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Happy New Year's to you both and I look forward to seeing these two grow in the year ahead of us <3
“Still flunk the math,” Jedao said cheerfully.
That was true. While Jedao had excellent geometric and spatial intuition, he had never developed better than scrape-by competency at the algebraic underpinnings of calendrical mechanics. Kujen had considered fixing the dyscalculia, but it was more convenient not to.
[cracking knuckles] Director's edition and explanation of the math under the cut! Please enjoy!
I'm going to split this into three sections because it's SUPER LONG.
APPENDIX A - CONCEPT
Firstly I want to talk about the 'thesis' I had for this one. Of course the general idea was relating the characters back to multivariable calculus, but the bones I was building on from the beginning was this:
Jedao has dyscalculia;
He doesn't understand and therefore couldn't have killed Kujen (who we are representing by the end with a triple integral);
Je2 has his dyscalculia fixed by Kujen. Together with Cheris and Hemiola, who were lacking the parameterization/context necessary to 'solve' him, they're able to evaluate the integral to 0.
I wanted a three-part structure:
The first to establish the characters,
the second to establish Kujen and his 'backstory',
and the last to introduce Je2 and finally Kujen's death.
The epilogue where Kujen taunts Jedao about his dyscalculia is actually a last-minute addition! I felt like ending on the pinned moth didn't hit hard enough (you're free to disagree, I feel after a few rereads that I was wrong about this). The dyscalculia connection would've been before Section 3, so that Jedao 2 homing in on those curves actually strikes a direct contrast.
By the way, all the typesetting was done myself in LaTeX, or taken from my professor's course notes :D Nothing from textbooks. I had to cut a lot of sections because it was getting long and unwieldy. I realize I cannot make people who haven't taken math since high school understand calculus no matter how hard I add explanations. BUT it's a shame that the second narrative underneath this goes unnoticed!
APPENDIX B - EASTER EGGS
We'll begin with the non-math bits. There's some parts of this that I knew I wanted to include from the beginning and I want to talk about them here!
The moth crawling over top of the paper is Kujen, the insect pins are the 'silver lances'. One detail you may have missed is that the eye patterns on his wings go wide at the end when he's pinned and killed.
I've pinned moths/butterflies/mantises before; you really do pin them this way, you don't pin them through the wings. I've taken creative liberties with the paper stripes you use, but the idea here is they've trapped him in invariant space.
2. All the characters we see book quotes from get interlude tags unique to them. Je2's interlude tag is a ripped and torn version of Jedao's. This is to distinguish them from each other and also to establish Je2 as a character distinct from Jedao/Cheris.
If you look closely, you'll see that the quote from the top of page 3, the section on velocity, is from Je2 before he's even introduced!
3. The general theorem for this, of course, is E as the shape bounded between the surfaces over the region D. I changed those to h(eptarchate) and k(ujen) here to represent his influence and 'work done under' the hex/hexptarchate.
4. Evaluating an integral - this is about Kujen's 'backstory' which, well, he didn't ever tell Jedao and therefore Cheris about. Along the x-axis we can see the discontinuity at age 12 and the rest of the sketch as baby Kujen as a non-integrable complex function.
I think it is so so so so so crazy that he was a child sex slave?!
5. This figure is one I made in MatLab. It is actually from a section I had to cut about the Fourier Series. So if it seems weird that it's right in the middle of the section on double integrals/triple integrals, it's because it's completely irrelevant to that but I didn't want to waste the art....... My copium is that it connects to Je2 talking about the approximation of curves 😭
Anyway, all the numbers at the bottom along the x-axis are important dates to Kujen! See if you can figure out what they are! They're from the general timeline in Hexarchate Stories.
6. This snake servitor here is Hemiola!
7. This page is just a spacer. I did very deliberately place the moth so it obscures the more irrelevant text and points its antennae directly at the crazier parts of his story, though. It also acts to answer the 'question' Hemiola raised earlier in the weave. Where did Kujen learn to dance? Well. 12 year old sex slave, it seems.
APPENDIX C - THE MATH
I'm realizing now how much context for this you miss if you DON'T have a general idea what's going on in the background, mathwise, and also how much more you get out of this if you DO. So let's get into it!
First of all a shoutout to this really good insight from @cloud-piercer. It's pretty much what I wanted people to understand! I'm going to go page by page and just explain specific thoughts and theorems used.
Math time. I think it's best if you interpret what I'm trying to say yourself, and people have gotten it mostly correct in the notes, but there's some overlooked parts I wanted to point out :)
This page is primarily meant to introduce Kujen and the rest of the characters as mathematical concepts. It's nice that Zehun helpfully says 'axis' here because now I can bring Jedao in with the right-hand rule (and associate him with the Cartesian coordinate system, something that will pay off later). Additionally, I wanted to end with the idea that it's impossible to understand/predict/evaluate an object - in this case Kujen - with only points of reference (observed actions). Also, fun - in vector fields, the directions of greatest and least change are orthogonal to the level curves.
2. Setting up the fact that Cheris (with Jedao's memories) and Hemiola separately have pieces of Kujen's context, but they don't have the whole picture of him yet. They don't know his parameters! They don't have a reference frame for his actions.
3. I talked about this page earlier, but here's why we are getting into why integrals now. Integrals are used in math and physics for a lot of things, so it's very flexible as a metaphor. If you're simply evaluating the integral of a function f(x) on [a, b], you're trying to evaluate the area under the function (bounded by the function, the x-axis, and the intervals). (Convention is to integrate with the x-axis as the bound since you're doing it in vertical slices, but you CAN integrate horizontally, and you do this for specific cases!)
Here, we're using the definition of the integral under the function as the 'work done' under the function - quite literally the integral is what happened with Kujen in the interval between his birth and him getting into Nirai Academy Prime at age 14. It's not that simple, though, because he's a complex function and there's a point of infinite discontinuity at x = 12. :)
4. This makes me laugh because we got this hammered into our heads all the time in class.
5. Kujen performing psych surgery on himself as Fubini's Theorem! The gist of this is that for certain integrals, you can evaluate the double integral as an iterated integral instead. This means you can redefine the way the integral is expressed (switching the order the integration gets evaluated). This is useful when you've got a complex integral. You can often redefine the integral and it becomes much simpler to evaluate. So here is Kujen redefining himself.
7. Section 3 - we're moving into the last act and establishing Je2 as a character now! We're moving onto vector functions! The specific idea here is parametrization and prediction of motion - if you had full parametrization you could model and predict anything.
This is the first time we see Kujen as a 'normal' vector function. Also, 'Almost all properties of a vector function are inherited directly from its component functions' - setup for later!
This bit here, this is an important bit. We're setting up Je2, we're bringing CHERIS and HEMIOLA back into the conversation, we're getting into CYLINDRICAL COORDINATES and TRIPLE integrals. The math tells a story here! Multiple!
Cylindrical coordinates - Je2 has the same z-axis (shared axis with Jedao) but different X and Y. We established Jedao as being in Cartesian coordinate system, remember? Je2 is cylindrical coordinates.
Hemiola, Cheris, and Je2 now have the context they need to kill Kujen. They each have one of the three parameters for evaluating a triple integral.
I also loved the insight from anon here: 'jedao 2 isn't the original curve but behaves the same way and that's what matters to kujen far more than those pesky constants that get erased when u take a derivative. applying a mathematician's eye for simplistic beauty to The Mind Of A Real Human Person is totally exactly what kujen does. jedao is presented as simplifyable, derivable, integratable from the start vs kujen whose graph is not a function Until he does psych surgery on himself'.
8. Domains of integration! That's what Cheris and Hemiola were missing! This page really drives me crazy. We know Jedao has dyscalculia. We know Kujen fixes Je2's dyscalculia. With the context given to Je2 willingly by Kujen, this also provides context that Jedao (and therefore Cheris) never had on Kujen before. It's the last parameter they're missing, metaphor wise, to 'pin him down.'
The pins are a secret tool that will help us later (the next page) :)
9. Stoke's Theorem and Green's Theorem: This is another one of those sections I cut for length, after realizing the math would've flown over everyone's heads.
Stoke's Theorem: the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the particular vector function around that surface.
Green's Theorem: the line integral around a simple closed curve C is the double integral over the plane region D bounded by C.
Essentially what they're doing here is taking Kujen as a triple integral and evaluating him with all the context and parameterization and tools they have now, and in the end he evaluates to zero.
This is another section that was cut, but originally he would've integrated to zero as an odd function over a symmetrical axis. I ran out of room to show this, and made do with the math collaged behind the moth/Je2 snippets...
Evaluating to zero: remember, an integral can be considered the 'work done' beneath a curve! And in the end here they've undone Kujen's life's work.
Finally, epilogue. I would've placed this before the Je2 establishing page, but there's a specific reason I wanted this at the end:
The thesis is Jedao has dyscalculia and therefore couldn't have understood/killed Kujen.
Kujen both FIXES Je2's dyscalculia and provides him the context Jedao/Cheris was missing.
Cheris is good at math, but Jedao was missing context and therefore so was she. Hemiola, Cheris, and Je2 all had the separate pieces of domain knowledge/parameterization they needed. It only could've been all three of them working together.
I think it's just... overall, I wanted this to end on a punchline. Through all this math and metaphor, Jedao never could've understood Kujen anyway.
ADDITIONAL
There is a LOT that didn't make it into the final cut of this mathweave. I sent a few drafts to friends and with the exception of, like, my one astrophysics friend who does calculus daily, they all agreed there wasn't any need to include a ton of math explanation/formulae within the comic structure itself. I think it is sad that a lot of the math narrative therefore went over people's heads. I'm EXTREMELY glad that there's a small minority who looked at the math and were able to pick out the meaning behind what I put where! It truly feels like they were shining a UV light over my comic and reading a second story in there, which makes me so so happy.
Parts that were cut, among others:
Fourier series approximations - this would've been about Jedao 2 and Cheris' contexts coming together about Kujen. 'The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed.' Cut for length and because I'd have to tie that back into the triple integral thing.
Explanations of Stokes & Green's Theorems, as well as properly working through the math in the final pages. Cut for length and because nobody wants to read all that.
Kujen as integrating to 0 as a result of an odd function over a symmetrical axis. Cut for length and because I'd have to go through and actually make sure people can follow the integration from beginning to end, and that's tricky for professors to do on a blackboard in front of 3rd university students, let alone a Tumblr audience.
CONCLUSION
That's all, I think! I'm sure I'm missing something but I'm writing all this on my phone in the airport right now. Let me know your thoughts. I really did put an insane amount of thought into this. Finally, if you liked this, check out my kujedao falconry webweave here.
Attack compilation of everything I've made for the first half of Artfight! Splitting these up into two posts - turns out I've had far more motivation than I originally thought I'd have for the event!
Zephinn, a fae creature shackled to a theatre director but is given a huge amount of leniency due to their star performances and good behaviour, who takes up the position of translator to the leader of a rebel cause in their spare time.
Zephinn, a fae creature shackled to a theatre director but is given a huge amount of leniency due to their star performances and good behaviour, who takes up the position of translator to the leader of a rebel cause in their spare time.
