Over the years, Hideo Kojima has, because of his Metal Gear Solid games, become one of my favorite video game designers. He's also certifiably bonkers, mixing in discussions of American militarism-as-neo-colonialism in a game where you fight giant mechs alongside a mostly naked sniper who can't speak because of a parasite that uses language to spread (and thus serves as a vehicle for Kojima to discuss how English becoming the global lingua franca is in turn another form of colonialism).
Point is, I'm always stoked to see what he's making.
A new trailer for Death Stranding, his first post-MGS game, dropped last night. Like the handful of other trailers for the game that have come out, it's weird and near indecipherable, with little information on what it's like as a game. And at eight minutes long, it's a pretty long trailer.
To the point where it's less a trailer and more of a short film unto itself. It's very self-contained, missing a lot of the “what comes next”-ness of trailers. While it does evoke a desire to figure out what's going on, but that's hardly the point.
There is little narrative in the traditional sense. Sure, we have a protagonist in Sam and a beginning, middle, and end; but it's not about him doing something. Rather, the trailer presents a tableau of a scene, a moment for you to experience and are the better for having done so. The trailer presents the sublime, something quite beyond our comprehension but beautiful in its terror. It's less about the catharsis and more about the process of watching Sam and his compatriots attempt to fend off these unseen creatures in a mysterious, physics bending world.
So in that sense it's a lot like the movie Lady Bird.
Lady Bird is about a girl in her senior year of high school, her relationship with her mother, her relationship with herself, and that messy transition from seventeen to eighteen. It's a tender story, told with a full heart and helpings of honesty. It's reliant less on vying for that big, cinematic climax than it is on capturing a very particular moment in time for a very specific person.
And like the trailer for Death Stranding, it captures the ephemeral. Things happen, and then something else does. Lady Bird isn't trying to say something bigger about the world, it's just trying to tell its story (as Death Stranding’s trailer weaves its vision of terror). There's no One Big Moment that defines protagonist Lady Bird’s life. Rather we see snapshots of a very specific person. Because of its honesty and specificity (Lady Bird’s idiosyncrasies are at once wholly unique and beautifully universal), we, as an audience, are allowed to experience a part of a life. One that, having seen, we are more for having done so.
It's a fairly common anti-structure in indie-darling movies; you can see it done well in Drinking Buddies and Lost in Translation. Boyhood doesn't know what it's trying to capture besides “uh, time passes, I guess” and so fails to capture anything. Meanwhile Monsters sets its journey against an alien presence to heighten its exploration of loneliness and presentation of the sublime. Ken Liu’s short story “The Paper Menagerie” captures a difficult relationship. And it's what Death Stranding’s trailer does so well.
I will campaign for narrative until the sky falls. But stories can be about moments too. The key is to make the audience feel something. As a reader/viewer/player I engage in fiction not because I want to sit idly by as something happens, but because I want to be taken on a journey. I want to feel something, sorrow or joy, something funny or something epic. Lady Bird didn't need a Big Epic Conclusion to make me feel like a teenage girl. And Death Stranding doesn't need flashy gameplay to present the sublime in a fracking video game trailer.
- Branded as "Fire" and "Lava"
- Dominant color: red
- Jungular—for lack of an actual word
- Branded as "(The) Jungle," "Swamp" and "Air"
- Dominant color: green
- Branded as "Water" and "Sub"
- Dominant color: blue
- Branded as "Earth," "Rock" and "Onyx"
- Dominant color: black
- Branded as "Ice" and "Frost"
- Dominant color: white
- Branded as "Stone," "Rock," "Sand," and "Dust"
- Dominant color: brown/yellow/orange/tan (strikingly inconsistent)
- Attempted to a degree as stand-alone in the Slizer/Throwbots series (with Energy/Electro, Spark and Flare)
- Seen as an accompanying element in BIONICLE (with the Inika)
- Dominant color: purple/orange
- Attempted to a degree as stand-alone in the Slizer/Throwbots (with City/Turbo, Judge/Jet and Blaster) and RoboRiders series (with Power)
- Seen as an accompanying element in BIONICLE (with the Toa Metru)
- Dominant color: yellow/teal
So this is what I've been building for the last couple of days. (Helped that Andrew sent me back some of my older mosaic plates, because I really needed to steal some pieces from them.) This is a scene from the Skybound season where Jay and Nya get back together at the end of it, which is one of my favorite scenes. However, the background in the screenshot was meh, so I swapped it out for a more unique skyscape shot of Ninjago City, including Borg Tower.
As for interesting techniques used here, I used some SNOT designs to get the eyes and mouths right on their faces. Plus, while the characters are made using bricks, the background is made using plates, which add some depth to the image. (The white wind circle is actually two plates tall.) The most challenging part of this was the medium azure background; I was able to fill in a lot with larger plates, but getting all the corners filled in took some effort and whipped out my supply of 1x2 and 1x4 plates in the color.
Anyway, I'm pleased with how it turned out, and it'll show up at Bricks Cascade in 2018. Probably won't sent it on he BioniLug circuit, since it's not quite Bionicle related, but I haven't seen too many Ninjago mosaics so maybe this'll stand out.
-Rollor’s Reach: Citadel-
He stepped into the dungeon, one hand on the hilt of his sword, ready to draw it at a moment’s notice. More than one Knight had been led to a trap down here; the Faith might be gone, but not everyone on Okoto was a fan of theirs. The Battle for the Dawn had been largely forgotten, in the aftermath of the War of Five Kings.
In the flickering light of the torches, he could make out two familiar faces.
“Ehks,” he said, nodding to the Protector of Ice on the left, before turning to the Protector of Jungle on the right. “Piruk.”
“Kingslayer,” Piruk said, his voice raspy from disuse.
Ehks merely nodded.
“Why are we meeting down here?” Jed asked. “I thought you were decoding the histories.”
The two Protectors shared an uncomfortable glance. Ehks shifted in place, before seeming to realize that Piruk was content to let her explain. “We… were. We’re not sure we should continue.”
“Don’t be ridiculous.”
“I’m not, I’m being serious,” she said. “What we’ve been reading about… it’s fragments of fragments, Jed. I think most of it was lost before the Mask Makers came to Okoto, or maybe during their war. Early enough that I’m beginning to suspect even Voltex’s notes on them are barely enough to give us an outline.”
“Continue decoding them until the task is completed,” Jed said, turning away. “Until then, there’s nothing for us to do.”
“You’re wrong, Kingslayer.”
Jed sighed at the sound of Piruk’s voice, glancing over his shoulder. “Why?”
“What we have decoded,” Ehks said, pausing to draw in a shuddering breath that had Jed turning back to face them properly. “It… there’s no time. There’s too many signs, and that’s just what we’ve translated so far. We need the rest of the information, and we need it now.”
“I need more detail than that. Don’t be cryptic.”
“As best we can tell, there is a war on the horizon,” Piruk said. “The texts refer to it as the ‘Great War’. They say it is the only war that will ever matter.”
“There’s been references to endless death,” Ehks continued. “Prophecies upon prophecies. But it’s all in bits and pieces, and all we know for certain is that it will be here soon.”
Jed opened his mouth to reply, but Piruk cut across him: “the texts do refer to one location very clearly. A ‘Temple of Histories’, located in the south of the Barren.”
“There aren’t any temples there,” Jed said slowly. “We’d know if there were… and they weren’t there before the War of Five Kings or Voxumo’s Rebellion, either.”
“We believe it’s been gone for a long time,” Ehks said, frowning. “Lost before the Long Night, possibly even before Ekimu.”
Scowling, Jed rubbed at his temples. He sighed, and shook his head. “I’m too old for this. Gods. I still remember the War for the Throne. Simpler times... just a small war, with none of this world-ending, weird mystical stuff. Except for….”
“…the Temple of Time,” Piruk said, completing the unspoken thought as Jed trailed off. “That is my suggestion. We need to access this Temple of Histories. I suggest we find a way to recreate Khan Nato’s journey into the past, using the completed Mask of Time.”
“The Mask of Time is missing. It has been for years.”
“Then we find it,” Piruk growled, “or we die.”
I’m thinking of taking another crack at this series with Technic Survival,
The team aspect is that each team shares a pool of items and currency, and at the start of each round they need to get together and determine what they’re going to buy, and which fighters will be using which of their items that round. (Team PMs will be provided of course.) Maybe they could also give suggestions to each other on how to proceed in battle? But the downside comes in why this is called “Survival”: when a team accumulates enough losses amongst them (I’m thinking twice their number of fighters, so if a team of three hits 6 losses), that entire team is eliminated from the game. We keep going until only one team remains.
I should probably get to the combat system, since it’s been completely revamped from previous entries in the series. Like in Coliseum, we’ve got stats, though I’m thinking the manager will get to decide the spread, and the stats in question have changed. Now it’s down to just Muscle (for physical attacks), Spirit (for elemental attacks), and Stamina (essentially health). When a match begins, players will take turns executing maneuvers—let’s say Player 1 starts with a Muscle-based maneuver, like “Tahu will swing his sword”. I’ll compare Tahu and his foe’s Muscle stats and RNG a number, decide how damage is dealt, and then Player 2 gets their turn. Maybe “Kopaka fires a blast of ice,” which means I’ll compare Spirit stats and RNG a number. As for how that RNG is going to work…
Let’s start at the baseline, or how the RNG would work if both fighters had an equal stat. I’ll RNG a number 1 through 5, with these results: 1 (attacker loses 3 stamina), 2 (attacker loses 1 stamina), 3 (both lose 1 stamina), 4 (target loses 1 stamina), 5 (target loses 3 stamina). Having a difference naturally modifies these values—for example, let’s say Tahu has 12 Muscle and Kopaka only has 10. There’s a difference of 2 in Tahu’s favor, so if he attacks using Muscle, I’ll RNG a number 1 through 7, with these results: 1 (attacker loses 3 stamina), 2 (attacker loses 1 stamina), 3 (both lose 1 stamina), 4 (target loses 1 stamina), 5 (target loses 2 stamina), 6 (target loses 2 stamina), 7 (target loses 3 stamina). The point advantage adds chances for middling damage. Naturally, if Tahu’s Muscle was lower, he’d gain chances that his attack would backfire and cause middling damage to him.
The goal of a fight is to reduce your opponent’s Stamina to 0 using these maneuvers. I’d also like to have players lose 1 stamina for every post they make, to add a sort of time limit to these battles. In addition, after I post the results of a maneuver, the other player will have 24 hours to respond—if they do not, their opponent gets a free shot (2 damage) and it becomes their turn again.
When a battle is over, the winner receives Widgets equal to [10 X s], where “s” is the number of Stamina points they lost over the course of the battle. (You won’t get much for an easy win, but a narrow victory will grant a huge payoff.) Widgets can be spent on items which grant their users stat buffs and passive effects. The loser gets 3 stat points to be allocated as their manager sees fit. While this does allow the loser to get a permanent increase, remember that items can be passed amongst a team, so they have an extra degree of versatility.
Other ideas I could implement to spice this up:
-I’d like to give each fighter a unique passive ability, but I don’t know how many I’ll have to come up with so I can’t be sure if this is feasible. Maybe a specific ability for each theme?
-Different choices for maneuvers, like Feint or Defend (which would probably have an additional benefit of nullifying the stamina drain for that post). These along with standard attacks would probably have a rock-paper-scissors relationship.
-And of course items are a whole thing. Some of the stronger ones will need to have drawbacks, so I’ll need to balance those.
-I suppose I should consider bringing back trainers, but this might interfere with the balance of giving winners Widgets and losers stat points.
-I should probably also try to think of further perks for team play. Maybe…team mates can make posts during your turn shouting encouragement, and that activates an RNG with a chance of giving a one-turn increase to the stat you’re using?
Hm. Again, to properly have a team vs team game, I’m going to need players, and I’m not convinced this will attract numbers comparable to Mafia and Realms. Of course, the more players, the longer each round, and the more work I’m going to have to do…a co-host could help, but there’s potential for confusion there, plus one fewer interested member to serve as a player. And my gosh, what if after the signup period I try to figure out the team size to make them all equal and find out we have a prime number of players?!
But not quite yet. Among other things, people will start going on recess soon enough, and they're all celebrating Christmas, but I'm thinking that right around New Year's Day, I might be sending out these letters, and additionally, the emails of all of the rest of Iowa's state congressmen are publicly available on the the government website. Until then, I will be thinking through my case so I can build it into something compelling. Iowa has every reason to change its anthem. The main challenge to changing it (and I must admit, my real-world experience tells me that it will be significant) is that people really don't like to change things. Politicians, amirite? Stodgy people who don't take their constituents seriously if they're not petitioning for something that fits into a readily recognizable political narrative. Pfaah!
Anyway, I'm pretty sure it should be self evident to y'all why Iowa's anthem needs changing. Here it is:
Wait, sorry, I had a technical error. This is Iowa's state anthem.
Ho. ly. Cow. I'm sorry if that gave you whiplash. And no, this isn't a joke. I'm not jigging your leg. That's actually Iowa's official anthem. There is literally onely one good thing about that song, and it's Maryland also rips off its tune from "O Christmas Tree", so we don't have to feel so bad. And Maryland has Michael Phelps, Savior of the Universe, King of the Impossible, Medals Be Upon Him, so by a very distant stretch, I guess I could say that Iowans have something in common with The Phelps that most other Americans don't.
But seriously? "O Christmas Tree"? Iowa already has a reputation for being bland and forgettable, and continuing to use this song that nobody cares about only contributes to that. Look at the Kansas theme: "Home on the Range". See? Isn't that much better? People actually know that one and like it! It gives Kansas a good reputation. Plus it's the home of Dorothy Gale and Clark Kent, so they're cultural legacy is set.
Here's an idea: ifi you're going to borrow music, at least borrow music that has something at least tangentially related to their state. West Virginia made "Take Me Home, Country Roads" their national anthem, because they were sensible and realized that people love John Denver and that it basically was the song that best represented their state. West Virginia would be cool if it used that song, so it did. Because why settle for something else when you already have something perfect wrapped and packaged for you? Oklahama is likewise wise, picking what else but the titular song from "Oklahoma!" to be their state standard. Do you see how easy that it?
It might be tempting to give Indiana a similar treatment by paying John Williams some royalties to use that one march of his as their anthem, but as it happens, Indiana already has a decent ballad from the 19th century.
Iowa, however, does not have a decent original ballad, so what are Iowans to do? What piece of music out there is indispensably associated with Iowa?
Ah, remember that Star Trek theme that I just shared? As every Iowan knows, the future Captain James Tyberius Kirk was born in Iowa. Ergo, Star Trek is a quintessential part of the Iowa's state mythology. This is what I personally would make Iowa's anthem. I just need Kim Reynolds to call Jerry Goldmith's estate, and Paramount Pictures, and basically whoever else she needs to sweet talk and pay off in order to get permission to use it when representing the state of Iowa. Aaaaaand BOOM! You have a great national melody. All you need to do from there is replace all of those nonsense "O Iowa! You're so pretty! Ain't we nice? Don't we all like our state? Iowa! O Iowa!" lyrics with something more sensible and imaginative so that you aren't singing about the exact same things as all of the other states. Create lyrics that actually describe what makes Iowa unique! Tell us about how Iowa was founded! Give people a sense of history and identity!
There you go. Except when I write my letters, I will probably have to drop all references to Star Trek if I want to be taken seriously by the politicians who may or may not bother reading them. Which is why I need to take some time to think through a serious proposal and maybe come up with a draft for a song myself, even though I'm not a musician. So by that, what I really mean is that I need to write up some poetry having to do with Iowa's history and then ask a friend who's a musician to compose a hopefully good proposition for a melody to go with it, and then my friend can be a co-signer in my petition. I'm thinking of a march, definitely in major. Hopefully Goldsmith or Williams-esque without overtly lifting anything from them.
And then I'm going to fail, but when I inevitably try again sometime down the road, I will be far more researched and have a serious game plan for changing Iowa's anthem. Because as small and as trivial as it seems, at least I will feel that I made a difference as one person, and I can confidently say that this would be something in history that wouldn't have inevitably happened without me.
Happy Thanksgiving everyone!
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.
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