#not tagging c c bc its tangential here

[adamantiline-blog]

I posted 279 times in 2021

100 posts created (36%)

179 posts reblogged (64%)

For every post I created, I reblogged 1.8 posts.

I added 96 tags in 2021

#dream smp - 25 posts

#waite draws shit - 19 posts

#technoblade - 15 posts

#philza - 10 posts

#hbsmp - 7 posts

#hoisin twitter archive - 6 posts

#ill informed sword talk - 4 posts

#my beloved - 4 posts

#hell yeah hell yeah hell yeah - 3 posts

#dream mcyt - 3 posts

Longest Tag: 130 characters

#i think its because i was consuming probably about equal amounts american and british media while i was relearning how to not talk

My Top Posts in 2021

#5

sick and tired of pig/human buff/noodle techno design discourse i wanna know everyone’s takes on what The Fuck is going on with his legs

so like presumably we’ve got red pants/shorts and then the uhhhhhhh chain mail/fishnet/checkered patterned whatever and like either maybe two toned boots or hes got like, leg? warmers? anyways

i feel like, i should be pretty confident that the bottom black part is shoes right?

but

they don’t have any soles????

like, they’re the same as his hands hooves are just out; biblically accurate techno doesn’t wear shoes? you’re telling me this man has 4 distinct leg sections and none of them are shoes? king how many layers of legwarmers do you need?

but

that isn’t fucking all-

id like to direct everyone’s attention back to the “pants”

are they pants? are we sure? because they’re a very similar color to his coat

“but waite, they’re clearly separated from the coat by the trim“

oh are they?? are you fucking sure????  dear reader have you ever noticed that the trim just stops at the front? it doesn’t wrap around

are His PANts attached?!!? To hIS coat? is IT a jumpSUit???

whatisthetruthimlosingitoverhere

716 notes • Posted 2021-08-13 08:27:48 GMT

#4

Taking a quick break from spreading rapier technoblade propaganda to instead spread messer nihachu propaganda

What is a messer?

Its a big knife, just a hugeass fucking knife, sword sized but knife shaped which is the vital distinction. Messer is like literally german for knife.

C!niki should have one and heres why i think that:

German

I think this one is self explanatory

The history of the messer

So basically messers exist to exploit this lil old loophole in 15th century german laws where most people aren't allowed to make or have swords but they are allowed knives. So a messer, in a very "behold a man" sort of way, is technically a knife bc of the construction of the handle and only having the one sharp side. (I think there's more minutia to that distinction but im too lazy to look it up rn, handle, sharp on one side is the broad strokes)

So anyway remember how during the Schlatt administration Niki specifically got her taxes raised n stuff? Now ok hear me out, hc that at some point during that time period Niki also gets disallowed from owning weapons for like, probably similar petty reasons to the taxes so she just, gets a Really Big fucking knife and goes on her merry way.

Knife like cooking like baking? Slice bread in bakery with giant knife?

C!niki is a baker so having her weapon b kind of tangentially culinary is p neat i think. Also the mental image of niki like, serving pie w her Huge Knife amuses me.

In conclusion niki nihachu deserves a giant knife, thank u for coming to my ted talk

1196 notes • Posted 2021-08-12 05:30:42 GMT

#3

So like, the new skin Phil's got on namemc huh? I know that logically probably I should be going buckwild over like is this another death flag? Wings got blown up canon??

But I will be completely honest, I just can't unsee the black as one big shirt and the green as overalls and thats the thing that my brain has latched onto today

1479 notes • Posted 2021-02-24 08:05:22 GMT

#2

"Don't talk to me, or walmart me, or Philza Minecraft, or my dog, or my dog, or my dog, or my dog, or my dog, or my dog, or my dog, or my dog, or my dog, or my dog, or my dog, or my dog, or my dog, or my dog, or my dog ever again" -Techno probably, canon killing the next sorry fucker who tries bothering the tundra cottagecore retirement bois

1862 notes • Posted 2021-01-10 08:17:46 GMT

#1

I love bird mumza as much as the next guy however if there is not enough panda!death goddess art out there I guess I will make it myself.

Well yknow,, shes American so shes got the right to uh, b-bear arms?

3751 notes • Posted 2021-06-05 06:55:33 GMT

Get your Tumblr 2021 Year in Review →

[bestworstcase]

#I'm so happy to see someone articulate #why Raps pulling rank in RatGT was so damaging and hurtful #b/c I often see that scene framed as a#''she's coming into her authority and role as queen!''#but it never sat right w/ me and I couldn't pin why

so these are some of @pnumbra-rbs​’s tags on this post and while it’s a bit tangential to the point of that post i do have sOmE tHoUgHtS about that framing of the scene in the great tree, bc i also see that take floating around now and then and it always makes me go hrm.

broadly speaking, rapunzel’s character arc in rta does indeed involve coming into her authority. she enters the story as a socially-stunted, naive person struggling to grasp basic social norms like “don’t touch people without permission,” and she exits it as a competent, compassionate ruler who is comfortable and content in her role as the leader of her country.

but narrowing our focus down to just the argument in the great tree, it just... is not correct to say that this is the moment when rapunzel steps into her power or begins to come into her own as a princess and future queen. it is Rapunzel Making A Big Mistake, and the narrative intends for it to be taken as such. 

let’s break this down. first, why is it a mistake? well.

is publicly yelling at a subordinate good leadership? is it a healthy, productive use of authority?

consider the argument in RATGT from this perspective for a moment. strip out all the emotional context surrounding the argument, and at the core of it what we have here is a disagreement over the safety of the group. adira’s position is that the group is exhausted and must rest, regardless of the fact that they are in enemy territory. cassandra’s position is that the great tree is too dangerous to make the benefits of stopping to rest worthwhile, and they must press on until they find a safer location.

both are somewhat valid perspectives. hector is likely to catch up to them regardless of what they do, and it’s better to fight well-rested on chosen ground than to be caught off guard when you’re tired. adira doesn’t spell it out, but her plan is basically to get some rest, then stand and fight at the top of the tree when, not if, hector catches up. meanwhile, cassandra’s plan is to get out of the tree as fast as they can and hope that hector doesn’t catch up with them until after they escape. who’s right? it depends on your estimation of the threats posed by hector and by the great tree itself.

(personally, i’m with cass on this one. a high ledge in enemy territory that has already proven to be very dangerous in its own right is not ground i would ever choose as a battlefield unless i had absolutely no other choice. and while everyone is tired after a long day, i like the benefit-to-risk ratio of taking a few minutes to stretch, drink some water, and eat some food before pressing on a lot better than the benefit-to-risk ratio of stopping here for the night and staking everything on a battle in this location.)

now... cass doesn’t handle herself well in this argument. at all. i think she, unlike the rest of the group, is picking up the subtext of adira’s proposal (that they will fight hector here), but rather than rationally express her concerns, she goes off on a paranoia-fueled rant accusing adira of wanting to get them all killed. she comes off as hysterical and unhinged because she’s speaking out of frustration, stress, and exhaustion. her behavior is absolutely inappropriate. 

for rapunzel, as the leader, what is the correct way to handle this situation? 

good leadership is not just about making choices. it’s not just about the judgment calls. it is about management. which means that as a leader, if a member of your team is having an emotional outburst like cass, you cannot respond in kind. it sucks and it’s kind of unfair, but them’s the breaks.

in this situation, an Ideal Good Leader would a) perceive that this paranoid outburst is coming from a place of real concern, and b) respond to it with the intention of pulling out the underlying meaning. eg: “i see that you’re upset, and i’m trying to understand. what’s going on? / where’s your head at? / where is this coming from?”

in canon, rapunzel’s actual response is “come on, cass, listen to yourself,” which is an invalidating (and somewhat condescending) statement that makes cass feel unheard, so cass gets more upset. the situation escalates. by comparison, Ideal Good Leader’s statement directly acknowledges cassandra’s distress and invites her to explain her reasoning. this would help to lower cassandra’s anxiety (by showing her that her concerns are being heard), which would enable her to better articulate her real point.

thus, the emotional side of the argument is defused, and a productive conversation becomes possible. perhaps a compromise (such as stopping for a few minutes to recharge and have a quick meal) could be reached, or else rapunzel can do her own cost-benefit analysis of adira’s proposal versus cass’s and make a reasoned decision about what they should do. cass feels heard, no one’s feelings get hurt, and rapunzel makes her choice based on what she thinks the safest course of action is, instead of angrily defaulting to the opposite of what cassandra thinks they should do.

and thus, there’s no need for an “i’m going to be queen, i’ll make choices you don’t like, and i need you to be okay with that” conversation—because, in this scenario where rapunzel makes good, healthy use of her authority, cassandra wouldn’t leave this conflict feeling hurt. she isn’t upset, in canon, because raps didn’t do exactly what she said to do; she’s upset because she got shot down pretty brutally and she feels like rapunzel doesn’t value her insight, advice, or judgment. if rapunzel had given her the space to calm down, feel heard, and feel like she was allowed to contribute to the conversation re: what to do, i think cass would’ve truly been okay with whatever rapunzel ultimately decided to do.

but of course, rapunzel isn’t an Ideal Good Leader, she’s rapunzel, and she’s still in a place where she doesn’t grasp the full reality of her own authority, and she’s thinking of cassandra as her friend, not as her subordinate. so she escalates the situation by mistake and we get... what we get, in canon. cass ends up demoralized, their friendship is permanently damaged, and the battle with hector goes predictably badly, with cassandra sustaining a horrific, debilitating injury in the process. (this, btw, is fundamentally why rapunzel is at fault for cassandra’s injury. it’s not about the spear vs incantation decision; it’s about this decision, this disagreement, the way she handles this fight.)

and second... how is this framed by the narrative?

well... our first clear signal that we are not meant to see this as anything but rapunzel making a big mistake is that the show explicitly draws a parallel between her behavior here to gothel’s behavior at the beginning of tangled. in tangled, gothel screams “Enough with the lights, Rapunzel!” with the intention of shutting rapunzel down and browbeating her into submission. in RATGT, rapunzel screams “Enough, Cassandra!” because she’s frustrated and upset and cass is acting irrational—but the emotional effect this has on cassandra is the same as the emotional effect gothel’s calculated outburst had on rapunzel in tangled. it’s an abuse of authority in both cases. 

and on top of that, look at how the group reacts—not to cassandra’s outburst, but to rapunzel’s: 

they’re all shocked, unhappy, and uncomfortable, because this is... a shocking, upsetting, uncomfortable thing to witness. (have you ever been on the sidelines while your boss or a teacher singled out a coworker/fellow student to yell at them? yeah.) and when cassandra looks to them for support, none of them say anything... because rapunzel is the princess and if this is how she wants to handle this conflict then, well, that’s her prerogative and they’re just going to cringe and grimace on cassandra’s behalf instead of sticking up for her. 

and of course, no punches are pulled in illustrating how deeply this harms cassandra herself. we see how upset she is in the moment, we see how subdued she becomes when rapunzel comes to talk to her, we see the smile she forces when rapunzel signals that she isn’t interested in hearing cassandra’s feelings, and of course, this is what triggers “waiting in the wings,” which is a song all about cassandra’s pain—how she feels overlooked, neglected, unvalued, and unheard.

logistically speaking, this argument is a lot more complex than “cassandra right, rapunzel and adira wrong.” and emotionally, rapunzel’s reasons for reacting the way she does are perfectly understandable; it’s her first time ever dealing with a situation like this, she’s still muddling through leadership with no idea what she’s doing, and hearing cassandra call her “obliviously naive” was probably legitimately triggering because, hey, what did gothel always call her!

(though i tend to think that the point cass intended to make before raps cut her off was “what do you mean you can’t do that? are you so naive that you can’t see that you’re in charge? of course you can do that!” vs what i think rapunzel and a lot of the fandom took it as, ie cass saying rapunzel is naive for trusting adira. i like interpreting it that way because it dovetails so neatly with everything else happening in this argument re: rapunzel’s authority.)

but... despite this, the narrative expects us to sympathize with cass here. not because she’s right (her underlying concern is legitimate, but her argument as she presents it is... not right), not because her behavior is appropriate (it’s not), but rather because rapunzel’s leadership in this situation is terrible, exacerbates the conflict, and harms cassandra emotionally. 

and as for rapunzel’s growth into her role as leader, this... is kind of another queen for a day moment. in QFAD, rapunzel experiences a difficult choice for the first time: she is forced to weigh the plea of one person she has a personal relationship with against hundreds or thousands of people who are looking to her for leadership in a time of crisis. and in QFAD, she makes the hard, but right, choice by prioritizing the needs of the many. 

and in RATGT, rapunzel is introduced to another common type of crisis that leaders must be able to handle: an emotionally-fraught interpersonal conflict between two of her advisors which she, as the leader, must mediate. and unlike in QFAD, this time, rapunzel bungles it—and sees the consequences play out in vivid detail. RATGT is a vital learning experience for rapunzel. she couldn’t grow into the kind of leader she becomes over the course of s3 without making the mistake she makes in RATGT. and that’s why it just doesn’t make sense to say that this incident is rapunzel “coming into her authority”—because she’s not, she’s making a mess of things and then dealing with the messy, confusing, hurtful fallout of having done so.

[janiedean]

Is it bad that I really enjoyed reading the kinda fandom history/fanw*nk history posts abt j*nsa? XD Tbf I'm also a bit addicted to fanlore articles. XD Anyway I think in the tags of one post you mentioned j*imexs*nsa starting as an antijb thing? Would you mind elaborating on that? Sounds like an interesting, albeit bonkers, story, too.

dw I don't think it's bad I mean... we all enjoy catching up on wank we weren't part of (and my friends the things I could tell you abt spn fandom or when I was around at the sidelines for the worst rpf fic ever..) but anyway *deep breath* SURE THING LET'S SAY IT I mean....... it's not like it hasn't been years and I can talk abt it so whatevs, under the cut because reasons, also I'm saying now that a lot of this was what I deduced from circumstances and you've got to take my word for it unless someone who was around then remembers it and can confirm my impressions bc it was a lot less obvious than the j*nsa stuff

but going from the premises again:

I got a tumblr in like may 2011, read asoiaf june-july of that year, started being active on the tags around august-september 2011 so it was before S2 aired and 95% of the people that were bnfs/around then aren't on tumblr anymore/are here rarely these days so you've gotta take me at face value

in the main asoiaf bnf group there was this one jc shipper who like... I mean she's still active on twt and blocks everyone shipping jb on sight but good bc let's just say she's a piece of work not to mention that she spent months/years writing meta abt how jb was like fanfic stuff and had no book basis and how if you liked j but not c you were a misogynists which was wildly popular

I would like to add that person is basically... let's say everyone in jb fandom who ever got harassed knows exactly who I'm talking about and I'll leave it at that and was not for ship and let ship let's put it like that

so basically this person was on my dash bc I followed ppl who followed her and I saw a lot of jaimes*nsa stuff from her

in the sense that like... she was doing with it what I did with jonc/brynden back in the day when I was wanting to convince the world they were the crackship that was promised as in posting abt it/spamming it/discussing how it made canon sense/more canon sense than jb anyway and the likes, and like there was a fairly numerous amount of stuff for that ship back in the day - like before S2-3 aired jaimes*nsa had pretty high fic numbers/fanwork numbers considering that in canon they haven't interacted once

and that went hand in hand with keeping on trying to burn ground around jb before S2 aired/before they met in canon with that meta

and like... it's not like it started as an antijb thing in itself bc I'm sure that there must have been someone into it before the show aired same as there were ppl who as I was reminded lately wrote brienne/loras hatesex and so on, but the explosion which was during S1-3 (then it lost a bit of traction I think mostly bc jb exploded and sansa/tyrion happened on the show and the ppl involved kind of went more on twitter and didn't produce that much content anymore tho it still has numbers™) was also continuously fanned by person above + her followers who most likely were like... genuinely into it but again the antijb meta was around quite enough at the same time, so

also like... let's just say that back in the day I was the main t*robb ship contributor and I also wrote jb though less but jb was smaller as a fandom so I did post when the ao3 tag was at like... 30 fics or smth so I separately had my foot in both ponds and like until S3-S4 any t*robb fic where jb was the sideship would get tangentially less views/reviews than ones where they were not which in itself wouldn't be a problem bc obviously maybe ppl wouldn't ship one of them, but then there was this uuuh specific case

where someone who followed me for the t*robb who also followed her at some point asked me in private if a 22k t*robb fic that was the sequel to... my most famous/popular jb one let's put like that had a lot of jb in it because they were die hard jc people and it was triggering to them (which... I mean fair enough but in retrospective using that term instead of 'you know I really don't vibe with jb and I'd rather not read it' felt a bit like they were trying to guilt trip me ngl), I said that they showed up together in two scenes and were in the background and at most it was 1k worth of fic and the answer was that it was way too much so they were going to have to skip and like again their prerogative but idk that conversation kind of left me like is this some kind of roundabout way to say that it'd be better if I didn't sideship them? idk but until S3-4 I had the distinct feeling that at least for my own stuff having both ships together meant less readers and ngl I didn't update the infamous t*robb wip fic for a year because I was planning to make jb fairly prominent as a sideship there and I was there like 'oh god what if I get wank over it' and I didn't but again I would like to state it wasn't just because of one person, it was... the general atmosphere that seemed unfriendly unless you stayed on the jb tag/kept both separate

and I would like to add that the moment S3 happened and it sailed anyway bc it was going to was when the targeted harassing of anyone who was active on the tag started ssssso :|

basically like... I have a lot less evidence to go on re ^^^^ than the j*nsa stuff because that is well-documented the rest is my very humble opinion about the background of when j*aimesansa started getting popular but like it also was v. prominent in between jc people (same with the jcsansa threesomes) who were also following that person - I would like to also add that a bunch of ppl who shipped both were absolutely lovely and nice to interact with in other situations and not harassing anyone about it and again nothing against ppl into it as a thing but the moment of its peak was in conjuncture to the burning ground around jb moment driven by that specific person so *shrug*

[scripttorture]

1/? I have a character who has been caught up in a war between planets ever since he was a child. He was out into hiding from the age of 10 to 16, before watching his younger brother killed by the person prosecuting them and elder sister sell her planet (she's heir basically) to save his life and swore herself loyal to the person to save herself.

2/? (She isn't loyal, but she'd be killed otherwise.) The character is then sent to grow up on a different planet, with his mother who figureheads a resistance against the people who took the characters sister and killed his brother. That's basic backstory continuing the character eventually gets captured again, and it taken to a prison. The character is tortured in the prison bc he killed several very important people and cut off the hands of another. 3/? Its seen (by the torturers i suppose, or at least the woman ordering them to do so) as rightful punishment. I havnt quite hashed out exactly what the torture is other than he definitely by the end has rather severe nerve damage in his hands from the shackles and chronic pain/weakness in one of his legs from something or another. Anyway the characters sister was put in charge of this prison, 4/5 and has no choice but to stand by and watch as the character is tortured. She does her best to make sure he isn't killed and the character knows she has no choice but to let them hurt her bc she is just as much of a prisoner as him, albeit in an entirely seperate way. She could stop the torture, and she could get him out, but she would be killed for it and he knows it. Im just wondering if he would blame her, 5/5 because she is in charge and could stop it. But she would be killed and it would likely end with them both dead. She cares for him when she can which isn't often bc she isn't exactly allowed too. Would he blame her I suppose? She has never hurt him, but lets it happen.

-

Alright I understand what you’re going for here.

 It’s not the kind of situation that’s common enough for there to be systematic studies. Most of the time torturers and their victims don’t have a close relationship. It’s much more common to find cases where they were strangers or acquaintances prior to torture then close family or friends.

 This doesn’t make this a bad idea. It just means that there aren’t definitive answers. I’m working from a handful of anecdotes and extrapolating from other things.

 Even if this was a more common situation I don’t think you’d find many definite answers because individual variation would probably play a huge role.

 Torture changes things for survivors in a lot of unpredictable ways. While we know the possible symptoms what any individual ends up experiencing is unpredictable. And how well people cope with mental health problems, and how that in turn impacts their relationships is dependant on the person. Someone’s personal experience, friends, support network, work, general knowledge and a host of other things can effect these sorts of outcomes.

 Having that person also be tangentially involved in the survivor’s torture complicates things even further.

 What I’m trying to say is that there are a lot of plausible outcomes here and I think that makes this a writing question rather then a realism question. So the real focus is: what works best with the character?

 Blame is definitely possible in the scenario you’ve created but it doesn’t have to be straight-forward or simple.

 For instance the character might blame her while knowing logically that there’s nothing else she could do without putting both of them in more danger. And that could make him feel conflicted about blaming her, possibly feeding into self-blame as well. He could openly blame her, or he could hide his feelings for a variety of reasons.

 He might feel angry, that she’s ‘safer’ or that she can’t protect him. Or just because she ‘stands by’ and watches him at his worst. He might even come to hate her.

 But it’s also possible that he wouldn’t associate her with the torturers or guards and would view her more as a fellow (though perhaps favoured) prisoner. He might pity her. He might feel sympathetic towards her plight.

 He could plausibly have no strong feelings towards her at all.

 Whatever emotional response you think is best it’s important to tie it to what’s come before in the story.

 However you look at things he’s been away from his sister for a long time. It’s not clear to me how much time they spent together growing up (they could have been apart since he was 10 from the sounds of things).

 If they spent a lot of their childhood apart they may not have a close relationship to begin with. I don’t think that would make a particular response more likely but it could mean he has a less intense response to her presence generally. If they weren’t close before then he might not feel her presence is particular significant.

 If they were close then I think it’s a good idea to look back over the story. Read their interactions again and try to get a clear picture in your mind of what their relationship was before.

 Whatever happens you’re writing the process of how that relationship changes. And it’s really helpful to have a clear idea of where you’re starting from first. I personally find it helpful to have a clear idea of where I want to end up as well but some people prefer a more exploratory style where they find out where the characters end up as they write.

 It doesn’t matter which approach works better for you, what matters is that the intervening steps, the process of the relationship changing, are clear and understandable to your readers. And preferably pack a heavy emotional punch as well.

 So if blame is the result you want (if it isn’t use this as an example and apply the same process to the emotional response you want) think about what aspects of their relationship could feed into that.

 If they had a competitive or slightly antagonistic relationship then it might feel natural for him to place some blame on her. After all it’s probably an established pattern from their relationship. If he saw her as a protector and relied on her to keep him safe then this might feel like a huge betrayal.

 If they had a really loving, tender relationship then you might want to lean in to the illogical nature of the response. It might even be a good idea to have the character acknowledge (internally or verbally) that this isn’t a sensible response. And yet this does not make the feeling go away.

 With a more distant relationship did he feel like she betrayed her people or her family by ‘giving up’, regardless of how desperate the situation was? Or did he (as a kid raised in the rebellion) mostly view her as a prisoner?

 If he saw her as a prisoner and felt pity for her would that vanish as she stands by while he suffers? Or would it seem to confirm what he already thought; that she’s helpless, powerless.

 Find some part of their previous relationship that you can tie to this new set of feelings. Or acknowledge that it’s not a sensible response and have the character deal with more complex feelings as a result.

 Mostly try to resist the idea that there’s a ‘right’ response for your character to have.

 Try not to suggest in the story that there is one ‘proper’ response for a survivor to have. Because they are a varied bunch. People can live through more or less the same thing and come out with very different attitudes or perspectives as well as symptoms.

 The response you write should be the one that works best with your characters and the story you want to tell. Don’t feel you must use blame. Instead think about whether it adds to your story: does it create interesting character moments, obstacles for the characters or feed into the plot?

 You’re the person who knows what’s best for the story and what will work best with the characters. Be open to multiple options. Take your time and think through what works best.

 For the character himself it’s possible (may be likely) that he’d already have some trauma symptoms before he’s captured.

 I get the impression you’ve probably already seen the Masterpost on common trauma symptoms, but here it is for the new readers. :)

 For the physical injury pattern you’ve got multiple options.

 I think that really severe nerve damage suggests something more then shackles. Unless something went wrong.

 The easiest way to get both injuries in your character would be a suspension torture that was more common historically. Victims had their hands tied together in front of them, were hoisted anywhere between a few feet and two meters in the air and then dropped.

 This causes nerve damage in both hands and could cause breaks or fractures in the legs. Either could lead to chronic pain.

 Suspension without the drop would still cause nerve damage in about 15-20 minutes.

 Nerve damage is less common with restraints but it is still possible. Ratcheting cuffs that can tighten are more likely to cause nerve damage, especially if they’re applied too tightly over a long period.

 Other dangerous things that can happen with those sorts of restraints being too tight- Broken wrists and reduced circulation leading to painful swelling in the hands (look up ‘finger milking’ in my tags for more information).

 Over longer periods (multiple hours with the cuffs tight enough to cause swelling in the hands) blood clots might form and that uh… really dangerous. Basically if large blood clots start forming in a limb due to reduced circulation then they either block the blood vessels (which kills the limb and leads to amputation) or the clot gets swept back into the body when the restraints are removed. The clot usually then lodges in the brain or the heart causing a stroke or a heart attack respectively.

 I’d say suspension probably works better for your purposes.

 Standing stress positions can lead to chronic pain in the legs. But it often also effects the back and usually effects both legs.

 Falaka might work. It’s beating the soles of the feet with an implement. Depending on the implement it can be clean, scarring or even lethal. With a harder implement like a wooden stick it can lead to fractured or broken bones in the feet.

 But even when falaka is performed in a ‘clean’ manner it can lead to chronic pain. It causes a thickening of the tendons in the soles and also causes tiny bone fragments to detach inside the feet. It’s unclear how long these bone fragments stick around but they’re detectable by MRI for a few months with the right method.

 You could also just go with the idea of the leg injury being the result of a specific attack or accident. A broken knee perhaps, after a beating or a fall. Not all injuries in torture scenarios are ‘deliberate’, in the sense that they weren’t necessarily intentional. Because torturers are not as in control of the situation as they’d like people to believe.

 I think I’ll leave it at that for now, but if you have any further questions don’t hesitate to come by when the askbox is open. :)

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[thousandmaths]

A Lemma: Geometric Series

This is the second post in a five-part sequence (1 2 3 4 5) devoted to the Rogers-Ramanujan identities, inspired by Sun Kim’s talk at 2017 Midwest Combinatorics Conference. Her explanation was the first time that I really felt that I understood these identities, and for me it shined a light on their beauty, which I had never seen before. This sequence is an attempt to pass that on to you :)

[ As usual, I am using LaTeX in this post which means that it will look (much) better if you click through to my blog instead of reading it on the dash. ]

------

“Lemma?”

In this post, we actually won’t be looking at the Rogers-Ramanujan identities at all. Instead we’ll be doing some algebraic work that will allow us to start to get rid of those pesky fraction bars. A piece of work that is useful for, but somewhat tangential to, the main inquiry of a mathematical study is often called a lemma.

In this post we will be proving the following statement about the evaluation of the so-called “geometric series”:

Lemma. As formal power series, $\displaystyle\frac{1}{1-x^N} = 1+x^N+x^{2N}+x^{3N}+x^{4N}+\cdots$.

This sort of thing seems useful because you may remember that in our expanded form of the Rogers-Ramanujan identities, we had lots of things that looked like $1/(1-x^N)$ floating around. However, this lemma turns out to be a fairly versatile little result that shows up all over the place.

(Also, it happens to be my favorite piece of math in the entire kindergarten-to-calculus curriculum, and I’m glad I found an excuse to dedicate an entire post to it :P)

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Proving the Lemma

The proof of the Lemma is formally straightforward, but it’s a little mind-bending if you haven’t seen similar arguments before.

The main idea here is the “obvious” fact that $\frac{a}{b} = c$ means that $a=bc$. In other words, if we want to show that $\frac{1}{1-x^N}$ is the same as this infinite sum $1+x^N+x^{2N}+x^{3N}+ x^{4N}+\cdots$, then it’s good enough to show that

$$ 1 = (1-x^N)(1+x^N+x^{2N}+x^{3N} + x^{4N}). $$

The left-hand side is about as simple as you could expect anything to be. The right-hand side looks a lot more complicated, but we notice that at its core, it’s a multiplication problem. Let’s try to do the multiplication.

We’re not going to sit around here all day and multiply $1-x^N$ by infinitely many things. Instead, we’ll try multiplying it by the first couple of terms, and see if we notice a pattern.

If we multiply $1-x^N$ by $1$, the first term in the other factor, we get $1-x^N$.

If we multiply $1-x^N$ by ${\color{red}{1}}+{\color{orange}{x^N}}$, the first two terms in the other factor, then we get ${\color{red}{(1-x^N)}}+{\color{orange}{(x^N-x^{2N})}}$. We see that we’ve got a ${\color{red}{-x^N}}$ and a ${\color{orange}{+x^N}}$, so those will cancel and we’re left with $1-x^{2N}$.

If we multiply $1-x^N$ by ${\color{red}{1}}+{\color{orange}{x^N}}+{\color{green}{x^{2N}}}$, the first three terms in the other factor, then we get ${\color{red}{(1-x^N)}}+{\color{orange}{(x^N-x^{2N})}}+{\color{green}{(x^{2N}-x^{3N})}}$. Again, we have a ${\color{red}{-x^N}}$ and a ${\color{orange}{+x^N}}$, so those will cancel. And we also have a ${\color{orange}{+x^{2N}}}$ and a ${\color{green}{+x^{2N}}}$, so we’re left with $1-x^{3N}$.

You may suspect what’s happening. If we multiply $1-x^N$ by the first few terms of the other factor, then we get $1-x^{kN}$, where $k$ is one more than the number of terms we considered.

There is a somewhat more visual way to understand this, which also helps with the conclusion that we obtain in the infinite sum setting. Ordinarily, multiplication of $1-x^N$ by ${\color{red}{1}}+{\color{orange}{x^N}}+{\color{green}{x^{2N}}}+{\color{blue}{x^{3N}}} + \cdots$ gives something like

\begin{array}{rrrrrrr} {\color{red}{1}}&+&{\color{orange}{x^N}}&+&{\color{green}{x^{2N}}}&+&{\color{blue}{x^{3N}}}+\cdots  \\\  -{\color{red}{x^N}}&-&{\color{orange}{x^{2N}}}&-&{\color{green}{x^{3N}}}&-&{\color{blue}{x^{4N}}}+\cdots \end{array}

But then if we shift the bottom row to the right one space, we get

\begin{array}{rrrrrrrrrr} {\color{red}{1}}&+&{\color{orange}{x^N}}&+&{\color{green}{x^{2N}}}&+&{\color{blue}{x^{3N}}}&+&\cdots  \\\  &-&{\color{red}{x^N}}&-&{\color{orange}{x^{2N}}}&-&{\color{green}{x^{3N}}}&-&{\color{blue}{x^{4N}}}&+\cdots \end{array}

and now we can clearly see that, as long as we take infinitely many terms, everything is going to cancel, except for that first $1$. And this is what we wanted to get from our multiplication, so we’re done.

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Nitpicking (optional)

There is a tiny piece of the lemma that I haven’t said anything about: it’s that little bit of text up at the top that says “As formal power series,”.

These words are related to something that may have concerned you about the visual argument: since we shifted one of the terms over by one, what about the “last term”? It’s an important but subtle point that, depending on how exactly you are making things rigorous, can have multiple different answers.

The idea behind saying a “formal power series” is that it’s sort of a magic word for constructing a setting in which infinity never “stops”. Rather, for every term, there is a term that comes after it. If you apply this principle to the hypothetical $x^\infty$, it’s easy to see that you would still get a cancellation, with the term that comes after it $x^{\infty+1}$ (and such a term must exist, by definition!).

The benefit to working with formal power series is that we can just kind of ignore what’s going on “at the end”, but this comes with a big cost. In a formal power series, you technically can’t plug in a number for $x$. I know, it’s weird, but that’s the way it works. To see what sort of trouble you can get into by plugging in numbers, consider the Lemma and try to plug in $x=2$. Then on one side you’ve got $\frac{1}{1-x}=-1$, sure... but on the other side you have $1+2+4+8+16+32+\cdots$ and clearly* if you add up all these numbers you get infinity. You don’t get a negative number, at least.

If you go back to the first post of the sequence, you’ll notice that the Rogers-Ramanujan identities that I stated also have this little warning tag attached to it, So we’re not ever going to think of these things in any other way.

You may find this existentially disturbing: if we cannot say that the identities are true when we plug in a number for $x$, then what are we actually doing here? There are two answers to this question (neither of which, I might add, I found very satisfying the first time I heard them, so... sorry in advance). First of all, you can plug in numbers for $x$, it’s just that for the addition-only understanding of the identities, it is neither easy nor important to do so. Second, our understanding of the identities will be enough to say the following: if you expand out both sides in terms of powers of $x$ (i.e. “as formal power series”), then the coefficient attached to each power of $x$ will be the same on both sides.

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[ * Readers who know too much may be aware that there is a sense in which $1+2+4+\cdots=-1$; this is true in the $2$-adics, for instance, or with Euler summation, or by analytic continuation. But such readers will also be aware (or at least, should be informed) that you’re not really adding things together at that point. You’re just playing games with sequences of numbers. ]