when it died, it curled up into a tight little ball.

something kind of uncomfortable: you do that when you sleep.