#field theory

[disposable-semicolons]

I've seen a lot of students disliking complex numbers

because of their "weird" property of being two dimensional, or being generally led astray by their intuition telling them something like "a+bi" should somehow "simplify" if it "really is one number". So, I am going to throw you a curve ball to think about. Forget about real numbers for a bit, at least forget about decimal representations and stuff like that.

As a quick reminder, a rational number is any fraction p/q where p and q are whole numbers (and q≠0). The golden ratio

φ=(1+√5)/2

is irrational, that means there is no integer fraction equal to it. Since φ is irrational, so is any b⋅φ where b is rational (unless b=0). It also means that any combination of rational numbers a,b yields a unique number a+bφ. You might already be able to guess where this is going. While a+bφ would not give different numbers than just a if we allowed real a, restricting a and b to be rational numbers means that this set of numbers is "two dimensional" in the exact same sense as the complex numbers are. For those with a linear algebra background one would say that ℂ is a 2D vector space over ℝ while our set {a+bφ|a,b∈ℚ} forms a 2D vector space over ℚ.

That's not the interesting part though.

The fun part is that this number set, just like the complex numbers, is closed under multiplication! This is because

φ² = 1 + φ (Eq. 1)

by definition of the golden ratio.

Therefor we can multiply what I will call golden numbers (for obv reasons) and obtain another golden number:

(a + bφ)(c + dφ) = (ac + bd) + (ad + bc + bd)φ

What about fractions then? Well, we can take Eq. 1, shuffle a bit and divide by φ:

1/φ = φ⁻¹ = φ - 1.

Now instead of immediately generalizing I think I would like to give people a bit of a chance to have some fun with this on their own first. So here's a question which you can go ahead and try to answer for yourself, it's fun.

Does every golden number a+bφ≠0 have a golden multiplicative inverse?

Or in equations: Let a,b,c,d be rational numbers and let φ be the golden ratio. Find for any given a+bφ≠0 a c+dφ such that

(a+bφ)(c+dφ) = 1,

if it exists.

Since every golden number is also a real number, we know that (a+bφ)⁻¹ exists, so the question is if the golden numbers are not only "two dimensional" like the complex numbers, but also a so-called (number) field, that is a set of "numbers" that behave as we expect under addition, multiplication, subtraction and division. The rational numbers are a field, integers are not (because 1/2 for example is no longer an integer), complex and real numbers on the other hand are.

Well, you can dwell on that for a while, have fun! Next time I might talk about other similarities to complex numbers like "golden conjugation" and the "golden unit hyperbola". Have fun!

[brent-emery-pieczynski]

Force Fields, Behind the Fog of Maths: Sheldrake–Vernon Dialogue 87

[hanafubukki]

Thinking about Skully and his obsession with Jack.

Jack is his idol.

You know what else Jack has? A Sally.

Someone that Skully is missing.

And who else can be Skully’s Sally than Yuu?

Yuu/MC, who unlike the others, thought he was a gentleman and didn’t freak out like they did.

So it would make sense when things go downward, Yuu might be kidnapped or used by Skully. Especially if he tries to take over Jack’s position.

[kizzer55555]

Core Gems

So when a ghost becomes injured, they have a last ditch defense where they retreat into their core. And I mean, injured badly where their body is rip apart to the point they can’t hold a solid form anymore. And they basically go into a hibernation state until they are strong enough to form again.

Ellie, Danny, and Dan are all injured in a final battle against the GIW. The organization was destroyed and the ghosts were safe but the halfas ended up being so injured that they reverted to core form and then went to sleep for a bit. When they woke up, they were still weak but at least recovered enough to gain consciousness. And realize…they are in some kind of auction…in the middle of a heist. It appeared that two furries (one in a bat costume and one in a cat costume) were ducking it out. And they…they were a necklace. All three of them had been turned into a necklace with their cores as gems accompanied by sapphires, pearls, and opals. And frankly gorgeous craftsmanship as the metal was crafted around their cores as if to cradle them and the other gems.

Unfortunately, they were too weak to take a form properly, they could still feel the strain on their bodies. But at least they could still communicate through their auras. Then the cat lady punched a hole in the glass container surrounding them and grabbed their necklace.

However, the bat grabbed the other end and it resulted in a sort of tug-a-war. Meanwhile, Danny, Ellie, and Dan were having a back and form commentary on the situation and what they should do. Completely unheard by the other party.

In the corner of their eye, the three halfas finally noticed a third contender. Some kind of clown who was…hold on…holding a gun?! And it was pointed straight at the two fighting furies who had yet to notice him. The ghosts’ protective instincts went into overdrive and they frantically tried to shout, yell, move. Just do something to warn the two but their cries fell on deaf ears. All they succeeded in doing was faintly glow which immediatly caught the attention of the fighting duo. The two turned to look at the strange necklace but right at that moment, the clown fired and a gunshot rang throughout the auction room. Having no other options, Danny and the others poured every ounce of ectoplasm they had to try and phaseshift, making the two furries intangible as the bullets passed right through them, but in their shock, the two jumped away in opposite directions and accidentally ripped the necklace apart. Gems and pearls went flying and the three cores bounced along the ground.

Luckily, the two finally noticed the clown and went to deal with him and his minions who had appeared. Seemingly putting their fight on hold and forming a temporary truce. The three halfas could only watch as the battle finally wound down, ending with the cops barging into the place and arresting the clown and his grunts, the cat managing to escape with half the scattered gems and pearls from the broken necklace along with a few other jewelry pieces (none of their cores though) and the bat leaving through a skylight.

The auction continued and in the end, despite being broken, their necklace seemed to have caught someone’s interest. A man named Bruce Wayne bought up every piece of the shattered jewelry wear. The auctioneers appeared relived that the item managed to sell in the end and gratefully gave it to him.

Bruce had no idea what happened at the auction, but he could have sworn that some of the gems faintly glowed right before he and Selina were shot. If the necklace was some sort of magical item, then he needed to understand exactly what has been brought to Gotham. It was unfortunate that Selena had taken some parts of the necklace but he utilized his vast wealth to make sure all the other parts ended in his possession. Now he would take them back to the mansion for examination.

[riessene]

my take on fields of mistria's second mystery romance ;^)

[wheredidiputthatlightsaber]

Soooo I'm feeling like Caldarus is going to be the "antagonist" based on his "every hero needs a villain" line at the star festival, and the discussion of how souls can be intertwined throughout time in different ways?

The fact that he /his power was sealed away and has no recollection of his past?

"Caldosian artifacts" so many of which indicate war- the ambiguity of the Dragonsworn and the Priestesses- not to mention whatever Juniper is doing in Mistria.

BUT NOW- he's fallen in love with the farmer, and with the people of Mistria.

[unhinged-as-hell]

Alright here's most of the theories I've seen for ghasts:

Gonna be real with you, crying over ghasts was not on my 2025 bingo card-

Bonus because people were feelin bad bout all the ghasts they've killed:

Edit: there's more in reblogs

[a-path-by-the-moon]

[communistkenobi]

a prof in the department retired last year and was like just raid my office for whatever you want idc im not cleaning it out so it was like book whalefall. and because nobody in my department cares about theory i got to take all of his books on states and revolutions. I’m finally getting around to looking through them and he had some gooooood fucking books dude, I got Lineages of the Absolutist State by Perry Anderson, States and Social Revolutions by Theda Skocpol, Social Origins of Dictatorships and Democracies by Barrington Moore Jr, an edited collection on apartheid, and a book about settler colonial policy in Canada

[midnight1nk]

So, this week's episode...

[spoilers below cut]

Man I tell you, another heavy day in my production job. And on top of that, pollen season. There goes my allergies smh. Well, at least I get bit of a break, have some food with me, and I can finally watch today's episode. Let's see what we got! *sees thumbnail* 😶 .....I haven't even pressed on the video and my theorist senses are tingling already

"wait what?"

Alright, so I'm going to need you to follow along with me here, cool? Now, doesn't 4's pose seem familiar to you? Well, it should because that's exactly the same pose as the ringmaster 4 render from the early version of the WOTFI website. We all know it was really Mr Puzzles behind it all.

And now look at the ads! It certainly feels very similar to what happened in the "Mario PC Virus" episode, and y'know how I already pointed out how it could connect to the goop!4 theory. BUT y'know what other video had ads?

That's suspicious. That's sus. 💅 /ref

"Don't you think it's Ben trolling again? Or just taking creative liberties?"

No bc the thing is, it's the thumbnail and it's the Team who gave him the prompt in the first place. Ofc Ben can take creative liberties, but essentially it has to follow the prompt. How have I not started the episode yet?! *turns to the Team* I'm watching you.... /silly

(the following is my live reaction:)

and there's our beloved intro *claps like an excited lil kid*

A convention? oooooh

LOOK AT THEM AAAAA

ofc 3 talking about his villain self 😌↕️

Clench being a VTuber was not on my bingo card but honestly, good for him (and I do love Tari sticking out from the table hehe)

and MELONY! you got what Axol's been working on, Two Piece. Gotta love the continuity, dude. The cover looks great!

Melony: "At least, I hope so..." 🥺 Oh honey, Axol would've been so proud of you and what you've done. I can already imagine him singing praises to it and say "look at what my girlfriend and I have done! :D"

YES, let's spread the word!! ofc Mario and 4 would help, they care so much about their friends

ngl Mario's got a good tactic.

and there goes 3's eye lol (don't think about goop!4 don't think about goop!4 don't think about goop!4)

*wii sports theme plays*

YES YES YES oh I'm so here for it

look at them go :D

nahnahnah, Team. You think you can do a speed-up moment and for me not to pause it frame by frame? smh /silly

luckily for the rest of yall, I got you:

and yes, it did have some repeats that I think they're very interesting: "turns you muscular SUPER chad", "makes you rich", "become SWAG", "get your a bunch of friends", "become a member of society", "become a SIGMA", get a romantic partner ("girlfriend" (the one with Mario) or "boyfriend", "you WON'T NEED SLEEP", "become AWESOME"/"RESPECTED"

Ofc it could just be how much 4 wants to emphasize the benefits of getting the manga. But it certainly feels strange, doesn't it? *writing notes down like a madman*

also a win for the skittle squad

and ay, it's got 4's stamp of approval!! (curious that the Team used the old model instead of current one 🤔)

hehe look at 4 pouting while he sits 💙

SMG4: "This is how it works!" 4, bud, are you ok?

ok first off, WE GOT THE OG 4 MODEL BACK?! WHAT? and two, Mario has a point. 4 is taking things too far and really needs a wake-up call. I have a feeling as to why 4 may be acting this way....

oh. oh okay.

So they decided to bring parallels to show how much 4 and Mario's fighting is affecting her/reminding her of her family separating. Hmm. Alright, Team, which one of you decided to strike that dagger into my heart? 😭 /lh

And Mario walked away just like Melony's mom did *head in hands*

and that's 4 alright. He would say that "everything's fine" when it's clearly not smh

4, buddy no :(

SMG3: "The real villain is society..." Well you ain't wrong, 3

Wait a second. Enhance..... enhance.....

I see what you did there, Shadow 😔↕️

love that Toad's like right here too

OMG was that 4's "date" from the "Find Luigi" episode? IT IS.

I'm tell yall, 4 has a secret art account and is selling his work as merch/prints. I mean, c'mon

NO 4, people were waiting for Silksong for decades naurrrr

oh Melony, my girl :(

*pauses episode* ........they did not just do that. I'm rewinding that moment 2/3 times now, what do you mean I was right about my suspicions? WHAT DO YOU MEAN THEY JUST BROUGHT BACK IGBP?!

I need to walk away. I need to leave the room, do a cartwheel and dive into a lake, bc HUH?!

okok, let's breathe. goop!4 is gonna happen, and this is not just a "haha Ink is a crazy lil theorist wishing for this to happen". The fact that he said "nowadays" means that the situation is affecting him too. Ofc it would, that's his livelihood, his WHOLE PURPOSE as a meme guardian. But, other than that, it makes sense why he's doing this (I'll talk about it soon)

Mario: "SMG4, you're better than this! You've done enough damage." 😨 holy shit. it's like I got shot through the heart WTF. I might just cry in this episode

who keeps punching me in the FEELSSSSS 😭 poor Melony, those guys were absolute jerks smh (desperate times do call for desperate measures *ready to make swiss cheese*)

*head in hands* :(

at least they can apologise

*points at screen* hehe that's me :) I'm just a reporter

did I hear the peanut gallery audio from ace attorney? 👀 (<- ace attorney fan, don't mind Ink)

The Crew came? 🥹

EVEN SHROOMY?

OUGH that got to me, dude. They all came to support her and enjoyed the manga. I'm tearing up, dude.....

what did I say? Melony may have not had both parents as they were separating, lost Axol to the events of the Genesis/Revelations arc, was betrayed by who she thought was a friend who understood her, and despite it all, she's been so strong. Having friends, her found family...

man, this hit me hard and it's not even from the allergies, I tell you that

the truest supporters are the ones who care

this episode wants me dead /very pos

what's the lesson, Mario?

SMG4: "Is that we shouldn't care so much about fame and profit and instead be proud of our friend's work? No matter how successful it is?" yeah pretty much... "No"? wydm "no"?

....nahnahnah, you can't end the episode there, what the hell do you mean by that?

You can't close it off like that! What about my suspicions!? You can't just move on like that in front of my theorist self, c'mon!!

sitcom laugh track and everything, smh /silly

Congrats to SteveAlexAri12 for your art being featured in the end credits for this episode 🎉 cool art of WPNZ

um. Team, any particular why? just asking /j

.・-: ✧ :--: ✧ :-・.

Wow, this. This was absolutely fantastic omg. The writing in this was so good and I loved the tidbits of animation we got, like when Mario and 4 were picking up the pages to clean! This was somehow brilliant, and this is coming from a long-time viewer.

It' crazy's awesome that they brought Wolfychu back to do some lines this episode, and I was right about the Waggy cameo from Shadow (YES!). Also, EVAN! You make the cover of the manga? It looks SO GOOD, dude!! As always, great work with the writing on your end 👏

I've already mentioned Melony but we have to talk about the boys. Both Mario and 4 had good intentions to help their friend out. "I would do anything to not lose my friends" and that's exactly what 4 did. The algorithm has been affecting him so much that he didn't want Melony to go through it either. So, he went overboard with the crazy advertising and such, but he still cared for his friend. And a touch of self-doubt and insecurity from IGBP is back because he thought he wasn't doing enough. What good of a friend is he if he can't help out? The savior complex in him would say that he isn't good enough to help Melony, to "save" her from the low viewer retention. That he isn't enough, so he went desperate to have anyone take a copy of the manga. This isn't just about selling a book, it's that he's worth the friendships he has and they can always come to him if they need anything.

He doesn't want to lose them.

Ofc Mario cares so much about 4, they've known each other for years. Best of friends. If anything, Mario would know everything's not fine with 4's actions and thoughts in general. It's why the line, "SMG4, you're better than this! You've done enough damage" from Mario to 4 hits so hard. 4 was back to the "None of you understand the work I do" moment from Mar10, relapsing, and Mario needed to remind 4 of his true self, the one that cared for his friends without perfections. That instead of helping as 4 sees it, 4 was hurting Melony unintentionally, and he has to open his eyes to see that. Two sides, indeed.

Curious that the episode ended with the not "go cuckoo crazy" lesson instead of what 4 said. Goop!4 would eventually happen, bc at this point, yes. But what Mario said is exactly what 4's mindset is rn. 4 is bottling up his emotions and trauma and as long as he doesn't think about it for too long, everything's fine. As long as he doesn't reach that breakpoint, he's fine. In some way, he learned his lesson after IGBP but not quite. For one, he still doubts himself with the guilt of IGBP on top of it. And two, nothing is letting him move on from his trauma. His in-universe audience isn't letting him, Puzzles during WOTFI wasn't letting him, the memories he's been suppressing wasn't enough.

Why does he go through the extreme? Bc he thinks that it would solve everything for him. Like how he locked Puzzles in prison, thinking that Puzzles would remain there when we all know a prison escape is bound to happen. Like the Meme Factory itself was an elaborate trap. Whatever it takes to protect himself emotionally, as he's under the line from having a meltdown, he's fine. But what is that line? What is his breaking point? The thing is he wouldn't known, as shown in this episode, thinking he would still fix it. Everything's fine, he's fine.

And y'know what the cherry on top is? That, other than 4 and Mario enjoying the manga, Boopkins and Luigi were the first to be interested in Melony's creation, regardless of popularity. Very representative of Kevin irl and the deeper significance of IGBP.

....whoops sorry, I went on rambling for too long. I mean, c'mon, you should know the drill by now *points at bio*. If anything, this episode is the biggest piece of evidence for the goop!4 theory, I'm sure our local co-CEO of the theory Funkii (hey moot!) and other goop!4 enthusiasts would agree, YESSIR. Overall, this was such a sweet episode and I can't wait for more. That's all from me, and remember: numbers always go first!

*taps mic* ...well chat, how do we feel knowing that the star trio all canonically wore maid outfits? (well technically Madoka outfits from 4 & Mario but still)

[marihem]

@fransweek 2024

Day 1: Blushing (literally)

[spidercoris]

eye uhhhhhh i miss eiegegh😭😭😭😭😭😭i miss when they acted like kids😭😭😭😭euuuegerheuiwiwoeoeAAAABBBRNRJRIEI😭😭😭😭😭😭😭😭😭😭 i miss when they had fun together hhhrhrgggeAUUUGGHHH😭😭😭😭😭😭

red let them have a happy ending my heart hurts for them

[novasintheroom]

ALSO I think the other character that hasn't been revealed yet is another dragon, and their name starts with an "Ald," as in "Aldaria???" Because we have "Caldosians" and "Caldosia," which sounds an awful lot like "Caldarus," and it would MAKE SENSE that he and this other dragon feuded, ergo the "ancient war" between Aldaria and Caldosia!!

[senjuop]

The key to the void is to disconnect from the 3D, the physical world.

I wanna share some info I got from reading dr. Joe dispenza book called “becoming supernatural”

*sorry I have to show this in pictures bc I tried to make a post but ig there were too many words and there’s some error*

[folgzstudies]

Sunny evening with axiomatic quantum field theory