#follow and reblog everyone!

[sona1800]

Our @kai-lee08 @j-oneproduces is back with her new account @bunnystruggles Do follow and reblog everywhere! 🫶

[girldraki]

[gayfranzkafka]

The Nisenan tribe in my local area has the opportunity to purchase 232 acres located on a historic Nisenan Village site called Yulića, but they have a limited time (until April 4, 2024) to raise the needed funds. You can learn more about the fundraiser and donate here.

Especially if you have ever enjoyed any of my writing, like Rosencrantz and Guildenstern Aren't Dead, it would mean the world to me if you'd consider donating what you're able.

I'm also happy to take commissions and donate the funds in full to the fundraiser or to write anyone who sends me proof of their donation oneshots upon request; dm me or send me an ask if you're interested!

Reblogs are appreciated to spread the word. Thank you <3

[shima-draws]

Hey. Hey you know what would be cool and fun. First post of 2024. 1000 notes. We can do it. Yeah? Yeah??

Take a picture of my dog as an incentive

[daily-odile]

1,,,, 100 days,,,,,,,,, and 800 followers,,,,,,,,,,,,,

From the bottom of my heart, thank you everyone.........!!!!!!!!!!!!!!!!!!

[anyataylorjoys]

seeing so many fellow creators' gifsets getting under 180 notes with an average reblog to like ratio of 1:3 is so disheartening. less and less people are putting out content cause it seems that no one cares anymore.

If you truly care about supporting creators at all, I encourage everyone to have a tracked tag in this economy whether you make content or not, you probably have mutuals who would love to share things with you and can't.

[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.

[aropride]

if you watch tiktoks/other forms of video content out loud in public, why

for the purpose of this poll "tiktok/other forms of video content" includes youtube videos, podcasts, instagram reels, etc, but not phone calls or facetime. "public" is just anywhere where there's other people- public transport, the library, a bathroom, the store, wherever).

also "i forgot my headphones" isn't an option because the question is still "why not wait until youre at home or have ur headphones again, or just turn the sound off," not the literal circumstances if that makes sense

[postazkabansiriussupremacy]

Surprise. Pt. 3 Post Azkaban!Sirius x Mom!Reader

You and Sirius have a chat.

Part One. Part Two.

Taglist: @box-of-kinderjoy @projectdreamwalker @goldenharrysworld @navs-bhat @sagestack

You aren’t sure what you expected Sirius to look like after more than a decade in Azkaban, but this is much worse than you pictured. The dirty, malnourished, manic-looking man standing in front of you is a far cry from the healthy, handsome Sirius you once knew. The sight of him in this horrid state is enough to grow a lump in your throat.

You always had complex feelings about Sirius after he was sent to Azkaban. It was easy to be angry at him for betraying everyone and causing the death of James and Lily, but you were more than angry. You were devastated, to say the least. It’s impossible to say how many nights you cried yourself to sleep. Even knowing what he’d done, it was hard to imagine the one you loved rotting away in Azkaban.

The first few years without him were hell on earth. It wasn’t long after he was incarcerated you found out you were pregnant, and whilst everyone in the wizarding world was partying and celebrating the downfall of You-Know-Who, you were curled up in a ball sobbing and wondering how in the world you were going to do this all on your own.

It was difficult, but you managed. As the years went by and Estelle grew bigger, things got easier. You were able to push Sirius out of your mind and go on without him, but not without continuous effort. With every life change and new milestone reached, you couldn’t help but wonder how differently things would be if Sirius were there too.

For Estelle, you tried your very best to make sure she never wanted for anything, but your heart ached at the thought of her never knowing the love of a father. Estelle used to ask about him (“Why don’t I have a dad?” “Where is he?” “Is he dead?”), and you were never sure what to tell her.

You aren’t proud of it, but as her questions persisted, you lied to her. You lied and told Estelle you don’t know who her father is. She stopped asking about him after that.

You don’t know why you lied. It would’ve been much simpler to tell the truth, but maybe a small part of you wanted Estelle to blame you rather than blame Sirius for her lack of a father. It felt a little stupid, but you didn’t want Estelle to hate the idea of her father. You supposed it would be easier for her to accept her father doesn’t know she exists, rather than to accept her father is a mass murderer in prison for life.

Then you got an owl from Remus Lupin—someone you hadn’t heard from in over a decade—asking to have lunch and talk. You were surprised but receptive. You assumed he met Estelle at Hogwarts and he wanted to know of her lineage under the guise of catching up, and you were half right.

After having Estelle in class and putting two and two together, Remus decided to get in touch with you to tell you the truth about Sirius.

After taking in all of the new information, you felt numb. It’s a lot to take in—learning that Sirius is innocent, and Peter Pettigrew of all people was the one to cause all of this pain.

You came home, politely asked Estelle how her day was, and barely heard her as she told you about the stray dog she found today. Too lost in your thoughts, you ‘listened’ to Estelle’s rambling for about ten minutes before realizing she was talking about Sirius’s animagus.

It had to be Sirius. Why else would there be a giant, wolf-like black dog hanging around your house?

You pretended to Estelle that you’d never seen the dog before, and maybe he belongs to some of the muggles that live further up the road. You carry on your evening as normal, quickly changing the subject anytime she began to talk about the dog, and had her go to bed at a reasonable time.

Only when you were sure Estelle was asleep did you come outside.

You suppose you’ve been staring too long as Sirius speaks up first. It’s hard to read his expression, and his voice is deeper than you remember. “I’d ask how have you been, but clearly you’ve been busy.”

You try to swallow the lump in your throat, urging yourself not to cry. After meeting with Remus and immediately coming home to Estelle, you haven’t had any time to process the information you’ve been given.

There were so many times you’d asked yourself “What if Sirius were still here?” and then immediately pushed the thought away, reminding yourself he’s a horrible man. A traitor and a murderer.

Only he’s not. He’s none of those things.

He’s suffered terrible consequences that he’s done nothing to deserve, and that’s heartbreaking. The last thirteen years of his life were ripped away from him and he was sent to live in horrid conditions, just because he and James chose to trust Peter with something they shouldn’t have.

A heavy weight of guilt drops into your stomach. Sirius had done nothing wrong and yet everyone—including yourself—thought he got what he deserved by being locked away. You hardly even thought to question whether he was truly guilty or not.

Your throat tightens and your lip quivers, and you step forward to wrap your arms around his waist. You can feel the bones underneath his skin, and you sniffle, feeling a couple of tears escape from your eyes.

Sirius takes a moment to respond, a little shocked by your sudden hug and crying. He supposes it’s not unwarranted though.

He reciprocates your hug, one dirty hand cradling the back of your head and the other wrapped around your torso. It’s almost strange how natural it feels. He rests his chin on the top of your head and faintly smiles. You smell good, and it’s wonderful to get such an unexpectedly warm welcome.

Although he has Remus to thank for that. If Remus hadn’t reached out to you first, Sirius imagines this meeting would be going a lot differently.

After a few moments of letting yourself cry into his chest, you finally speak, your voice cracking a bit as you do so. “You smell like shit.”

Sirius gives you a tight squeeze and chuckles quietly, “You live in a cave for a year and we’ll see how you fare.”

You purse your lips and feel more tears forming. He’s been living in a cave? Your throat feels tight as you breathe, “I’m so sorry, Sirius… For everything.”

“There’s nothing you could’ve done.” He responds immediately. There was no way for you to know the truth, and even if you did, it probably wouldn’t have changed anything. “…Did you know you were expecting when it happened?”

You shake your head. “No. Didn’t figure it out until a few weeks after you’d been gone.”

“I-… I can’t find the words to tell you how sorry I am.” Sirius whispers. His adam’s apple bobs. “I can’t say when, but I promise I’ll come back to the both of you.”

“I know you will.” You say quietly, nodding. You knew from the beginning he wouldn’t be able to stay, but it still hurts nonetheless. New tears fall onto your cheeks and Sirius’s prisoner robes.

You cry less for yourself and more for him. Even though he’s successfully crawled his way out of Hell, he still can’t rest. Sirius hasn’t known peace in over a decade, and there’s no telling if he ever will again.

Sirius is the first to pull away. Trying to remain strong for your sake, he clenches his jaw as he looks down at your tearful face. He uses his thumbs to wipe the tears off your cheeks, then he pulls your face forward for a kiss. You waste no time reciprocating, your hands moving to the back of his head and tangling in his greasy hair.

Once again Sirius is the first to pull away, ending the kiss too soon for his liking, but knowing he needs to go. He’s been here far too long. He kisses your forehead. “I love you, and I love Estelle.”

“I love you, Sirius.” You reply, looking into his eyes. They’re the same eyes you see every time you look at your daughter.

“This isn’t goodbye.” He says kissing your forehead once more. He steps off your property and out of the confines of the anti-apparation wards. He gives you one last look, then winks. “You look absolutely stunning, by the way.”

You scoff, a stupid grin forming on your face as he disapperates.

You stare at the spot he left from, wiping your tears away.

Realistically, you don’t know if Sirius will be able to keep his promise. You may never see him again. There’s no telling if his name will ever be cleared, but you hold onto hope, and you will wait for him.

[derpiedoxie]

Tw: eye contact

I paint what I see

[dynalope]

Tumblr is an amazing website because someone will post my exact opinion and belief but do it in such a condescending and annoying tone that it makes me want to disagree with them.

[dyketubbo]

my message to da world: feminize everyone in dsmp now. your moral duty

[itsahotminuteinbetween]

Venting on the Internet:

I know this is probably only going to get a couple of notes but I need as MUCH data as possible so PLEASE VOTE AND REBLOG!

[millenniumscreampuff]

💖

[hellsite-hall-of-fame]

How do you feel about staff's intent to remove reblog chains, considering your blog's basically an amalgamation of reblog chain long posts?

haha I’m actually legitimately terrified and trying really hard to not think about it :)

(seriously like i don’t wanna make a thing of it, but i’ve gone down the posts about it and like wtfff?!? why why why?!?? like I get their reasoning for it but it’s not accurate to the actual users of tumblr?? i’d rather scroll though 20 long posts than use the examples of what this would look like that I saw. but if this is wrong let me know…. but yeahhh i’m scared.)

[why-are-all-the-users-taken]

REBLOG IF UR PART OF THE MISS PEREGRINE'S HOME FOR PECULIAR CHILDREN FANDOM!!!