arc-length

I think the reason why Infinity Train flows so well is because it’s structured like a full length movie, only 10x faster.

It’s got the 3 acts structure and full character arc of a full length movie, but every sequence takes 1 minutes instead of 10 minutes.

Formulas to Remember for the Math Section of SAT

Algebra

Percents:

Percent Change= Change/Original × 100

Part = Percent /100 × Whole

Examples:

75% of 300 is what?

Solve x = (75/100) × 300 to get 225

45 is what percent of 60?

Solve 45 = (x/100) × 60 to get 75%

Example: 30 is 20% of what?

Solve 30 = (20/100) × x to get 150

Average Speed = total distance/total time

Distance =rate × time

 

FOIL: (x + a)(x + b) = x ^2 + (b + a)x + ab

Difference of Squares:

a)     a ^2 − b ^2 = (a + b)(a − b)

b)    a ^2 + 2ab + b^ 2 = (a + b)(a + b)

c)     a ^2 − 2ab + b^ 2 = (a − b)(a − b)

Distance Formula:  SQRT (x2 − x1) ^2 + (y2 − y1) ^2

Mid-point of the segment AB: x1 + x2 /2, y1 + y2/2

Slope of the line: y2 − y1 / x2 − x1 = rise run


Geometry

Opposite Angles are equal, so a+b and 180 degrees.

All the “a’s” are equal and all the “b’s” are equal. Any “a” plus any “b” equals 180 degrees.

 

Right Triangle Formulas***

Triangle area= ½*b*h ***

Angles in a triangle add up to 180 degrees.***

 

Circles

Area of a Circle = πr2 ***

Circumference = 2πr ***

Full circle = 360◦ ***

Length Of Arc = (n ◦ /360◦ ) × 2πr

Area Of Sector = (n ◦ /360◦ ) × πr2

 

Total number of degrees in the interior of an n-sided polygon= (n-2) × 180 where n is the number of sides.

For example, an eight-sided polygon (or octagon) has:

(8 – 2) · 180 = 6 · 180 =  1080 degrees

 

Arithmetic

Arithmetic sequence formula: an=a1+d(n-1)

a1=first term, d =distance between the numbers, and n= term to find

Example: Find tenth term in 3,7,11….

a10=3+4(10-1)= 63


Geometric sequence formula: an=a1×r(n-1)

a1=first term, r =rate  between the numbers, and n= term to find

Example: Find 6th term in 3,6,12…

a6=3×2(6-1) = 3×2 5=3×32= 96

Functions

I was putting together a cheat sheet for studying for the math section of the SAT. Hope this helps some of you! :)

*** on the SAT given formulas, but still good to know by heart. 

  • ban ryu: alright then han sung. since you say you're that smart, solve this without using any calculator. find the arc length of three x times the square root of seven minus two from x equals zero to one half.
  • han sung: *stares at equation while deeply thinking* well, there are two 'twos' in the equation- minus two and the denominator from the one half. the number two sounds like the letter 'u', and u is between the letters 't' and 'v' in the alphabet.
  • han sung: but tvs aren't really relevant anymore because everybody has computers now, so it's kind like... if you have two tvs, what is it even-
  • han sung: *out loud* four.
  • ban ryu:
  • han sung:
  • ban ryu: ... that's right...
Beneath The Waves

Submitted by: Arc (http://arcanity.tumblr.com/)

Length: Short

Nothing calms me more than the sight and sound of the rolling waves. The smell of the crisp, salty air. I love everything above the waves, from the gulls to the clouds. I love everything below the waves, from the smallest mollusk to the largest whale. The ocean is home to me.

Or, it used to be.

Now, the ocean holds no welcome for me. The waters send chills through me when I merely think of them. I have violent nightmares, and anxiety that haunts me persistently.

I was out on my small boat in the deep sea, quite a ways from land. I had just passed the point in the area where the ocean floor dramatically drops, and the water visibly darkens due to the change in depth. I come to this spot often and relish the silence, the calmness of the waves gently splashing against the boat. Today, however, I felt uneasy. I felt… exposed. Nervous.

Suddenly, the silence broke. A thundering, guttural resonance was all around me, shaking the boat, disturbing the waves. My first thought was the foghorn of another boat, but quickly I dispelled it because there was none in sight. And, this sound was much, much deeper. Organic.

And it was coming from far below me. Something utterly massive was in the depths beneath me, calling. I could feel its presence, waiting for something.

The noise continued, and I could do nothing but lay in the boat quivering, clasping my hands over my ears to no avail.

I am a rational man, and not afraid of the unknown. But the titanically large creature below me made me lose my wits, and left me feeling like a helpless, terrified child. It can’t be a whale, the call is too… monstrous. Too primordial.

And then the noise stops.

I sit in my boat, paralyzed, listening intently for nearly an hour, in complete silence once again. I know it’s still there, I can feel its presence. Finally, I steel myself to start the engine and turn back to land, and resolve to never go out in that part of the water again.

However, just as I was beginning the journey back, the noise starts again. No, a different noise. Coming from a different place. The same creature, but farther out, deeper down. And then the noise erupts below me again as well.

My God.

It was not alone.

I have never gone back to the ocean, and I never will. I sleep fitfully, and am always haunted by the noises echoing in my head.

The ocean is not my home, for I fear what lies beneath the waves.

Credits to: Arc (http://arcanity.tumblr.com/)

Study With Me: Line Integrals

Hey guys! I’m currently studying for the Mathematics Subject Test of the GRE, which I plan on taking in the fall. One of the ways I like to study is by explaining the material to someone else. I currently have weekends off from research, and since Saturdays are for the boys, it leaves Sundays for GRE preparation. 

Because of this, every Sunday, I’ll explore a different undergraduate topic that could appear on the Mathematics Subject Test. This week: Line Integrals.

I’ll talk about the following:

  1. What is a line integral?
  2. How do you calculate a line integral?
  3. An Example

As a brief note, this post contains LaTeX code and will be much easier to read when viewed directly on my blog, where the code will compile!

What is a line integral?

Let’s first recall what we already know about integrals. We’re used to integrating functions of one variable over an interval [a,b]. We can think of this as integrating over the path on the x-axis from a to b, and the value of the integral as giving the area bounded by the curve y=f(x) over the path [a,b]. 

But! We can also integrate over paths that aren’t just straight lines along the x-axis. The resulting integral is called a curve, contour, or path integral. Most commonly, it is known as a line integral. 

In this post, I’ll be talking about line integrals with respect to arc length. 

Before we get into it, I’d like to start by defining what it means for a curve to be smooth. A curve, C, with parameterization r(t) = <x(t), y(t)> is smooth if the derivative r’(t) is continuous and nonzero. Additionally, we can say C is piecewise smooth if it is composed of a finite number of smooth curves joined at consecutive endpoints. Basically, this means I have a bunch of curves $C_1, C_2, …, C_n $ that are all individually smooth and Each $C_i$ has its endpoints connected to $C_{i-1}$ and $C_{i+1}$. 

Back to line integrals. Suppose we have a function f(x,y) and a smooth curve, C, in the x-y plane. We want to think about breaking C into n tiny pieces of arc length $\Delta s_i$. For each of the tiny pieces of C, choose any point $P_i = (x_i, y_i)$ and then multiply $f(P_i) = f(x_i, y_i)$ by the length $\Delta s_i$. This process is fairly similar to how we define integration for the case where the path is a line on the x-axis. We want to sum up these multiplied terms for all n terms. If the value of that sum approaches a finite, limiting value as $n \rightarrow \infty$, then the result is the line integral of f along C with respect to arc length. Below is a comparison of the single variable case integrating over a path [a,b] on the x-axis (left) and the line integral with respect to arc length over the curve C (right). 

Note the notation used for the line integral. If we’re integrating over a path C, we write C at the bottom of the integral. 

What does this mean geometrically?

The value of this integral is the area of the region whose base is C and whose height above each (x,y) point is given by f(x,y)

How do we actually calculate the line integral?

First, parameterize C. That is, for a parameter t, find the equations x=x(t) and y=y(t)  for $a \leq t \leq b$. We consider C to be directed, which means we’re saying that we trace C in a definite direction, which is called the positive direction. Basically, we’re saying that t runs from a to b, so A = (x(a), y(a)) is the initial point and B = (x(b), y(b)) is the final point. 

Since we have $(ds)^2 = (dx)^2 + (dy)^2$ (think Pythagorean theorem), we can write:

$\frac{ds}{dt} = \pm \sqrt{( \frac{dx}{dt})^2+(\frac{dy}{dt})^2}$

which can then be rewritten as:

$\pm \sqrt{(x’(t))^2+(y’(t))^2}$.

We use the + sign if the parameter t increases in the positive direction on C and the - sign if t decreases in the positive direction on C

So, we have:

$\int_C f \,ds = \int_{a}^{b} f(x(t), y(t))\frac{ds}{dt} \,dt$.

An Example

Determine the value of the line integral of the function f(x,y) = x + y^2 over the quarter-circle x^2 + y^2 = 4 in the first quadrant, from (2,0) to (0,2).

Solution below.

Keep reading

anonymous asked:

Okay so I personally don't get why people are super upset about gon and killua separating? Maybe I'm a terrible person but I thought it was a good thing. After the Zoldyck arc, Killua was like "okay I'll stick with you until you find your dad/I find something that I want to do." And HE DID he has a purpose now besides following around Gon!! It's protecting Alluka!! Idk, what do you think? That's how I always saw it and I see posts of people being really mad/sad and I get confused

Mmmm okay this is a really good ask! Here are my thoughts:

I was…devastated, pretty much, after I got to their separation in the manga. It’s really upsetting I think because after spending so much time with Killua and Gon, after seeing how much they adore and care about each other, after witnessing the strain and devotion of their relationship during the Chimera Ant arc and the lengths Killua goes through to save Gon- it’s just kind of overwhelming to witness all of that for them to part ways at the end of it.

And a part of you knows this isn’t the end end, but you don’t know how long they’re going to separate or if they ever properly talked about everything that happened with the Chimera Ants. We don’t know if they keep in contact after they separate, either.

So I think a lot of hxh fans (myself included) get upset that they separate because they don’t stay together despite how incredibly important they are to each other. There’s also the ambiguity that we don’t know what’s in their future or if they share a future at all.

On the other hand, I totally agree with what you said. It’s good that Gon and Killua separate. It’s important that they grow and seek out what they want in life because if they’re together, that won’t happen.

Protecting Alluka, traveling the world with her- that’s what Killua wants to do. It’s the first time he’s thinking about his goals and he’s putting himself first and that’s so so so good. He only focused on Gon for so long and he needed to get away from that. And Gon? Gon needs some time to figure out what he wants to do now. Killua can’t help him do that.

So, as painful and heart shattering as the killugon separation is, it is the best thing that could have ever happened to them at that point in the series. Killua and Gon couldn’t help each other grow anymore. They had both completed what they set out to do- Gon finding Ging and Killua finding something he wanted- and now they have new goals. 

When they’re older and more mature, after they’ve had time to reflect and grow and live, then they can reunite and see what they want in a relationship from each other- be that friendship or romance or whatever. But that time is not right now and it might not be for a while. We just have to wait and see.

anonymous asked:

Never have I ever written Loki transforming himself to look like Steve to be with Tony. Inspiration from Dark World.

Pretty sure I’ve never written that. 

My instinct would be for Loki to have transformed himself for nefarious purposes, but fell for Tony on accident. Maybe he was trying to hide from Thor in plain sight? Maybe he heard about Civil War and just wants to kick Tony in the teeth by making him think he’s rekindled his friendship with Steve then reveal that Steve had never returned for Tony or tried to fix their friendship. Maybe Loki’s reason wouldn’t be nefarious at all? Perhaps, something happened to Steve, and for reasons it can’t be known Steve is incapacitated, so Loki is pretending to be him so Steve’s status remains a secret. 

Now granted, there is of course the darker route, which is Loki 100% pretended to be Steve to get Tony. If that were the case, I’d go pretty dark. Loki would be an absolute villain with no redemption arc. Depending on story length, Tony could break free, or Tony would remain Loki’s forever/until death.

[Never Have I Ever]

anonymous asked:

Am so pleased you loved Mad Max: Fury Road! I would love to hear your thoughts on how the female characters are portrayed compared with in Avengers: Age of Ultron. Also wonder if you have seen Anita Sarkeesian's tweets proclaiming that Mad Max isn't feminist, which I couldn't disagree with more.

There were a few different parts to your ask, anon, so this ended up turning into a loooooong reply.

The main appeal of Mad Max for me personally is that it’s an amazingly well made action movie without any of the—pardon my French—unselfconscious dickwaving that makes the majority of action movies alienating for me as a female viewer. In fact, the film very clearly addresses how destructive toxic masculinity is. I wish I could say ‘big whoop’, but unfortunately this level of self-conscious criticism in mainstream films is incredibly rare, especially in the action genre. So just off the top of my head, here’s some of the things I feel Mad Max does differently from Age of Ultron (and the majority of other action movies out there) with respect to women and gender. (Mild spoilers under the cut.)

Keep reading

the-foley-knoll-horror replied to your post “it’s been ages since I cared about what steven universe was up to but…”

Jasper was my favourite character and they just kinda had her fuck off out of the series when they set themselves up to do something really interesting with her. I hate that show’s writing so bad.

I think the biggest problem with steven universe’s writing is that after gem drill (and also kind of before then but especially after that episode since it would’ve been a good moment to end the series imo) everything gives off the feeling that the writers had a lot of worldbuilding plans from the very start of the series that they want to incorporate somehow, but didn’t think they’d actually get this far into writing the series. 

su is incredibly diverse and thus constantly at risk of getting cancelled if they get too gay for the network’s liking. they already have the most fucked airing schedule in human history and it’s probably a miracle it’s not cancelled yet. because of that the writers never seem sure of just how many things they can insert that they’ll actually be able to get back to. if they bring back lapis they’ve got to be able to dump her in the barn just in case they won’t have any more episodes to actually develop what they want to do with her. if they bring jasper back they can’t let her stay around for too long because there’s no certainty of whether they’ll be able to do an arc longer than the length of a stevenbomb. new characters that are introduced have to be written to the sidelines as soon as possible too because they either wrap up their arc in the five episodes the network decided to grant them or they might not be able to finish their arc at all before the show gets cancelled. they’re forced to imply things a lot without actually getting into them for fear of being cut short when they do go into them, but it results in a whole lot of implied cool things that are then never explored which is understandably frustrating.

on top of that I also feel like the show is starting to encompass stakes on a scale that was unplanned for. anyone who’s been in the fandom™ for a while knows the posts that go “episode one: steven thinks his powers come from ice cream. episode x: everything is terrible” and while I definitely don’t dislike this kind of escalation on principle, you do very much need to know what you’re doing and also have a clear end in sight when escalating things on this scale or you end up with some dragon ball nonsense. steven universe doesn’t seem to have an end in sight because it has no idea when it might be forced to end and the show as a whole suffers for it.

anonymous asked:

Story-wise, are we around the halfway mark of the comic? And do you ever relate to any UT characters? Just curious. And, just a theory, but i suspect that the spoiler kittens music track may involve Pap & Frisk confronting Flowey after learning of his true colours, with Megalo symbolizing Pap thinking of Sans? Awesome work, especially with the characterization & OST! I hope you're having as much fun with IF as your fans are!

Hmmm… Tentatively, I’m gonna say this is around the halfway point, yeah! Hotland’s gonna be a fairly long arc, after which New Home’s gonna be notably shorter (only slightly longer than in the original game). There is no Photoshop Flowey fight, but I do have some extra stuff planned after Toriel and before the True Lab…

Annnnd post-True Lab is gonna be bonkers and a full-length arc in its own right.

As for relating to ‘em… sssorta. Alphys feels relatable in some respects, and Papyrus to a degree.

Thank you for the kind words, btw!

Everyone, please help Kagami-kun.
  • Kuroko: I'm deeply sorry for calling everyone in here.
  • Kise: That's fine, Kurokocchi!
  • Aomine: It's not that I'm doing important things and it's been a while since you asked for my help.
  • Murasakibara: Anything for Kuro-chin~
  • Midorima: Takao said it's important to help my former teammates and since I don't have anything important, I decided to go.
  • Akashi: Anything for my Tetsuya.
  • Kuroko: Thank you, everyone.
  • Akashi: So, Tetsuya, what can we do for you?
  • Kuroko: Finals are coming up. Since Winter Cups are over, we are currently working to increase our grades... But knowing how stupid Kagami-kun, it's impossible for him to achieve learning things within a month. So I beg all of you to teach Kagami-kun.
  • Everyone: Eh?
  • Kuroko: It's a burden but... I'm only an average person.
  • Kise: Wait, wait! So that's why Kurokocchi called us?
  • Akashi: I even declined the offers some of the counselors offered for me...
  • Midorima: I missed my usual training because of this.
  • Murasakibara: I consider all of you are lucky since I ditched Muro-chin for our snack hunting weekend.
  • Kagami: Tatsuya wouldn't mind though!
  • Everyone: ...
  • Aomine: And why are you here, Kagami?
  • Kagami: This is my house, Aomine!
  • Aomine: Ah, right.
  • Kuroko: Everyone, please, pardon that tiger for a while and listen to me.
  • Kise: Go on, Kurokocchi.
  • Kuroko: Murasakibara-kun is good at Physics and Math, I suppose?
  • Murasakibara: Hm? I do.
  • Kuroko: Midorima-kun is good at Biology.
  • Midorima: Certainly.
  • Kuroko: Kise-kun and Aomine-kun... uh.
  • Aomine: Hey, Tetsu! I'm good at something too, you know!
  • Kise: I'm good at English-ssu!!
  • Kagami: I don't need you to teach me that!
  • Kise: You need to since you are in Japan now!
  • Kuroko: I guess I can teach Kagami-kun Japanese since I'm pretty average on that.
  • Akashi: That leaves to English and Social Studies to me, Tetsuya. Don't worry, I hate slacking off so I guarantee you will succeed, Kagami Taiga.
  • Kuroko: Aomine-kun and Kise-kun can head home.
  • Aomine: Wait!!
  • Kise: We can do something else-ssu! We can!!
  • Kuroko: How about keeping on guard in case Kagami-kun ran off?
  • Kagami: I won't run since this is my house.
  • Kuroko: Just in case. Shall we begin?
  • ---
  • Murasakibara: Kagami, it's easy if you memorize sine, cosine, and tangent. Also, that's not how you use Pythagorean Theorem~
  • Midorima: Are you even listening to the teachers?! Atoms are the small ones!
  • Akashi: Do you... even learn English in abroad?
  • Kuroko: Kagami-kun... please, listen to me.
  • Kagami: Ugh! I hate this! You guys are giving me hard times! Kuroko?!
  • Kuroko: You're not listening.
  • Kagami: Then don't look at me like that!
  • Akashi: We don't have a choice, let's replace his brain with a newer and intelligent one.
  • Kagami: Wait!
  • Aomine: Ha! Sucks to be you, Kagami!
  • Kise: Kagamicchi, I feel bad for you.
  • Akashi: Oh, don't be envious, Daiki, Ryouta.
  • Kise & Aomine: Eh?
  • Akashi: I also got a brains for the both of you.
  • Kise & Aomine: WAIT!!
  • Akashi: How about we do a little quiz? Also, I want to test Shintaro and Atsushi for this.
  • Murasakibara: Eh~
  • Midorima: Akashi, I don't need such a thing.
  • Akashi: Oh, don't worry. It's just a little review. I trust both of you to get a full mark.
  • Kuroko: ...
  • Akashi: Uh, Tetsuya...
  • Kuroko: I get it, Akashi-kun.
  • Akashi: Alright then. This is Math.
  • Aomine: ...
  • Kise: ...
  • Akashi: Find the length arc with a 67 degrees and 3 radius. Round to the neare---
  • Aomine: Just hold a fucking moment, Akashi! Why are you doing both?!
  • Kagami: ... Even though that was English, it feel so foreign.
  • Kuroko: Was that even Math?
  • Midorima: ...
  • Murasakibara: Can I go and buy some snacks?
  • Akashi: Hold on a moment, you guys are not going anywhere until you answer my question.
  • Everyone: ...
  • Akashi: Now then, this study date shall begin.
  • ---
  • Imagine Kagami and Kuroko asking Kiseki to tutor them~
funny story

so last year during math class we were talking about circles and how to find the arc length (it’s a section of the distance around) and my 50 year old male teacher asks what we use to find this. no one was really in the mood to answer any questions that day, so I took it upon myself to answer it. now note, I’m usually pretty quiet in all my classes, but I’ll answer a question here or there. this is because I tend to stumble over my words and mess them up sometimes. so the answer was circumference, and I raised my hand to be called on. He calls my name and what do I say?

“You use the circumcision.”

my class just stared at me and a couple chuckles were heard from around the room. my teacher was mortified, and just replied “you mean the circumference?”

I said “yeah, what did I say?” and he just laughed and continued with the lesson. I had to turn to my seat partner and asked what had happened. she explained what all went down. now I avoid speaking in classes all together.

this week on Discovery predictions
  • “the butcher’s knife cares not for the lamb’s cry”
    • yo this is some TOS-era Long Dramatic Titling (see: for the world is hollow and i have touched the sky) and i am PSYCHED for what it really means
    • is Lorca the butcher?? are the Klingons butchering?? wait and see, kids, wait and see
    • my best guess: Klingons attack a Federation settlement and slaughter inhabitants; Discovery is the only ship that can go after them
    • Landry and Burnham sic the one creature from the other ship on the Klingons??
  • Tilly and Burnham have bonded™ a little bit more but Tilly still gets nervous around her
  • we learn Almost Nothing about Lorca’s menagerie because that’s the kind of season-length arc the writers wanna drag us down
  • Michael’s “i’m sorry” is for… the settlers who died? someone else who dies because of her actions or the actions of other starfleet crew?
  • Discovery does not get destroyed either by klingons or by impacting a star because hey this is episode 4 and they wouldn’t do that to us - right? right??
  • heads up that jason isaacs will be on After Trek so all you fellow losers who are head over heels for Lorca despite his shadiness y’all better watch after trek!!

cynicalclassicist  asked:

So why did D&D think Ramsay needed more focus then Stannis? Do the words Creator's Pet perhaps apply here?

I suspect it’s partly an outgrowth of the fact the writers don’t care about plots north of Moat Cailin, and partly general misunderstanding of Ramsay’s original dramatic purpose.

The writers have never been interested in Northern politics, Ned’s legacy, or Jon Snow’s fraught relationship with Winterfell. They’ve never understood Stannis or invested effort into Sansa. They’ve all but ignored the fact that the Ironborn have occupied large chunks of the North since season two. And those are the central pillars of the plot in the North (except for Sansa, as of AFFC, but I expect that to change in TWoW). These things are all factors that take multiple seasons of thoughtful writing to get across properly.

But Theon and Jeyne Poole getting tortured, those plots they could get their heads around. They saw nice, discrete, season-length character arcs in that. And so they built nice, discrete, season-length character arcs around Theon and Jeyne Poole Sansa being tortured. Since Ramsay is the chief torturer in those arcs, he’s on screen a lot. In terms of plot, with all the book plots I listed pared back to almost nothing, Ramsay has become a blunt dramatic instrument, there to be a roadblock preventing Yara from rescuing Theon and Brienne from rescuing Sansa and the vague, amorphous Northern society from participating in any sort of plot-relevant anti-Bolton activity. Need something accomplished between Moat Cailin and the Wall? Too bad, Ramsay time!

It’s a boring role, though, Ramsay. One-note and intentionally so on GRRM’s part. It would be easy to think you were wasting an actor. So they gave him more to do, dramatically speaking, a bit more nuance, forgetting why it was GRRM chose to make Ramsay Snow so utterly one-dimensional (and also a minor character). They have the time to expand Ramsay’s role, now, since they edited out Jon’s concern for his sisters, Stannis’ campaign in the North, and the fantastic stuff with the Northern lords convening in Winterfell. (With regards to Stannis, I suspect we’ll see Jon Snow take over his role, as we saw Tyrion take over Barristan’s and Dany’s narrative roles in Meereen. Ramsay, I think, is simply the instrument of removing Stannis as a military power in this case.) That’s how we get from the gleeful torturer in season three, through an increased focus on Ramsay’s daddy issues in seasons four and five, to poor sad Ramsay mourning his dead abuse victim girlfriend (she was never scared of him, honest!) in season six.

I suspect this is the case because this is pretty similar to what happened with Tywin Lannister in seasons one to four. Book!Tywin is nowhere near as one-note as Ramsay, but he is an ice-cold emotional abuser (and a lot of other bad things) without a shred of humour to him. Personally, I find scenes where Tywin interacts with his children painful to read precisely because Tywin does not give them anything, emotionally speaking, not the slightest indication that he’s affected by them and therefore that they matter to him. The emotional range he displays in public is very limited. That has to be hard to write for screen.

With Tywin, the writers took a shortcut to showing depth. Hence stern grandpa Tywin talks about Jaime’s dyslexia with Arya, going so far as to laugh at her “most girls are idiots” comment, displays actual passion when talking to Tyrion (even if it’s seething rage over his failed murder attempt), and reframes the Red Wedding as something he did to protect his family. 

This “depth” and “nuance” means that the nature of how book!Tywin abused his family on a day-to-day basis is necessarily dulled.

Likewise, misunderstanding why Ramsay’s a one-dimensional villain means the writers have missed the broader thematic point (I know, I’m as shocked as you are). Ramsay’s one-note evil is not an asset. As far as problem-solving goes, all Ramsay has is a hammer, and to him, every problem looks like a kneecap. Unsurprisingly, this is not a good way to win friends and influence people. It’s not a good way to keep allies. Ramsay’s gone past the Machiavelli point of “avoid being hated.” The point of a lot of the Winterfell stuff in ADWD is to see how people nominally in Roose’s own camp are plotting against him - not just to see Theon tortured more.

But yeah, between not caring and not understanding, that’s how we got Ramsay the Villain Sue. Thanks for the question, @cynicalclassicist!