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physicallyimprobable · 9 months ago

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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.

#math #numbers #human interaction #this took the better part of a day to write oops #although to be fair I haven't exactly been focused #Also hi Pyro! Welcome.#that silly fast food emoji post went wild #I've gotten 30 followers just from that one post #which isn't that many in objective terms but like it's 40% of my current count so #hello everyone #I might start reblogging things again now

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whateverisbeautiful · 1 year ago

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♥️ Ranking Richonne

#3: Have Your Mints (S6E10)

✨TOP 3!✨

Wow. 😍 Let’s just start there. Oh this moment is perfect and I wouldn’t have Rick and Michonne officially become Richonne any other way. This is one of those mesmerizing scenes that is guaranteed to make my heart flutter every time I see it no matter how many times I’ve seen it. The joy radiating from them in this extremely significant moment is beautiful to behold. This scene set off a 'canon' heard around the world and history was made as Rick and Michonne finally let all that love and desire for each other completely pour out. Three words best describe this gorgeous scene to me - euphoric, soul-stirring, and, of course, iconic...

With so much build-up from season 3 to this moment on their living room couch, it was perfect that all it took was some mints and finally holding hands for all that blazing passion between Rick and Michonne to be released.

After nine months of knowing each other, it was time for Richonne's canon era to be born. 🥳

And if you wanna talk about scenes that are instrumental to the making of Richonne, few are more instrumental than this one right here, where the two joyously embrace that the love of their life has been right in front of them all this time. 🥹

Ok so first, I quickly want to talk about the beautiful scene between Carl and Michonne on their porch in this ep. It is such a sweet profession of the love those two have for each other and their mother/son bond, and I adore that Judith was included in this scene too.

I always think about how Michonne and Carl hug at the end of the scene, but I had forgotten Judith is right there in that hug too. And I love this moment because it’s this great visual of the way Michonne is so clearly the Grimes matriarch to these two children. They’re hers. 🥰 She loves them so much and it’s clear they love her so much too (Also my ears hear that adorable baby girl say 'Michonne' at the start of this scene every time lol 😂)

And this moment with her Grimes kids just really makes it feel like all that’s left is for Michonne to embrace that she also has a whole Grimes husband waiting for her too. And thank goodness that’s exactly what is embraced at the end of the episode. 🤗

It’s always so telling that the scene starts with Rick clearly tired but plopping down on the couch instead of just going to bed. He’s waiting for her.👌🏽

And Michonne was waiting for him too as she entered with the baby monitor. Like Danai said, Michonne wants Rick home. 🥰

I adore the way this whole episode establishes the deep comfortability Rick and Michonne have built with each other. Like their morning and night routine in this ep just shows they really feel at home with each other.

I love how she gently kicks him and says move and, of course, this has Rick looking more awake as he gets up to make room for her on the couch. I like getting this insight into how they’ve spent these last two weeks and how Rick and Michone clearly enjoy starting and ending the day together. 🥰

And then I love that she sits so close to him on the couch and shows him their daughter practicing crawling in her crib. And the way they both smile and move their heads in sync. PARENTS. (Also just imagine them looking at a baby monitor of RJ together 😭😭😭)

I love Rick saying, "it’s good to be home" and that Michonne agrees - and truly home is more than four walls and a roof, home is with each other, and Rick and Michonne know it.

Rick is such a family man, and I love how he cherishes these familial moments just being home with her and their kids. It’s painful knowing he’s gone without that for several years when what he wants most in life is to be with them.

Side note: I just have to say, I feel like what Rick stans who dismiss Michonne and their kids tend to miss is that all that cool warrior stuff Rick does is stuff he does because he has to. And it's super entertaining and we all love him for it absolutely, but he savagely throws down like that cuz he has no choice in their world. When he’s at home decompressing with Michonne or having family fun days with her and his kids - this is the stuff Rick does because he wants to. Like this is who he most wants to be. Having these moments with them is what drives the character and motivates all those cool action moments that Rick-only stans love. His wife and kids are his "why" and the "why" is vital because it makes "what" the characters do matter. Rick's wife and kids are why he keeps going and keeps fighting. And that’s a big reason why I celebrate that Rick found such a wonderful love story and family with Michonne because it’s what this protagonist most wanted. A character’s core want is so important to a show. And what is always made clear is that what Rick wants at his core is Michonne and their family. So we love him so much for being a fighter but we also adore him for being who he truly is - which is a devoted family man. And if you really want the character of Rick Grimes to win in this story, then you don’t just want him to win the battles. You want him to have what he wants. And again, what Rick Grimes wants is Michonne Grimes and their family. That can't be denied. That can't be overlooked. And when anyone tries to dismiss it...Rick's character happily re-reminds everyone of this every chance he gets.👌🏽

Knowing how much family is a part of Rick and Michonne's core wants and needs, contributes to why it's so rewarding seeing the two finally go after what they most want in this 6.10 canon ep. 😌

I love the Richonne tones as they talk but don’t talk about their days cuz they both are open to hearing about the other's day but are on the same page about wanting to just relax for a bit.

Whenever Rick and Michonne are alone together there is often this desire to just be present in the moment which I appreciate. And just their ease as they decompress together - you know it means so much to them to have found a bond like this with someone who makes you feel safe, heard, and comfortable in this broken world.

Like to turn your brain off for a minute in the apocalypse is such a rarity, and it's refreshing that they can do that with each other. 😊

The way they can just sit side by side and silently enjoy being in each other's presence in the calm and quiet here - that’s real love. Also, Michonne’s curious face when Rick then dishes the mints out of his pocket will always make me smile. 😋

Rick says, "got you something" and hands Michonne the mints and I love the way it’s shot with the emphasis on their hands. It reminds me of the emphasis on their hands in Clear when she handed him a bullet and a spark was so clearly felt. And now, seasons later, they will finally act on that spark. 🙌🏾✨

It’s precious the way Michonne playfully takes the mints and the laugh they share, with Rick clearly happy that he could make her smile with this. I always liked that bringing home the mints was Rick caring to not just fulfill a survival need, but a personal want that Michonne had. He loves her, y’all. 🥰

And it’s sweet that at this point, Rick did this with zero ulterior motives. He didn’t know this would be the night. (Even tho earlier in this ep he does happen to tell Daryl "today is still the day" or something, which made me laugh cuz uh...little did he know😋)

As they smiled over mints on this couch, it was clear that making Michonne happy was the sole goal. And Rick seems genuinely pleased to have achieved it. 👌🏽

This moment also just made me think back to when Michonne playfully teased Rick with a razor in season 4, and Rick sorta had this nervous energy as he gave a little smile. For seasons, Michonne has always been the best at reminding Rick to smile, and what's sweet is that in this canon ep, Rick doesn't just do shy smiles anymore - no, now he grins from ear to ear over Michonne. 😊

And their banter is just the best as she amusedly asks if this is instead of the toothpaste and he playfully says yeah then tells her he actually did have a whole crate of toothpaste for her it’s just currently at the bottom of a lake.

I always love that he says he had a crate of toothpaste "...for you." I knew when Rick found toothpaste in that truck his first thought was Michonne, and I love that he can never help but reveal that she and her wants are important to him.

It is beyond sweet that in this world, Rick’s romantic heart still manages to gift Michonne and find ways to make her smile. Real one. I'll always love that he's so aware of how special she is. Rick gets gifts for Michone because he knows she is such a gift to him. 🥹

Then I adore the way Michonne looks at him as he hints at the whole lake debacle and then forever love her saying, "Oh so you had a day?" 😋 They are the best, and I like how they can really show their personality with each other. And Danai's delivery was perfect yet again.

Rick says, "Yeah all on account of your dental hygiene," and I love that one thing Rick Grimes is always going to find time to do from season 3 forward is flirt with Michonne. Apocalypse be damned lol.

And Michonne smiles, clearly a smitten kitten by that man Rick cuz who wouldn’t be. And Rick has these little content glances over at her as he smiles upon seeing her happy. 🥹 Again, her happiness genuinely means something to him, and I appreciate these loving glances for further conveying that.

And as Rick smiles, it really feels like it’s him knowing, not subconsciously but cognizantly knowing, that he’s in love with her.

No speculation needed. He's in love, your honor. 😌

And because those magnets can’t ever be denied, Rick proceeds to pat Michonne's hand and say, "Have your mints." And then Michonne ever so gently takes his hand.

And that’s all it took for history to be made. 🥳🎉

Truly the sparks between Rick and Michonne had been so strong for so long that all it took was a single moment of them actually letting a moment of touch linger for the fireworks to be set all the way off.

Like the shot focuses on the handhold, but had their been footage of R&M's hearts in this moment it would have for sure looked like this...

And I love how after Rick makes the move to touch her, Michonne takes his hand, and then Rick is ready for those mints to exit stage left as he more than reciprocates in fully holding her hand.

Once again their hands gave a great performance of their own, cuz this handhold really told a story. 👌🏽

Like starting with Rick instinctually patting her hand in this innate need to touch her and Michonne instinctually taking his hand as though she was no longer resisting what felt most natural for her to do.

And then the two of them letting this touch linger a bit as tho they have a decision to make at this moment - just pass this off as a (fiery) fleeting moment or not let go and instead let what is meant to be finally be.

You see Rick then initiate in showing he doesn’t want to let go as he more deliberately holds her hand. And I love that by not letting go of each other's hand, it’s all the two needed to know that in a figurative sense they also never want to let go of each other. 🥹

Then the way they stay looking into each other's eyes, communicating so much without words before they lean in - it’s beautiful and lets you know that what they're about to do is such a meaningful, personal, long-time-coming moment for both of them.

Michonne looks over at him, and I’ve noticed that for several TWD reactors, her look here is when they realized Rick and Michonne have not in fact been 'uggin' bumplies' before this moment and that what they’re about to do is actually a first.

So much of R&M's behavior in this ep made it clear they’d been a couple for a while. This physical part of the relationship was really just the final piece of the marriage puzzle finally locking into place.

The way Rick smiles right at her while looking so sure about this. 😍 Like there’s just zero uncertainty about how he feels about her. He knows confidently that she is what he wants. They say when you know, you know - and Rick knows. She's the one for him. He's been knowing.

Then it's precious that in this shot of Michonne, you can practically see the exact moment she realizes in Rick's eyes that he's been wanting this just as much as she has. Danai plays this moment just before the kiss so well, communicating everything with her eyes.

So then Michonne smiles at him with those pretty eyes glowing as they finally lean in for their iconic first kiss.

🤩😍🤩😍😭🤩😍🤩😍😭🤩😍🤩😍😭🤩😍

Now you know it's physically impossible to not happy dance real quick. i gotta do it. 😋

This first kiss is absolute GOLD. 🔥

I really do see that “of course” moment that Andy and Danai talked about cuz it just looks like they’re thinking of course it’s always been you.

As I’ve said before, to me this couch moment is not where Rick and Michonne first realize their own feelings for each other, as I think they been knew that, especially within the last two weeks - but rather this couch scene is where they finally realize, at the exact same time, that the feeling is very much mutual.

So still holding hands, Rick and Michonne lean in for that kiss that is utter perfection and just keeps getting better.

I love that Richonne's canon era started with a handhold that unsurprisingly turned into the most passionate first kiss of all damn time. 😊

And y'all, I’m watching this scene back to write this post, and I have to admit the moment they first kiss I literally gasped like it’s not my thousandth time seeing this moment. 🤭 Just goes to show...

I love the way the kiss starts off just so slow and intimate and of course ramps up. Cuz you know once these soulmates open this door it ain’t ever closing again.

Every good adjective comes to mind when thinking of this kiss. And let me tell you, this is the first kiss of two people who have wanted to kiss each other for a long time. 💯

Also, the fact that this was the actors not wanting to overthink it and just let the moment play out as organically as possible and this is the level of passion that came from that...no words. Just applause.

I adore everything about this equally hot and heartfelt moment. The natural progression. The quietness and the music. The way they can’t stop looking at each other. The way they can’t stop smiling at each other. (Their smiles throughout might be my favorite part. #PureBliss.) The unscripted impromptu moment the show kept in between Andy and Danai, with 'Rick' gently saying wait a minute because his gun holster was in the way, and 'Michonne' laughing. (The best soulmates are best friends too, and I feel like their reaction to this little 'hiccup' moment inadvertently ended up nicely reflecting that. 😊)

The way they so intently and lovingly take each other in. The way they both equally feel like “finally.” The steamy passion as it goes from a first kiss to more.

The hunger for each other. 🔥 (Especially from Slick Rick. Truly, I forget sometimes that we’d ever seen Rick kiss before this because the passion Michonne brought out of that man was different in the best way. 👌🏽 And I love that the second Rick had the greenlight, he let Michonne know exactly how much he's been yearning for her).

The way it was so clear that this person in front of them is their best friend but also is so much more - this person is their soulmate, the greatest love of their life. 😭

It was and is amazing to see the immense love and desire they have for each other pour all the way out after all the build-up throughout the previous seasons. I'm forever in awe.

And no matter who claims to not see the chemistry (which to me requires an urgent optometrist appointment if someone still can't see it tbh) it'll always be factual that the characters of Rick and Michonne deeply love and desire each other in every possible way.

This special scene immediately proved that all that chemistry and tension sensed from Rick and Michonne pre-canon was not only really there but would be amplified perfectly once they acted on it.

Whenever I think about Richonne's story from the very beginning to this canon moment here, it never fails to warm my heart and make me emotional because it is just absolutely moving that their journey led them to each other in this way.

This moment will forever be beautiful and a big deal both for the characters, the series, and honestly for television as a whole. #NotAnExaggeration.

The scene was a gorgeous moment of love prevailing in a show that is more often grim and gruesome. And it's a rewarding moment for these two characters to receive a major win by officially letting their romance ascend to new heights as they embrace being the married couple they've been for the longest.

Richonne's canon moment more than delivers and even gives us a bit more when they cut to that work-of-art scene featuring Rick and Michonne asleep in bed together, all tangled up cuz magnets don’t ever quit even when they’re sleeping. 🧲😊

And even with Jesus' bold self barging into their room, they both show that the Get Things Done Grimes are still ready to stand on business at a moment's notice when they swiftly hop out of bed with their signature weapons.

See, Rick and Michonne are great at being the cutest love birds, but they're also great at being...

We got us a ship that can do both. 😋

And while Gimple was behind my least favorite decision the show ever made - killing Carl (😞) - he's also behind my absolute favorite decision TWD ever made by putting Richonne together as a romantic couple. So Gimple, for your Richonner heart, I'm grateful. 👌🏽

Also, hearing Andy and Danai discuss this whole couch moment is truly as swoon-worthy as the scene itself. Like every time I come across their heartfelt reflections on this moment, my heart completely melts. 😍

Again, we’re so fortunate to have Danai and Andy as both the talented actors bringing these characters to life and as the passionate, thoughtful captains of this glorious ship. The way they always value and champion Richonne is just the best. 🥰

Everything about Rick and Michonne's iconic canon ep let us know that the power couple to end all power couples was finally here and here to stay. And I will always treasure this impeccable, beautiful, passionate, and game-changing scene where Richonne officially took flight. 🥳💜