#that silly fast food emoji post went wild

[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.

[ikesenmotonari]

yay that ikesen oc thing. she’s a multifandom oc but hey im not creative so i might as well JAM her into another universe lets go lads

idk who to tag but if u have an ikesen oc go for it i wanna know bout em!!!

i was tagged by @nyktoon-ikemenlove thank you sweetheart!!!

Age? Height?

“Hi! I’m Melody Wyverne, but my friends if I had any would call me Mel, Mells… that fun stuff! I’m nineteen and I’m five foot two!”

She’s petite, on the curvy side. She’s 5′2″, or 155 cm; she is only nineteen and wants to go home. lmao

What’s your fashion like? [Time travelers – pre & post-wormhole!]

“Er, pre time-jump I was a fan of sleeveless hoodies and high-waisted jeans. I didn’t go out much, so I dressed comfortably. Now I wear kimono and hakama… it’s pretty. I sure hope it didn’t trouble people too much to make these…!“

The top one is an older doodle. thonk emojis

Where are you from?

“I’m from Canada, actually. I lived alone for a bit then my dad reached out to me and asked me if I’d like to spend some time with him in Japan. So uh, I just wanted to talk to him again and thought this would be a good way for me to intern somewhere… it might as well be his business right? Then the storm thing.“

Her mom is French, her dad is English. She’s got a ridiculously high IQ level and no social skills whatsoever, so she knows most Japanese and speaks it pretty okay despite her North American kind of accent. Is that a thing? thonks

Feudal era – pros and/ or cons?

“There are pros?!“ She’s not very happy being surrounded by blood and death…

Pros? Cool clothes. Cons? HAVE YOU MET THESE PEOPLE?!

If you’re not in your homeland/time, do you want to go home?

“Gosh. Absolutely.“

What’s your home life like?

“Well, I’ll run you through a usual day! Ah… well, I’d eat something, then play with my cat, talk to Avery for a bit, then… yeah I’d spend the rest of it at the garage, just working on my projects! It’s… I like working. It’s not healthy, and I’ve been told… but it’s nice, you know?“

Melody doesn’t go outside.

She scarcely leaves her property and the only close friends she has is a cat and a cyborg (a man with no arms). Her parents split years ago and she wasn’t properly socialized as a kid. Being raised as a certified genius? It’s… lonely. She doesn’t know how it’s affected her, but being thrown back in time is forcing her to look at things differently.

You just got your dream job! What is it? / Or, what’s your line of work?

“I’m a programmer on the side, but first and foremost, I make prosthesis for people. I have a background in medicine and engineering I suppose…”

Any other hobbies or skills? Do you use them / how do you use them in the Sengoku period?

“Not really… I can barely take care of myself as is, haha! Erm. I can’t use my skills much at all in this era. Other than some simple automatons, there’s also guns I can piece apart, but I’m more hesitant with those… all I can do is use my expertise in biology I guess. There’s a lot of injuries going around, and I’m glad I’m not completely useless.“

Where is your base of operations? Azuchi Castle? Kasugayama Castle? A pirate ship? Running all over the woods or in a secret monastery? Some other cool place?

“I woke up in a forest near Azuchi and stayed in a neutral tea house for a while before Lord Masamune pretty much dragged me to the castle. So I stayed there for a few months, and the warlords grew on me, but… I… got kidnapped. So I guess my base of operations is a pirate ship.”

Oh?

“…I don’t like it. Don’t get me started on their captain…”

How do you feel about killing and violence?

“I just wish there’s another way past it all. I don’t think I’ll ever understand the logic behind killing someone. I… I’m a doctor. Not a murderer.”

Have you learned to fight? If so, what’s your weapon and/or fighting style of choice?

“Nope, actually. But there’s some good guys on board that might be willing to teach me! At least, I hope so. The only thing I’m leaning towards are guns, unfortunately, but they’re the closest thing to a modern mechanism I can get. Eep… I don’t know how to fight! I don’t want to!”

What are you fighting for?

“Um, my life?!“ It changes. Eventually. ;3

What are your feelings about authority?

“As long as they’re not bullies? Fine.” She pauses. “Lord Nobunaga was different though. I didn’t understand him fully, but he doesn’t really make fun of me or anything. He’s just curious I guess. I don’t tend to question authority unlike some pirates.”

How do you handle someone invading your personal space?

“I just get really uncomfortable and back away if I can. I can’t bring myself to yell at people unless they’ve seriously whittled down on my patience. I didn’t know I had it in me, actually… huh.“

…do you invade people’s personal space?

“Ha…. haha! Yeah, sometimes. I get nosy okay? I didn’t even know what personal space was until I was eleven!”

Are you more open, or more reserved? Are you secretive?

“Avery says I’m an open book. I have nothing to hide, no secrets to keep. There’s not much that happened to me before, so I guess I’m some kind of blank slate? Gee, that sounds so harsh…”

Is this the first time you’ve been truly in love?

 “Love? Have you spoken to these people?”

Eventually? Yeah. Yeah this is her first time.

What’s your style as a lover? (interpret this as innocently or not-innocently as you please ;) )

She’s sweet, affectionate and balanced. She knows when to handle time with her partner and time working on whatever independent activities. Though she’s quite a dense person overall and won’t realize if she’s even fallen for someone, she deeply and wholeheartedly trusts them. It takes a while for her to know this.

Also, she’s 99% submissive and slightly masochistic. As well as a rope bunny.

What are your favorite ways for someone to show you love?

Touches, fleeting ones. Tight hugs, any kind of embrace. Show her something unconditional, undivided. She’s been isolated for so long she convinced herself she won’t have someone to love, so she carries on merrily alone, not knowing how deprived she is of human contact. Hmm.

Take her on an adventure. Bring her out of her comfort zone. Show her what the world could be like… you’ll change her.

Do you use a petname or endearments for your lover(s)?

Not really, she would give nicknames if she could! She already says things like ‘honey’ and ‘dear’, but nothing too mushy unless it becomes super playful and joking.

How do you feel about…

Nobunaga? “He’s such a complicated warlord… I don’t know if I’ll ever understand him. But he’s shown some really keen interest in me. Is it because I’m a Westerner? Either way, Lord Nobunaga scared me and he still does. But after spending months in the castle and talking to him, he’s not actually as mean as I thought he was. I don’t know why he laughs at me though! I guess the things I do seem really silly to him!”

Hideyoshi? “Oh he’s super nice. He taught me how to make tea when mister Mitsunari and I put way too much leaves in. He tends to scold me for staying up though, but I can’t help my insomnia without my pills! It’s really comforting to know that Lord Hideyoshi is looking out for me though. The things he does reminds me of how Avery takes care of me, so I guess he’s like a big brother? Heehee.”

Masamune? “He was the one who brought me to Azuchi castle, and boy is he wild! His energy shocked me honestly, it’s like he never runs out of it. He’s so cool though! He kinda scared me too, and I’m pretty sure he can stab me once told to, but he’s been really playful. I didn’t know I’d have so much fun in Azuchi thanks to him. And he makes amazing food! Though he should stop bugging me to eat three meals a day…”

Ieyasu? “Aw man, I wish I can talk to him without him speedwalking away or trying to avoid me! He’s reading stuff about medicine, right? I’d like to know what he’s learning. It’s no doubt super different from modern medicine. I try to talk to him sometimes but Lord Ieyasu’s always busy… was it something I said?”

You know that question about invading personal space? Melody leans in to peer over at his books sometimes. She’s a bit too friendly to his liking.

Mitsunari? “I relate to him a lot. He’s really friendly and sometimes I’d spend my free time reading with him. Time goes by so fast though and either Lord Hideyoshi or Lord Ieyasu would step in to tell us we’ve missed dinner…”

Literally. They are so alike. They got along swell.

Mitsuhide? “S-Scary… have you heard of those rumors about him? I mean, in person he’s okay, but he’s just tall and intimidating to me. He did take me out to the marketplace once and show me around, which… well, nevermind. I guess he’s nice. Apparently Lord Hideyoshi told him to stop teasing me, but I don’t recall being teased?”

She’s dense. 

Shingen? “Big man. Very tall, and… I think he was flirting with me. Whatever it was, he’s… really.. erm, polite and all, and I met him along with Yukimura and Sasuke at the forest tea house before I was taken to Azuchi. I don’t know him too well, but with what Lord Nobunaga told me, he must be a strong warlord…”

Kenshin? “I never met the guy. With a title like the ‘God of War’, I don’t think I want to…!” Fear.

Yukimura? “He’s… he’s kind of… what’s the word, gruff, isn’t he? I met him along with Shingen and I guess it’s a good quality to be honest and stuff. I think we got off on the wrong foot.”

She spilled tea on him. You can imagine the rest. Sasuke came in clutch.

Sasuke? “He’s super great. I wish I had more time to talk to him! He kinda saw right through my attempt to hide the fact that I’m from the future and he gave me helpful advice on fitting in and staying under the radar. Eep, sorry I got dragged to the castle, Sasuke! We can talk about astrophysics another time…! Seriously, I’m so grateful to him. Without his advice I’d probably be in big trouble.”

Kennyo? “He went near the tea house once and was feeding the fish in the koi pond. He’s scary, but… he’s nice. I served him tea and some dumplings. I had no idea he hated Lord Nobunaga so much…”

Motonari? “Ugh… infuriating. I didn’t know I could be so angry at someone before I met this guy. The nerve he has, to kidnap me in broad daylight, spit on Azuchi defenses, and then use me as his impromptu surgeon for the high seas! Hmph. … I… I don’t know. The more time I spend with him, I feel like I’m not going anywhere - you know, I try to get along, I really do, but I think he’s keeping me away? Like, he just tends to stay shallow with people. As annoying as he can get… I want to know why. He’s shown how playful he could be, and his crew loves him. He gives me the same vibes as Lord Nobunaga, actually… intimidating, but I want to know more about him.”

Any other friends/notables?

“I have a friend named Avery and he’s awesome! He took care of me ever since I was eleven, he was twenty-one at the time and now he’s thirty. He’s like a dad to me. Oh, and this isn’t too important, but he has metal arms. I made those! Er, the updated ones, actually. Just to make sure they’re functioning like real arms and all. The prototypes were made by my mom and he was severely injured, so the surgery took a while. He and I grew really close after my parents’ divorce. He lives a few blocks down, and he has eight dogs. Don’t ask ME why, you should ask HIM. … I miss him. I hope he’s okay.”

She also has a cat named Charlie. He’s a grey ragdoll. That’s… that’s about it.

hooray for my shallow motonari headcanons with trust issues