Happy Thanksgiving everyone!
Happy Thanksgiving everyone!
Justice League stays true, for the most part, to what it advertised: a movie about five iconic superheroes who need little-to-no introduction getting together (right now, over me) in order to stop a villain who wasn't noteworthy enough to show up in the trailers, that may or may not have anything to do with Superman. There has been several different marketing tools employed domestically across the world that both imply that he is in the film, and things that imply that he isn't, and a lot of it is fake and deliberately misleading in order to get people speculating. So, out of respect to Warner Bros., I won't reveal whether or not Superman comes back or if they were gutsy enough to keep him dead.
With that in mind, what do you have?
Batman. Everyone knows who this is. The only difference to this Batman that I think needs introduction is that his main schtick this time around doesn't seem to be his batty obsession with his dead parents, but a sense of guilt over Superman's death. This is the first time that I've ever seen Batman depicted on film with friendship being a major part of his identity.
Wonder Woman. Everyone's favorite character at this point. The movie acknowledges some important parts from her origins movie earlier this year.
These two decide that they're going to start a team based off of cameos from previous films, consisting of the following three:
Aquaman: A bigger outsider than Batman ever was, and with a complicated backstory with Shakespearean family drama that's complicated enough that he's naturally the next one to get his own film. He reminds me a bit of the brooding Superman from Man of Steel, but it does fit the character a little better. However, he's the character who will probably endear people the least.
Cyborg: The most obscure of these characters, but at the same time can be summed up in one sentence. His father tried fusing him with alien technology to save his life, and now he's afraid of the technology that's taking over his body and possibly his mind.
The Flash: Barry Allen, everyone. Probably the most famous superhero outside of the Big Three. He says it as briefly as possible, that be got struck by lightning. Now he's fast, can go into some alternate dimension, and has the Speed Force. He doesn't have any friends, and it the most eager to join the Justice League. It should be noted that he is responsible for all of the movie's best moments.
They must fight Steppenwolf, a cool-looking villain played by a Shakespearean actor who delivers his few lines very well. He's of the Marvel variety, a forgettable villain; however, I personally really enjoyed him, if only because I really enjoyed the actor's performance. Steppenwolf's plan is to gather the Mother Boxes, an all-powerful force, and transform Earth into that red place that you saw in the trailers, which he would get away with if it wasn't for the Justice League.
That's it. That's the film in a nutshell. I think that it does a fairly decent job, and will keep people entertained. It doesn't have the gravitas and level of excitement and payoff that The Avengers did five years ago, but I think that people will be more satisfied if they go in thinking of it as a pilot episode for a DC animated show, since it has about a similar feel. There were certain moments that took me back to these kind of shows. Even Steppenwolf, as underdeveloped as he was, reminded me of villains who show up in a pilot episode to get introduced as a larger series villain who will get more development later.
I would also recommend this movie especially to people who haven't seen Man of Steel and Batman v Superman, on account of this film deciding to ignore major issues from them. It's very clear that DC wants to do a course-correct and wishes that they never made the first two films the way that they did, meaning that they're not only changing tone, but they're changing characters to fit their beloved comic-book counterparts with no in-universe reason.
Other pros and cons:
Great cinematography. Everything directed by Zack Snyder looks gorgeous. In fact, I don't say this too often, but it looks even richer in 3D.
Poor sound editing. There were times when sound played an important part in storytelling, and it really should have been edited to make the film far more immersive.
The editing! This is where most people complain. It's very obvious that there are quite a few deleted scenes, because the scenes that remain, especially in the beginning, don't directly flow into each other and interrupt the momentum that the film is trying to build up. However, each scene on its own is pretty cool. The other editing problem is in Warner Bros.'s mandate to keep the film under two hours, including the credits. So the film feels like it's about an hour and a half long (hence why I compare it to their animated pilots), and that just wasn't long enough to build up some important conflicts and play off of character chemistry. I feel that the second act in particular could have had several extra scenes to help build up to several key character choices. The inevitable extended cut of this movie will probably drastically improve on this. However, it would have been nice to see all of these extra story on the big screen.
The music pleased me. Greatly. They distanced themselves from everything having to do with Hans Zimmer and embraced a lot of their more classical music that I hear from their television shows and their video games, and it took me to a nostalgic place. You hear hints of the original Batman theme, and the original Superman theme, and Wonder Woman's theme gets a makeover so that instead of playing on an electric cello, she makes her entrance to trumpets, which I think takes her good theme and makes it great. The best new piece of music easily belongs to the Flash. It plays whenever he goes into speed mode, and I truly loved it.
The costumes were great. The Flash once again gets my praise, because his costume is almost exactly what I always imagined that it would look like.
I will defend this film against comparisons to The Avengers, since most of the comparisons being made stem from similarities in the comics. Steppenwolf is compared to Loki on the basis of them both having horned helmets, but they both had those in the comics. The Parademons have been compared to the Chitauri, but the Parademons have been in the comics longer. The Mother Boxes as a generic source of power has been compared to the Tesseract, but again, this is ancient comics stuff. Most other comparisons after that come off as stretches, for me. Like, the fact that Wonder Woman knows who Steppenwolf is, and Thor knows who Loki is. Ahem, that's lame.
The Lord of The Rings is at once both one of my favorite books and one of my favorite film trilogies. And I don't really feel the need to write another sentence justifying that.
In any case, I reacted with some consternation upon finding out the Amazon was, having attained the rights to Tolkien’s world, developing a new series set in Middle-earth. On the one hand, we get to return to that world. On the other, it's hard to top Peter Jackson’s interpretation of that world – how else could Minas Tirith look if not like that?
But then, revisiting Middle-earth means a chance to do some things differently. Like maybe making the world look a little more inclusive.
The Lord of The Rings is very white. That's not so much a judgement as it is a fact. It doesn't make it any worse as a work, it's just how it is. So if we're telling new stories, let's ask why not and mix things up and cast some people of color as these characters.
Now, my own knee jerk response is “hey, let's make all the elves Asian!” because that way you'll be forced to have an Asian actor on screen anytime an elvish character is in play (and also we’ll get Elrond, half-Asian). But equating fictional races with real life ones becomes real hairy real quick. It runs the risk of feeling like stereotyping and, in the case of my own “make all elves Asian” orientalism and exoticism. Because if they don't look like the normal, clearly they must be other, so let's make them not-human. That line of thinking falls back on to the white-as-default mindset, where if you need a normal Everyman you make him a white guy. And let's not do that.
Because if we're diversifying Middle-earth, let's let everyone be everyone. Let's have black elves and surly Asian dwarves, let's have Latino hobbits and an Indian shieldmaiden of Rohan.
Because why not.
The Lord of The Rings, and a lot of high fantasy with it, falls into the trap of looking a lot like Western Europe in the Middle Ages. Which, I suppose, is fair, given that Rings is the forerunner of modern fantasy and that in writing it Tolkien wanted to give England its own myths to rival those of Greece. So of course it's gonna portray a very (white) England-inspired place. But that’s done, and it doesn't excuse modern fantasy works (and the upcoming Amazon show would indeed count as a modern fantasy work) from being very white and European.
Cuz there's nothing in The Lord of The Rings’ mythology that precludes a more diverse cast. Sure, you'd have to ignore Tolkien’s descriptions of characters as fair and golden-haired, but that's not a loss. Heck, even adding more women makes sense; we've already got characters like Lúthien and Galadriel who've kicked butt in their time. Eowyn’s given the title shieldmaiden so she’s probably not the first. There’s no reason not to.
This is a fantasy world with magic rings and enchanted swords (and, y'know, elves and dwarves and stuff), there is literally no good reason why everyone has to be white. The only reason a black elf or Asian dwarf sounds so odd is because it's outside what we've internalized as normal for the genre. We're simply used to seeing these archetypes as white. And that's s gotta change.
And where better for that change to happen than in the world of The Lord of The Rings? This is the book that elevated fantasy from children’s books to something taken seriously. It's what inspired the world of Dungeons & Dragons, it's the basis for just about every modern work of high fantasy. This is a chance to shift the framework, to redefine how fantasy usually looks.
I love The Lord of The Rings (and The Hobbit and The Silmarillion). Why can't I, someone who's reread the books countless times, quoted the movies in the opening to his thesis, and dominated Lord of The Rings bar trivia, get to see people in those stories who look more like me?
Also, Route 1 has Munchlax now. And sometimes they have Leftovers. I caught a Munchlax with Leftovers only five hours into the game. Rad.
Hm, what else? The Photo Club is fun, I'll probably be spending some time there, and the Roto Loto has proved quite useful already. Most of the changes I'm noticing right now are fairly minor, but I imagine that'll snowball as I go forward. Also, Mantine Surf is pretty darn fun. While you were required to do it, you were not required to be good at it, which was an immense relief.
Next thing I need to do is take on Lana's trial, which I'm...a bit nervous about. Pikachu's going to need to pretty much solo it and the only electric move he knows is Thunder Shock. Hopefully it won't take too long, I need that fishing rod to catch a Wishiwashi for my team.
(Also, Pokemon Bank will apparently be getting patched to work with USUM in late November. Trading with Sun and Moon works fine, though.)
Anywho I changed my profile pic, which gives me a grand total of one pic change and one name change in the seven years I've been here. All ya hooligans with your name contests every month give me vertigo.
Since then, things have been optimistic. As of now most of the larger fires seem to be at least 50% contained, which does not mean half way out. They just have preventative lines drawn around them using whatever techniques fire fighters use (or at least, that's how I am interpreting this). If the winds pick up strongly again the fires could get over the containment lines, but so far it doesn't look like those will come.
Though many people have been allowed to return to their homes, and some shelters have closed due to lack of necessity, no one is out of the woods yet. I don't know what the plan is to put out the fires, but it might be a while before they are 100% out. That said, the amount of help California is receiving is incredible. Apparently there are over 10,000 fire fighters working, and as of yesterday (I think) there were fire fighters from 17 additional states (Oregon, Washington, Idaho, Montana, Nevada, Utah, Colorado, Arizona, New Mexico, New York, Minnesota, Florida, South Caroline, Alaska, South Dakota, Wyoming and Indiana) and Australia helping out. Australia! Together they contributed 266 fire engines, 79 fire crews, and 56 other personnel. I am very grateful for the support that they have shown and for the work that they all are doing. If I weren't on the other side of the world I would sit outside the local fire station and cook pancakes all morning for them. My girlfriend works right next to it and will be dropping off a few batches of homemade brownies.
So while the battle is far from over, things are looking up, and I'm hoping they keep looking that way.
A wave, in general, is any function that obeys the wave equation. To simplify things, though, let’s look at repeating wave patterns.
The image above depicts a sine wave. This is the shape of string and air vibration at a pure frequency; as such, sinusoidal waveforms are also known as “pure tones.” If you want to hear what a pure tone sounds like, YouTube is happy to oblige. But sine waves are not the only shapes that a vibrating string could make. For instance, I could make a repeating pattern of triangles (a triangle wave),
or rectangles (a square wave),
Now, making a string take on these shapes may seem rather difficult, but synthesizing these shapes to be played on speakers is not. In fact, old computers and video game systems had synthesizers that could produce these waveforms, among others. But let’s say you only know how to produce pure tones. How would you go about making a square wave? It seems ridiculous; pure tones are curvy sine waves, and square waves are choppy with sharp corners. And yet a square wave does produce a tone when synthesized, and that tone has a pitch that corresponds to how tightly its pattern repeats — its frequency — just like sine waves.
As it turns out, you can produce a complex waveform by adding only pure tones. This was discovered by Jean-Baptiste Joseph Fourier, an 18th century scientist. What he discovered was that sine waves form a complete basis of functions, or a set of functions that can be used to construct other well-behaved, arbitrary functions. However, these sine waves are special. The frequencies of these sine waves must be harmonics of the lowest frequency sine wave.
The image above shows a harmonic series of a string with two ends fixed (like those of a guitar or violin). Each frequency is an integer multiple of the lowest frequency (that of the top string, which I will call ν1 = 1/T, where ν is the Greek letter "nu."), which means that the wavelength of each harmonic is an integer fraction of the longest wavelength. The lowest frequency sine wave, or the fundamental, is given by the frequency of the arbitrary wave that’s being synthesized, and all other sine waves that contribute to the model will have harmonic frequencies of the fundamental. So, the tone of a trumpet playing the note A4 (440 Hz frequency) will be composed of pure tones whose lowest frequency is 440 Hz, with all other pure tones being integer multiples of 440 Hz (880, 1320, 1760, 2200, etc.). As an example, here’s a cool animation showing the pure tones that make up a square wave:
As you can see in the animation, these sine waves will not add up equally; typically, instrument tones have louder low frequency contributions than high frequency ones, so the amplitude of each sine wave will be different. How do we determine the strengths of these individual frequencies? This is what Fourier was trying to determine, albeit for a slightly different problem. I mentioned earlier that sine waves form a complete basis of functions to describe any arbitrary function (in this case, periodic waveforms). This means that, when you integrate the product of two sine waves within a harmonic series over the period corresponding to the fundamental frequency (T = 1/ν1), the integral will be zero unless the two sine waves are the same. More specifically,
Because of this trick, we can extract the amplitudes of each sine wave contributing to an arbitrary waveform. Calling the arbitrary waveform f(t) and the fundamental frequency 1/T,
This is how we extract the amplitudes of each pure tone that makes up the tone we want to synthesize. The trick was subtle, so I’ll describe what happened there line by line. The first line shows that we’re breaking up the arbitrary periodic waveform f(t) into pure tones, a sum over sine waves with frequencies m/T, with m running over the natural numbers. The second line multiplies both sides of line one by a sine wave with frequency n/T, with n being a particular natural number, and integrating over one period of the fundamental frequency, T. It’s important to be clear that we’re only summing over m and not n; m is an index that takes on multiple values, but n is one specific value! The third line is just swapping the order of taking the sum vs. taking the integral, which is allowed since integration is a linear operator. The fourth line is where the magic happens; because we’ve integrated the product of two sine waves, we get a whole bunch of integrals on the right hand side of the equation that are zero, since m and n are different for all terms in the sum except when m = n. This integration trick has effectively selected out one term in the sum, in doing so giving us the formula to calculate the amplitude of a given harmonic in the pure tone sum resulting in f(t).
This formula that I’ve shown here is how synthesizers reproduce instrument sounds without having to record the instrument first. If you know all the amplitudes bn for a given instrument, you can store that information on the synthesizer and produce pure tones that, when combined, sound like that instrument. To be completely general, though, this sequence of pure tones, also known as a Fourier series, also includes cosine waves as well. This allows the function to be displaced by any arbitrary amount, or, to put it another way, accounts for phase shifts in the waveform. In general,
or, using Euler’s identity,
The collection of these coefficients is known as the waveform’s frequency spectrum. To show this in practice, here’s a waveform I recorded of me playing an A (440 Hz) on my trumpet and its Fourier series amplitudes,
Each bar in the cn graph is a harmonic of 440 Hz, and the amplitudes are on the same scale for the waveform and its frequency spectrum. For a trumpet, all harmonics are present (even if they’re really weak). I admittedly did clean up the Fourier spectrum to get rid of noise around the main peaks to simplify the image a little bit, but know that for real waveforms the Fourier spectrum does have “leakage” outside of the harmonics (though the contribution is much smaller than the main peaks). The first peak is the fundamental, or 440 Hz, followed by an 880 Hz peak, then a 1320 Hz peak, a 1760 Hz peak, and so on. The majority of the spectrum is concentrated in these four harmonics, with the higher harmonics barely contributing. I also made images of the Fourier series of a square wave and a triangle wave for the curious. Note the difference in these spectra from each other and from the trumpet series. The square wave and triangle wave only possess odd harmonics, which is why their spectra look more sparse.
One of the best analogies I’ve seen for the Fourier series is that it is a recipe, and the "meal" that it helps you cook up is the waveform you want to produce. The ingredients are pure tones — sine waves — and the instructions are to do the integrals shown above. More importantly, the Fourier coefficients give us a means to extract the recipe from the meal, something that, in the realm of food, is rather difficult to do, but in signal processing is quite elegant. This is one of the coolest mathematical operations I’ve ever learned about, and I keep revisiting it over and over again because it’s so enticing!
Now, this is all awesome math that has wide applications to many areas of physics and engineering, but it has all been a setup for what I really wanted to showcase. Suppose I have a function that isn’t periodic. I want to produce that function, but I still can only produce pure tones. How do we achieve that goal?
Let’s say we’re trying to produce a square pulse.
One thing we could do is start with a square wave, but make the valleys larger to space out the peaks.
As we do this, the peaks become more isolated, but we still have a repeating waveform, so our Fourier series trick still works. Effectively, we’re lengthening the period T of the waveform without stretching it. Lengthening T causes the fundamental frequency ν1 to approach 0, which adds more harmonics to the Fourier series. We don’t want ν1 to be zero, though, because then nν1 will always be zero, and our Fourier series will no longer work. What we want is to take the limit as T approaches infinity and look at what happens to our Fourier series equations. To make things a bit less complicated, let’s look at what happens to the cn treatment. Let’s reassign some values,
Here, νn are the harmonic frequencies in our Fourier series, and Δν is the spacing between harmonics, which is equal for the whole series. Substituting the integral definition of cn into the sum for f(t) yields
The reason for the t' variable is to distinguish the dummy integration variable from the time variable in f(t). Now all that’s left to do is take the limit of the two expressions as T goes to infinity. In this limit, the νn smear into a continuum of frequencies rather than a discrete set of harmonics, the sum over frequencies becomes an integral, and Δν becomes an infinitesimal, dν . Putting this together, we arrive at the equations
These equations are the Fourier transform and its inverse. The first takes a waveform in the time domain and breaks it down into a continuum of frequencies, and the second returns us to the time domain from the frequency spectrum. Giving the square pulse a width equal to a, a height of unity, and plugging it into the Fourier transform, we find that
This is one of the first Fourier transform pairs that students encounter, since the integral is both doable and relatively straightforward (if you’re comfortable with complex functions). This pair is quite important in signal processing since, if you reverse the domains of each function, the square pulse represents a low pass frequency filter. Thus, you want an electrical component whose output voltage reflects the sinc function on the right. (I swapped them here for the purposes of doing the easier transform first, but the process is perfectly reversible).
Let’s look at the triangular pulse and its Fourier transform,
If you think the frequency domain looks similar to that of the square pulse, you’re on the right track! The frequency spectrum of the triangular pulse is actually the sinc function squared, but the integral is not so straightforward to do.
And now, for probably the most enlightening example, the Gaussian bell-shaped curve,
The Fourier transform of a Gaussian function is itself, albeit with a different width and height. In fact, the Gaussian function is part of a family of functions which have themselves as their Fourier transform. But that’s not the coolest thing here. What is shown above is that a broad Gaussian function has a narrow range of frequencies composing it. The inverse is also true; a narrow Gaussian peak is made up of a broad range of frequencies. This has applications to laser operation, the limit of Internet download speeds, and even instrument tuning, and is also true of the other Fourier transform pairs I’ve shown here. More importantly, though, this relationship is connected to a much deeper aspect of physics. That a localized signal has a broad frequency makeup and vice versa is at the heart of the Uncertainty Principle, which I’ve discussed previously. As I mentioned before, the Uncertainty Principle is, at its core, a consequence of wave physics, so it should be no surprise that it shows up here as well. However, this made the Uncertainty Principle visceral for me; it’s built into the Fourier transform relations! It also turns out that, in the same way that time and frequency are domains related by the Fourier transform, so too are position and momentum:
Here, ψ(x) is the spatial wavefunction, and ϕ(p) is the momentum-domain wavefunction.
Whew! That was a long one, but I hope I’ve done justice to one of the coolest — and my personal favorite — equations in mathematics.
P.S. I wanted to announce that Equation of the Day has its own website! Hop on over to eqnoftheday.com and check it out! All the entries over there are also over here on BZPower, but I figured I'd make a site where non-LEGO fans might more likely frequent. Let me know what you think of the layout/formatting/whatever!
I was just informed that two of my RA's had made it their personal challenge to figure out what the weird circles on my door mean. They have apparently spent the last three weeks trying to crack what it says letter by letter.
They managed to do it. They still don't know what alphabet that was.
I'm amazed and stunned.
I GOT A JOB IN WASHINGTON AS A PASTRY CHEF AND I ACCEPTED IT AND NOW I HAVE TO APARTMENT HUNT AND FIND A PLACE TO LIVE WITHIN A MONTH AND THIS IS REAL ACTUAL ADULT STUFF THAT MY EDUCATION AND LIFE EXPERIENCES DID NOT PREPARE ME FOR.
WHAT IS? HOW DO?
But, like, everyone at work keeps telling me finding a place is the easy part and that I'll be fine and that is reassuring.
This is gonna be fuuuuuuuuuuuuuun.
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