August 6, 2017

Tacoma Review

AI Justice Warrior

Here's a story you've heard a thousand times before: it is the future. A very powerful corporation is obviously evil, but presents itself as a force for good. Things aren't what they seem, and there is a conspiracy to hide this fact. Also, is AI the same as human consciousness? Is it dangerous? Should it have the same rights as us?

Tacoma is full of such overused sci-fi cliches, and in lesser hands, it would be just another throwaway story about the evils of technology. But thanks to clever storytelling, informed by social consciousness and masterful character development, Tacoma manages to rise above the tropes and actually say new and interesting things about its well-treaded subject matter. The result is an exceptionally effective and touching plot-driven game - one whose more cringe-inducing moments are easy to forgive. 
Developer: Fullbright
Publisher: Fullbright
Release Date: August 2, 2017
MSRP: 19.99$
Rig: Intel i5-4440 @ 3.10GHz
16GB RAM
Zotac GeForce GTX 980 Ti AMP! Extreme

Tacoma is the second release by developer Fullbright, a studio known for the critically acclaimed 2013 game Gone Home, a game that. despite my best intentions, I simply could not stand. I thought its writing was amateur and that attention to detail was distributed in the worst way possible. It was a game where you could pick up every pen and kitchen utensil, but you couldn't have a single meaningful interaction with the world the developers built. 

You can still pick up a lot of random stuff in Tacoma, but this time around, we are allowed to do more than idly admire the well-crafted environments. To those not familiar with the studio's work, Tacoma is a walking simulator, a game devoid of the battle and puzzle mechanics of more traditional games which instead focuses on story and characters. A lot of games in the genre, Gone Home included, are content to let you walk around and pick up data logs that slowly unravel a story. But Tacoma takes a far more interesting approach to its storytelling. As contractor Amy Ferrier, you have been hired to investigate the events that transpired aboard the titular space station, which lost its oxygen supply, with the status of the crew members unknown. With AI Minny, which supplies us with what is probably the best and most criminally underutilized vocal performance of this year, and a device that looks kinda like a fancy 3DS, you are sent to Tacoma to retrieve whatever data remains in the station's systems.

This is done by examining captured security footage of the station's crew, but rather than regular video logs, you will be examining Augmented Reality records - letting you view events as they happened in their actual locations on the ship.
This mechanic - reminiscent of the detective cases from Arkham Origins, only good - isn't a simple gimmick, but integral to the storytelling. By pausing, rewinding and following different characters, the player can experience events from different perspectives, and find out more about the various crew members. Sometimes this allows you to fulfill a very practical need, like following a specific character to learn how to progress to a different location; sometimes it is used to establish characters and relationships, for example, by allowing you to view two characters in more intimate circumstances. Either way, it is always highly rewarding to experience all aspects of a particular scene. 

While watching AR footage, there will be times when crew members view their AR desktop, a sort of computer screen projected in front of them. This allows players to look through messages and other bits of information appearing on-screen, revealing more about the interactions between crew members, as well as the outside world - including families still waiting on Earth for their loves ones' return. Being reminded that characters have a life outside the confines of the station not only makes them feel all the more real, but also make the dangerous situation they find themselves in feel even more real.

The AR aspects are a fresh way of telling a game story, one that I don't doubt other games in the genre will iterate on in the coming years. But Tacoma, as befits a game by Fullbright, is much more than its mechanics. Gone Home was a topic of conversation due in no small part to its positive and realistic portrayal of LGBT characters, and Tacoma builds on this tradition while also presenting a cast diverse in terms of gender, race and religion. While not much is made of these differences, they do play a part, from things subtle as books found in one crew member's quarters, to a request by another to commemorate the genocide his family survived. These cultural aspects are strong enough to be noticed without being in any way stereotypical or cartoonish, as they have often been portrayed in lesser works.
But above all, Tacoma excels in touching on an aspect few works, informed by political liberalism, ever dare approach: class. Tacoma's main characters aren't high-ranking, "enlightened" executives, but working class people, ultimately victimized by an economic system that values profits over their lives. Its cast includes the type of working people you are likely to meet in either a factory or office environment: the rebellious professional, the union militant, the company man (and it is usually a man) seeking to ingratiate himself to upper management, complete with the sort of smarmy, cringe-inducing humor that comes with the territory, the admin worker trying to do her best despite the situation - they're all there, and they're all portrayed realistically, but also compassionately. Everyone will have different preferences when it comes to the cast members, but I doubt that by the time Tacoma's end credits role, anyone will have any genuine hate for any of them.

Sadly, this is also where I feel Tacoma's story falls a bit flat. Tacoma has been praised for presenting a world where oppression on the basis of identity no longer exists, but where working people are still victimized by capitalism. I do not believe one is possible without the other, and some would say that that is my view, and I shouldn't let it take away from my enjoyment of the game. Here's my problem: I come from a country where there is a staggering amount of workplace accidents in construction sites - over 50 deaths per year, which is more than 5 times than the amount of workers who die in workplace accidents in any other field of work. For comparison's sake, proportionally, that would be as if almost 2000 construction workers a year died in the US, about two times the actual figure (which is already alarmingly high). The reason for this is simple: an acute lack of safety inspectors enforcing regulations. The reason for the lack of inspectors? While one cannot say for sure, it is impossible to ignore the fact that Israeli construction workers are overwhelmingly Palestinians and Eastern European and Central Asian immigrants.

Tacoma has all the correct parts: a racially diverse workplace, a focus on working class people, and a story condemning capitalism for being callous with their lives. It just neglected to put these pieces together. And although I am impressed that Tacoma's story is so good and deep, far beyond what one can usually expect of video games, that I am even able to make this criticism, it is still a significant and disappointing weakness. If anything, its focus on the "oppression" of AI and whether or not they should have the same rights as humans smacks of the sort of timidness of games that discuss racism purely through the prism of the  oppression of elves or dwarves.
But while this weakness should be noted, it should not prevent anyone from giving Tacoma a chance. For aside from this point and some admittedly weak writing, as well as some wonky visuals - the limits of Unity, its defenders' protests notwithstanding, are quite apparent - Tacoma is not only a vast improvement over Gone Home, but a brave and refreshing story-focused game from a consistently brave studio. We deserve more stories of this caliber in games of this level of quality.

Final Score: 8/10

July 1, 2017

NieR: Automata and the Player-Character Dissonance



I was never really excited about NieR: Automata, not when it was first revealed, not when it was released, not even at any point playing it. But people I respect raved so much about the story that I had to check it out, and while I agree that it has interesting ideas, I... don't think they resulted in a good video game story. 

This video explains my disappointment with NieR: Automata, mostly through the prism of a single, early-game episode that I feel encapsulates Automata's approach to storytelling. However, be aware that there are clips from and allusions to later parts of the game as well, so be careful if you're worried about spoilers. 

There's also major spoilers for The Last of Us, and some really minor ones for Deus Ex (2000). 

Special thanks to reddit users Machinax and Aeratus for helping track down that one data log, and generally giving me a very pleasant first impression of the Deus Ex subreddit and reddit in general.

Also, I really apologize for the inconsistent voice-over quality. Never change your mic setup and then have no idea how to change it back in the middle of making a video, kids. 

Research Notes


Before I rushed to claim the term "Player-Character Dissonance" as my own, I tried sniffing around a bit. I did find a Pixel Scribe blog post discussing an unhyphenated "Player Character Dissonance", which is interesting, but seems unrelated; and a mention of an article discussing the hyphenated term in a blog post by The Astronauts. However, the link to the mentioned article is broken, and I have not managed to find any alternative source or archived version. A shame; the description makes it seem incredibly interesting.

April 19, 2017

I'd love to live in a world where Hollow Knight is "too familiar"

I quite like Rock Paper Shotgun and John Walker, but reading his Hollow Knight Impressions left me kinda puzzled. There's a lot I disagree with in the article, but the part that stood out the most to me was Walker's claim that the game is "too familiar". I feel like I either need to play the games Walker has been playing or smoke whatever he's been smoking, because, while it has its issues, I've played very few games of this level of quality.

Walker claims that Hollow Knight follows the Metroidvania formula too closely, and that it makes variations in the wrong places. Keeping in mind that innovation is often nothing but snake oil, these changes are exactly what make Hollow Knight stand out from other Metroidvanias. The need to consider abilities part of a loadout instead of a fixed addition to your arsenal, the tight, methodical combat, the way the game encourages exploration and finding your own path through the world - those are the things that make Hollow Knight very much a game of its own.

In fact, I feel like if Hollow Knight wasn't in 2D, far more people would recognize that it has more to do with - and please hear me out, because I know how terrible what I'm going to say is - Bloodborne. Not in terms of mechanics, but definitely in terms of lore and atmosphere. In fact, almost all the top comments on the article make the comparison to - sigh - Dark Souls. So it's not even me saying it, so I'm not a hack games writer or anything.

Not that the fact that something gets compared to Dark Souls proves anything. But in this case, if - and only if - we put mechanics aside, the comparison is apt.
So Walker decides that Hollow Knight belongs exclusively in a specific genre, finds its adherence to this restricted genre too strict, and then finds its variations lacking. To substantiate this, he compares the game to two other games, of which the two that I've played - Owlboy and Axiom Verge - are nothing like it in terms of mechanics or world-building (and are, in addition, quite dull). The whole critical approach just seems bizzare and misguided.

Having just finished Hollow Knight with 50 hours into it, I'll definitely say that it has its problems. Fast travel is minimal and awkward, so there's a lot of backtracking, and although this is more of a problem with me than the game, there are weird things to its action's rhythm that I never quite got used to. For example, sometimes using a special ability will allow you to move left and right, but not jump, which is a very weird and awkward limitation that nine out of ten times will make you crash straight into an enemy or a pit full of spikes.

Still, I'd say this is one of the finer games I've played in a while, perhaps the best Metroidvania I've played since the game that got that label started, Symphony of the Night.
The first thing you realize about Hollow Knight when you start playing is that oh my god, it actually looks like that. People who follow games are used to trailers looking much better than the end product, to the point where a lot of people will look at a game like Cuphead cynically, not believing the devs can actually deliver on the unique art style shown in promotional material. But Hollow Knight delivers, and my god is it amazing. It's like playing an animated movie of the highest quality, and as fashionable as it might be to say that graphics don't matter, I think art style and aesthetics can be as important a part of a game as anything else. Hollow Knight would still be a great game if it were ugly, but the way it looks adds so much.

The second thing you realize, after the first few bosses, is that this game is crushingly hard, mostly because it relies on a completely different skill set than the more obvious examples of challenging games. There are no combos, stamina or counters, and enemies rarely get stun-locked. Instead, surviving boss fights depends more on choosing the right loadout and carefully platforming around obstacles and attacks. It really takes the typical SotN combat to new places, and it's pretty glorious.

Trial of the Fool is still nonsense, though.

I'm not reviewing Hollow Knight - it was a total impulse buy, mostly due to it being an indie game and looking absolutely beautiful. But I felt like I had to write about it, because no one is talking about this game and that, to me, is downright criminal. Having done that, I'm now looking forward to reading Holly Jane Amareta's official Steam Shovelers review - you know, the hip new site everyone's talking about and where all the writers are really sexy.

April 15, 2017

Blog Underactivity / NieR: Automata, Empathy and the Player-Character Dissonance / Neil Druckmann Missing the Point of Uncharted Criticism

A Personal Note

This blog isn't in much use anymore. I post my songs and videos here, but not much else. The reason is that for the past few months, I've been writing for the very cool new site Steam Shovelers, and that's a .cool domain so you can't argue that it is cool. With my writing needs being mostly met by my reviews there, and my preference to talk about other topics in video form, there may not be a lot to see here. However, before making my next video, I feel like I need to organize my thoughts a bit, and doing a blog post seems as good a way to do this as any other. So there you go.



This section will begin spoiler-free, but after a certain clearly-marked point, spoilers begin. Tread carefully.

I'm a bit late to the party with NieR: Automata, and that might be my biggest problem with it. When I was done with the game - yes, the proper way - all I could think was: is that it? Is this jumbled mess of science-fiction cliches and half-baked gameplay ideas really what people have been raving about? Is this going to be another one of those times where I feel like an absolute maniac for being left very cold by a game a lot of people say is one of the most important video games of all times?

Well, yes and no. For one thing, experience shows that all this hyperbole needs to be taken with a grain of salt. Everyone was raving about BioShock Infinite back in the day, but once hype cooled off, people noticed that the plot makes no sense and that combat was kind of a mess. (*psst* I still kinda like it!

But more importantly, there are things to like about Automata that make me understand the love for it a bit more. Movement is clunky and combat gets repetitive, but I've seen worse, and while I can take or leave the main characters, the machines themselves are undeniably cute and likable. Yoko Taro built a beautiful, fascinating world. 

It's just that he chose to ask the most banal questions about it, and give the most predictable answers. 

SPOILERS START HERE
Does this unit have a soul? Can you offer me proof of your existence? War, what is it good for? 

We've heard it all before, and NieR: Automata does nothing new to justify hearing it again. Even if you ignore the more explicit plot and focus on the subtext, just the last couple of years have seen a slew of games that made brilliant use of meta-plot and fourth-wall breaks, like Undertale, Pony Island and IMSCAREDAutomata doesn't so much build on their insights as it makes poor copies of them, which I guess is at least thematically consistent with the poor copies its machines have made of human societies and customs.

That's what brings us to the one theme that Automata gets absolutely right: empathy. Because, after all, as the game asks explicitly at one point: why would these machines try to copy humanity so much even though in-game history shows it to have been a failure? 

Part of the answer lies in the best part of the game's plot, the main story quest The Machine Surge. Upon reaching the surface and contacting the Android resistance, protagonists 2B and 9S are ordered to commit what is in practice an act of genocide by destroying all the machines in the desert. The machines fight back, which 9S uses as a rationale to keep attacking them: 

"If they wanted help, why would they be attacking us?"
(Heard that one before!)

The machines then attempt to evoke empathy by voicing their feelings of fear and pain. When that fails, they flee, scrambling for a possible way to appease their attackers. They try anything from appealing to their conscience:

"You. Not. People."

To idle chat:

"Nice. Weather. Today."

When all this fails, the machines realize that they cannot defeat the Androids, nor convince to halt their attack, nor escape: 

"This cannot continue!"

And so, they fuse together to give birth to Adam - a machine's idea of what a human looks like, based on the closest living example they've ever witnessed, namely, the Androids.



It may seem like Adam was created as a weapon. He certainly has strong offensive capabilities. But . rather than attack, the first thing he does is ask you why you are attacking him

"An... droids... why... fight?"

His next step is to learn how to handle your aggression:

"Sword... dodge... projectile... deflect..."
(My god, the writing in this game)

And only then does he attack. 

And then Eve climbs out of his chest, because Bible reference, get it?

Adam eventually becomes hate incarnate, believing that the essence of humanity is conflict. But this isn't a perspective that's hard-wired into him. It's a result of the fact that the sum total of his experience with anything approaching a human is getting cut, shot and stabbed by 2B and 9S, and his resulting beliefs are later affirmed by a biased reading of human history.

The machines didn't make Adam to be a weapon - they made him because they thought that if the Androids saw someone more like them, they would be less inclined to attack. It's the next logical step after the orchestrated charade of family life that we witness when the protagonists first step down into the sand pit where Adam is eventually created: the machines already know that YoRHa is coming and will not yield, and in their despair, hope that if they're easier for the Androids to identify with, they might be spared.

But that never happens. Years of indoctrination have made 2B and 9S completely incapable of empathizing with the machines, an intentional ironic contradiction to YoRHa's glorification of humanity. The whole scenario is a brilliant commentary on the role empathy plays in our lives and how it is upended by prejudice and incitement.

It's also a perfect illustration of just how badly NieR: Automata's story fails as a video game plot.


You see, as 2B and 9S were massacring the desert machines, I could tell the game wanted me to feel the Androids' inner conflict between their sense of duty and the undeniable fact that these machines were saying and doing things that are uncharacteristic for unthinking murder bots. But all I could think was: why am I playing this game? I don't want to murder these machines that are begging for their lives. I wouldn't do it in real life, no matter how badly I was incited against someone, and I know this because I was faced with similar choices in the past. NieR: Automata told me all about why YoRHa wanted these machines dead, but at no point did it explain why I should want to aid them. 

There's a very important difference between character motivation in a movie or TV series and that of a character in a game: in both cases, there may be instances where someone has to play a character with whose motivation they don't identify. Actors who play villains generally don't want to nuke New York City or decapitate Gwyneth Paltrow - we hope - but they understand that their portrayal is vital to make the movie work. J. K. Simmons probably wouldn't be too thrilled about his role as Schillinger if the point of Oz was that being a Neo-Nazi is great. It's for the benefit of the story, and that's all the motivation an actor needs.

Also, I hear the money's pretty good.

In video games, players play a role, but they do so for their own benefit. Unless you're a YouTuber - in which case you should be ashamed of yourself anyway - chances are you're not playing a game thinking "wow, this would be really great for someone else to watch". You're doing it for your own benefit.

That doesn't mean, as some people claim, that games have to always be fun or entertaining. I'd be hard pressed to say I found 1979 Revolution: Black Friday or Detention fun, as much as I loved them. Nor is it necessary for a game to have a protagonist you could identify with: Yakuza 0, my favorite game this year so far, has you play as a couple of thugs who, personable as they might be at times, would do anything for money and power. But, to paraphrase Jim Sterling, you can't sell something on the lack of content. If a game isn't entertaining, and if you don't identify with its protagonist, there has to be some other reason for you to want to play, to enable the main characters' behavior.


Let's take a look at another critically-acclaimed game, to the point of utter hyperbole: The Last of Us. To keep spoilers at a minimum, the last part of the game faces you with an unambiguous fact: in order to cure humanity from the Cordyceps infection, a certain character, let's call them Character A, must be sacrificed. Character A agrees to sacrifice their life, and while this is a painful sacrifice, everyone understands it must be made - everyone except the character you play at that point, let's call them Character B. Unwilling to accept this sacrifice, Character B goes on a rampage, slaughtering many people who only have humanity's best interest in mind to save Character A.

It's a great character moment, but it's handled extremely poorly as game plot. No matter how closely you identify with either character, and no matter how many of us would do the same in that situation, there's no way you could justify these actions - and if you can, you should probably check yourself. What possible motivation could a player have to go through with this maniacal, selfish, murderous plan except to get to see the ending credits?

NieR: Automata has the benefit of being a much better game than The Last of Us, inferior voice-acting notwithstanding. But when it comes right down to it, it fails in the same way.

I expect to have a video covering these points, and maybe a few others, in the next few weeks. Chapter Select notwithstanding, getting all the footage I need this time might take a while. But I think it's a really important and under-explored aspect of the way we tell stories in games.

For now, here are a couple of links to articles that I've dug up during some preparatory research:

Pixel Scribe on Player-Character Dissonance in Dragon Age: Inquisition
The Astronauts on Empathy and Game Design


I really liked Uncharted 4, probably for the same reasons that longtime fans of the series apparently did not. It focuses less on wanton murder and more on making likable characters, establishing the relationships between them, and basing climbing sections less on trial-and-error nonsense and more on getting a feel for how the game's environment functions.

That's why I'm kinda disappointed at Neil Druckmann's response to a question in a Rolling Stone interview about the violence, or rather, the "ludonarrative dissonance" in the Uncharted games:

"...we don't buy into it. I've been trying to dissect it. Why is it that Uncharted triggers this argument, when Indiana Jones doesn't? Is it the number? It can't be just the number, because Indiana Jones kills more people than a normal person does. A normal person kills zero people. And Indiana Jones kills a dozen, at least, over the course of several movies. What about Star Wars? Han Solo and Luke Skywalker, are they some sort of serial killers? They laugh off having killed some stormtroopers. And in The Force Awakens, we see that a stormtrooper can actually repent for the person he is and come around, and there are actually real people under those helmets."

The only examples Druckmann can think of to justify Nathan Drake killing dozens of people to get some treasure are movies where protagonists kill either actual Nazis are very transparent metaphors for them. Nathan Drake isn't killing fascist soldiers to save humanity from fascism; he's killing mercenaries and guards to dig up treasure, a task at which he constantly seems to fail.

Again, this is a case where the character has a much clearer motivation than the people playing the game. It's very indicative of this problem in game writing that the really difficult questions about violence, as well as block-pushing puzzles, came from a publication that traditionally doesn't focus too much attention on video games.

March 14, 2017

How to Not Prove that Pi = 4

Today is March 14, which, due to 3.14 being the most common approximation for the most well-known mathematical constant, is celebrated as Pi Day. Some mathematicians frown upon that, since they know that Pi isn't actually equal to 3.14 and think that impresses someone, but any excuse to write about math is fine by me.

So before we get into anything else, we need to ask - what is Pi? A lot of people know that it's the ratio between the circumference and the diameter of a circle, but who's to say that ratio is the same for all circles? You can measure some circles and see that the ratio is pretty much the same, but who says that it's exactly the same? Who says we didn't just get lucky and pick a few circles that happen to have that ratio?

In other words, we want to prove that Pi is well-defined - that the definition we gave of Pi as said ratio actually means something.

Well, there's good news and bad news on that front. The good news is that there's a basic, intuitive proof that doesn't require a whole lot of mathematical background to understand. The idea is as follows:

Image shamelessly pilfered from ProofWiki

  1. Position two circle so that they're centered at the same point, and in particular, the one with the smaller radius lies inside the one with the larger radius. 
  2. Choose some integer, n, which is equal to at least 3, and divide both circles to n equal parts, like slices of pizza, i.e. by drawing straight lines from the center to the perimeter of the outer circle. 
  3. Now, for each circle, connect each points where the straight lines from step 2 intersect it with its neighbor. 
  4. We now have two families of triangles, one for each circle. The sides of each triangle are the length of the corresponding circle's radius, and the angle between the sides is the same for all triangles - namely, it is equal to 360 degrees divided by n, the number of triangles we divided our circles into. 
  5. In particular, each small triangle is contained in a large triangle such that the ratio of the sides is the ratio of the smaller radius to the larger radius, and the angle between these sides is equal. Laws of triangle similarity dictate that the ratio of the "bases" - the third edge formed by connecting two points on the perimeter of the circle - is also the ratio of the smaller radius to the larger radius.
  6. Now the magic comes in: as you can clearly see from the image up there, the larger n is, the more the sum of the bases of the triangles approach the circumference of the circle they're contained in. But we've seen that the bases of the triangles have the same length ratio as the radii, and since this is true for every n, in particular, the ratio of circumferences thus approximated must also be the ratio of the radii.
  7. Step 6 shows us that, if we mark the circumference of the smaller circle by $P_1$ and its radius by $R_1$, and do the same with index 2 for the larger circle, we have $$\frac{P_1}{P_2}=\frac{R_1}{R_2}$$ which is the same as saying $$\frac{P_1}{2R_1}=\frac{P_2}{2R_2}$$ thus showing that, indeed, the ratio between circumference and diameter is constant. 
Those were the good news. The bad news is that, alas, our approach can be used to render all of mathematics meaningless, for as the follow image shows, it can be used to prove that Pi is not roughly 3.14 but is, in fact, 4:


Which is a shame. I liked math. It's too bad that it has to Go Away Forever.

Well, as you can guess, the problem isn't so much with math as with our reasoning. Like I said, our approach can be used to show that Pi equals 4, but that just shows that our approach is wrong. Specifically, the issue lies in step 6, where we said that the lines formed by the bases of our triangles approach the perimeters of the circles and that therefore their lengths must also be the same. This reasoning is dead wrong, even if our conclusion was correct. Actually, math is full of examples of correct results one can reach with bad reasoning - my favorite is $$\require{cancel} \frac{64}{16}=\frac{\bcancel{6}4}{1\bcancel{6}}=4$$ Don't get me wrong - the approach of calculating a curve's length by approximating its length with a sum of lengths of straight lines that becomes ever-closer can be used correctly. In fact, it's pretty much what we're going to do. But as we can see, sometimes it works, and sometimes it doesn't, and to understand when it can actually work, we need to be a bit more rigorous. In fact, this is probably the simplest example I know of showing just how important mathematical rigor is.

So how do we show that Pi is well-defined? For this, just like with all good things in life, we're going to use Calculus - specifically, line integrals. The theory of integration is more than I can cover in a single blog post, but the idea is the same - calculate areas, lengths of curves or anything else by making these calculations for more basic shapes, and show that these basic shapes can be used to approximate the complex ones. The key is in the last step, though - understanding which approximations work and which don't, and most importantly, why. If I've made that point in this post, then my work here is done.

Just to be cool, though, let's show our calculus proof in full. We will calculate the length of half the perimeter of a circle with radius $R$ using the following parameterization: $$x(t) = t, y(t) = \sqrt{R^2-t^2}, -R \le t \le R$$ Why not use the more standard parametrization using cosine and sine? I'll leave that for you to answer, but think about how you would go about justifying each step of the following calculation using that parametrization.

Onwards, then. Let's use $P$ to denote the circumference of the circle. The integral we must now calculate is $$\int _{-R} ^{R} \sqrt{x'(t)^2+y'(t)^2}dt$$ which, after some calculations, is shown to be $$R \cdot \int _{-R} ^{R} \frac{1}{\sqrt{R^2-t^2}}dt$$ Now here comes the real magic. In this integral, substitute $v = \frac{t}{R}$. This gives us the integral $$R \cdot \int _{-1} ^{1} \frac{1}{\sqrt{1-v^2}}dv$$ But with $P$ as above, we have $$\frac{P}{2}=R \cdot \int _{-1} ^{1} \frac{1}{\sqrt{1-v^2}}dv$$ or equivalently $$\frac{P}{2R}= \int _{-1} ^{1} \frac{1}{\sqrt{1-v^2}}dv$$ Now check out this beauty! On the left hand side we have our coveted ratio, but on the right we have a definite integral - a number - which more importantly than anything else, does not depend on $R$, It's also kinda important that the function we're integrating is continuous and bounded over the closed interval of integration, which means that our integral exists.

Some of you may be tempted to say the primitive function here is $arcsin(v)$ and conclude that the left-hand side is indeed equal to $\pi$. But that would be naughty, as we have only now shown that Pi is well-defined, and that makes any reasoning in that direction fishy. But the important part is that the right-hand side is a constant, and from here, all the wonders of geometry, trigonometry and much of calculus are ours for the taking.

January 20, 2017

New Song - Terra's Theme (Final Fantasy VI cover)

Final Fantasy VI is widely claimed to be the best game in the series, even though everyone knows it's third after VII and IV. But all the same, it's a fantastic game, that to my great shame, I never actually beat. I got to that part - you know the one I mean if you played the game - and wanted to do everything you're supposed to do at that point, but real life intervened and it just fell by the wayside. I'm working on a full playthrough as we speak, I promise!

VI isn't just a fantastic game but, like every game scored by Nobuo Uematsu, has an amazing soundtrack. Terra's Theme in particular has always stood out to me as one of the most beautiful songs I've ever heard, and I've been playing a half-baked acoustic version of it I've improvised one day on and off for years now. Since I've been having some technical issues preventing me from working on full-blown electric releases, this felt like a great way to keep working on music while also paying tribute to a game and a soundtrack beloved by myself and many others. I've worked on it for quite a bit, and I ended up very happy with the arrangement I gave it. Hope you feel the same way!

Of course, as always, I would appreciate it if you spread the word.