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Akano

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  2. Akano
    Or, so some of my students in my Intro Physics lab think. Hopefully when you read the title you were ready to get your typing fingers ready to disprove me. You probably would have made an argument akin to the following mini-lecture.
     
    Gravity is a force between objects/particles proportional to the objects' mass. Newton's universal gravitation looks like this:
     

    Fg = - G m1m2/r2


     
    where G is a proportionality constant, the m's are the masses of the two objects in question, and r is the distance between the two objects. This is why we feel the Earth's gravity affect us, but we don't feel the moon's or sun's gravity affect us. They most definitely influence the Earth (since the sun causes our orbit and the moon causes the tides), but we don't feel the effects of their presence.
     
    So, if we have an object with mass m on Earth in free fall, its equation of motion is determined by
     

    Fg = m a = - G m ME/r2


     
    where ME is the mass of the Earth and a is the acceleration of the object. Note that, if we divide both sides by m, we find that
     

    a = - G ME/r2


     
    which means that the acceleration of an object in free fall has nothing to do with the mass of the object. In fact, you can see a video of this on the moon at Wikipedia's Gravitation page that shows Apollo 15 astronaut David Scott dropping a feather and hammer simultaneously. Since there is no air on the moon, the feather is not afloat longer than the hammer, and they fall at the same rate and hit the ground at the same time.
     
    Also, while I said earlier that gravity affects things with mass, it also affects light, which does not have (rest) mass. However, light has energy, and as Einstein showed with his Special Theory of Relativity, energy and mass are equivalent:
     

    E = m c2


     
    So, you can construct the relativistic mass of light, thereby finding the equations that govern the changing of the straight path of light in a gravitational field. Using Einstein's General Theory of Relativity, you can also view the gravitational field as a curvature of spacetime, which influences straight lines to be curved in the space near the massive object, affecting the path of light.
     
    Another interesting thing about mass: objects actually have two different masses associated with them: gravitational mass and inertial mass. Gravitational mass tells you how much an object interacts gravitationally, while inertial mass tells you how much an object resists a change in motion. In other words, more massive objects take more force/energy to alter their paths than objects with less mass. Here's the interesting thing, though: both these masses are equal, even though there really is no physical law stating that they have to be. The only reason we know these masses are equal is because empirical evidence says they are; there is no indication that these two masses are different to an appreciable/statistical extent.
     
    So, if you think that there are no unanswered questions in the realm of physics, you are sorely mistaken.
     

  4. Akano

    *Grabs Popcorn*

    By Akano,

    It's really amusing to me that everyone is freaking out about this canonization stuff, mainly because this is what happened to me several years ago.
     
    As someone who ended up disagreeing with new canon information during BIONICLE's initial tenure, let me assure you: the new canon is meaningless. Even if Greg Farshtey says that Toa can transform into gigantic dinosaurs and feast on souls, it would mean absolute bupkis. Why? Because the fun of BIONICLE isn't what's officially created, but what we, the fans, create.
     
    I have seen beautiful creations on this site in the forms of MOCs, comics, general artwork, movies, stories, and music posted on this site for the past 11 years, and it is far more impressive than just about anything that has come out of the official storyline (although, Time Trap is the best thing Greg ever contributed to the actual canon, and there's no way I can knock it). You want a story where Vo-Matoran, -Toa, and -Turaga are male? Or have no set gender? Go for it. There's a headcanon floating around about a transgender Tamaru, there were comics featuring the only male Ga-Matoran, and there were epics that featured Toa, Matoran, and Turaga of Muffins. Do you think Muffins would ever be canonized as a for reals element? Who cares?! It made for far more interesting plots and ideas than a Toa of the Green who has only ever been hinted at and never seen in action!
     
    Part of the fun of writing my epics – How I Became Me and The Inventor of Metru Nui (which I hope to work on again at some point, I swears!) – was twisting one part of the canon while leaving the rest be. Trying to get my OC to fit into the BIONICLE universe was fun, and when it was announced that Vo-Matoran are female (and blue, what's with that? If you've played Pokémon, you know that electricity is yellow bar a few minor deviations ), I didn't budge because I didn't have to.
     
    So have fun making mountains out of molehills, BZbloggers. I'm going to sit over here, lean back, and enjoy my bowl of popcorn whilst I watch the fireworks. Because I'm just that nice a guy. 8D
     

  5. Akano

    Physics Jokes!

    By Akano,

    In no particular order, some fun physics jokes. Ready? GO!
    My friend was trying to talk to me about atoms, but I got Bohr'd.
    Did you hear that Albert Einstein developed a theory about space? It was about time, too.
    Never trust an atom; they make up everything.
    The oddly pleasant feeling of looking down on a physicist as they finish the last of their drink. The strange charm of a top-down bottoms-up.
    Why does hamburger have less energy than steak? It's in the ground state.
    Why are physics books always unhappy? Because they're full of problems.
    Neutrinos make the worst friends; they rarely interact with anyone.
    In a quantum finish!

  6. Akano
    Today I want to talk about something awesome: Special Relativity. It's a theory that was developed by this guy you may have heard of, Albert Einstein, and it's from this theory that arguably the most famous equation in physics, E = mc2, comes from. I'm not going to talk about E = mc2 today (in fact, I've already talked about it, but it's not the whole story!), but I wanted to talk about two other cool consequences of Special Relativity (SR), time dilation and length contraction.
     
    First and foremost, the main fact from which the rest of SR falls out is the fact that the speed of light is the same for all observers moving with constant velocity, regardless of what those velocities may be. Running at 5 m/s? You see light traveling at the same speed as someone traveling 99% the speed of light.
     
    Wait, how can that be? This idea originally came from Maxwell's equations, which govern electromagnetism. When you solve these equations, you can put them into a form that results in a wave equation, and the speed of those waves is equal to that of light. This finding brought on the realization that light is an electromagnetic wave! But here's the interesting thing: Maxwell's equations do not assume any particular frame of reference, so the speed of the waves governed by Maxwell's equations have the same speed in all reference frames. Thus, it makes sense from an electromagnetic point of view that the speed of light shouldn't depend on how fast someone is traveling!
     
    Now, we're still in a bit of a pickle; if all observers see light traveling at the same speed, how do things other than light move? Think about it. If you're driving down the highway at 60 mph and the car next to you is driving 65 mph, they appear to be moving 5 mph faster than you, don't they? So why doesn't this work with light? If I'm traveling 5 mph, shouldn't I see light moving 5 mph slower than normal? No; the problem here isn't that the speed of light is the same for all observers, but the fact that we think relative velocities add up normally. In fact, this relative velocity addition is simply a very good approximation for objects that are much, much slower than light, but it is not complete.
     
    The answer to this conundrum is that
    . These two principles are governed by the equations 



     
    The first equation determines time dilation, and the second equation determines length contraction, when shifting from a frame moving at speed v to a frame moving at speed v' (β and γ are both physical parameters that depend on the velocity of the frame in question and the speed of light, c). From the first equation, we can see that the faster someone is moving in frame S (moving at speed v), the slower their clock ticks away the seconds in frame S' (moving at speed v') and the more squished they look (in the direction that they're traveling). These ideas are the basis for the famous "barn and pole" paradox. Suppose someone is holding a pole of length L and is running into a barn, which from door-to-door has a length slightly longer than L. If the person runs fast enough, an outside observer will see that the person running with the pole will completely disappear into the barn before emerging from the other side. But from the runner's frame of reference, the barn is what is moving really fast, and so the barn appears shorter than it did to the outside observer. This means that, in the runner's frame, a part of the pole is always outside of the barn, and thus he is always exposed.
     
    What if the observer outside the barn had the exit door closed and the entrance door open and rigs it such that when the runner is completely inside the barn, the entrance door closes and the exit door opens? Well, in the outside observer's frame, this is what happens; the entrance door closing and the exit door opening are simultaneous events. But in the runner's frame, there is no way for him to fit inside the barn, so does the door close on the pole? No, because the physics of what happens has to be the same in both frames; either the door shuts on the pole or it doesn't. So, in the runner's frame, the entrance door closing and the exit door opening are not simultaneous events! In fact, the exit door opens before the entrance door closes in the runner's frame. This is due to the time dilation effect of special relativity: simultaneous events in one reference frame need not be simultaneous in other frames!
     
    Special relativity is a very rich topic that I hope to delve into more in the future, but for now I'll leave you with this awesome bit of cool physics.
     

  7. Akano
    I'm taking a second pass at this one. Instead, I'm going to talk about chaos.
     
    Chaos is complexity that arises from simplicity. Put in a clearer way, it's when a deterministic process leads to complex results that seem unpredictable. The difference between chaos and randomness is that chaos is determined by a set of rules/equations, while randomness is not deterministic. Everyday applications of chaos include weather, the stock market, and cryptography. Chaos is why everyone (including identical twins who having the same DNA) have different fingerprints. And it's beautiful.
     
    How does simplicity lead to complexity? Let's take, for instance, the physical situation of a pendulum. The equation that describes the motion of a pendulum is
     



     
    where θ is the angle the pendulum makes with the imaginary line perpendicular to the ground, l is the length of the pendulum, and g is the acceleration due to gravity. This leads to an oscillatory motion; for small angles, the solution of this equation can be approximated as
     



     
    where A is the amplitude of the swing (in radians). Very predictable. But what happens when we make a double pendulum, where we attach a pendulum to the bottom of the first pendulum?
     




    Can you predict whether the bottom pendulum will flip over the top? (Credit: Wikimedia Commons)


     
    It's very hard to predict when the outer pendulum flips over the inner pendulum mass, however the process is entirely determined by a set of equations governed by the laws of physics. And, depending on the initial angles of the two pendula, the motion will look completely different. This is how complexity derives from simplicity.
     
    Another example of beautiful chaos is fractals. Fractals are structures that exhibit self-similarity, are determined by a simple set of rules, and have infinite complexity. An example of a fractal is the Sierpinski triangle.
     



     

    Triforce-ception! (Image: Wikipedia)


     
    The rule is simple: start with a triangle, then divide that triangle into four equal triangles. Remove the middle one. Repeat with the new solid triangles you produced. The true fractal is the limit when the number of iterations reaches infinity. Self-similarity happens as you zoom into any corner of the triangle; each corner is a smaller version of the whole (since the iterations continue infinitely). Fractals crop up everywhere, from the shapes of coastlines to plants to frost crystal formation. Basically, they're everywhere, and they're often very cool and beautiful.
     
    Chaos is also used in practical applications, such as encryption. Since chaos is hard to predict unless you know the exact initial conditions of the chaotic process, a chaotic encryption scheme can be told to everyone. One example of a chaotic map to disguise data is the cat map. Each iteration is a simple matrix transformation of the pixels of an image. It's completely deterministic, but it jumbles the image to make it look like garbage. In practice, this map is periodic, so as long as you apply the map repeatedly, you will eventually get the original image back. Another application of chaos is psuedorandom number generators (PRNGs), where a hard-to-predict initial value is manipulated chaotically to generate a "random" number. If you can manipulate the initial input values, you can predict the outcome of the PRNG. In the case of the Pokémon games, the PRNGs have been examined so thoroughly that, using a couple programs, you can capture or breed shininess/perfect stats.
     



     

    Dat shiny Rayquaza in a Luxury ball, tho.


     
    So that's the beauty of chaos. Next time you look at a bare tree toward the end of autumn or lightning in a thunderstorm, just remember that the seemingly unpredictable branches and forks are created by simple rules of nature, and bask in its complex beauty.
     

  8. Akano
    With the rebirth of BIONICLE, a great debate has arisen, and I am now able to put all of our qualms to rest. In my recent entry about making mountains into molehills, the comments brought up a serious issue: the color of the element of electricity. Specifically, I am a proponent of its color being yellow because I don't buy into the popularity of the color blue, and that Pokémon represents the element with the color yellow. I had many (read: a couple) people disagree with me on that sentiment, so I decided to use my math and science skills to better humanity to prove my point.
     
    I used the resource known as Bulbapedia to acquire the complete list of electric type Pokémon (pure and dual-type) and entered in their official Pokédex color. Using the advanced mathematical concept of "counting" to total up how many electric types are of each color, I obtained the following results:
    Yellow: 18
    Blue: 8
    White: 4
    Black: 3
    Gray: 3
    Red: 3
    Brown: 1
    Pink: 1
    Ergo, my scientific findings have brought me to the conclusion that electricity is yellow by a landslide. Have a good day, everyone.
     

  9. Akano

    Complex Numbers

    By Akano,

    Math is a truly wonderful topic, and since I'm procrastinating a little on my physics homework, I'm going to spend some time talking about the complex numbers.
     
    Most of us are used to the real numbers. Real numbers consist of the whole numbers (0, 1, 2, 3, 4, ...), the negative numbers (-1, -2, -3, ...), the rational numbers (1/2, 2/3, 3/4, 22/7, ...), and the irrational numbers (numbers that cannot be represented by fractions of integers, such as the golden ratio, the square root of 2, or π). All of these can be written in decimal format, even though they may have infinite decimal places. But, when we use this number system, there are some numbers we can't write. For instance, what is the square root of -1? In math class, you may have been told that you can't take the square root of a negative number. That's only half true, as you can't take the square root of a negative number and write it as a real number. This is because the square root is not part of the set of real numbers.
     
    This is where the complex numbers come in. Suppose I define a new number, let's call it i, where i2 = -1. We've now "invented" a value for the square root of -1. Now, what are its properties? If I take i3, I get -i, since i3 = i*i2. If I take i4, then I get i2*i2 = +1. If I multiply this by i again, I get i. So the powers of i are cyclic through i, -1, -i, and 1.
     
    This is interesting, but what is the magnitude of i, i.e. how far is i from zero? Well, the way we take the absolute value in the real number system is by squaring the number and taking the positive square root. This won't work for i, though, because we just get back i. Let's redefine the absolute value by taking what's called the complex conjugate of i and multiplying the two together, then taking the positive square root. The complex conjugate of i is obtained by taking the imaginary part of i and throwing a negative sign in front. Since i is purely imaginary (there are no real numbers that make up i), the complex conjugate is -i. Multiply them together, and you get that -i*i = -1*i2 = 1, and the positive square root of 1 is simply 1. Therefore, the number i has a magnitude of 1. It is for this reason that i is known as the imaginary unit!
     
    Now that we have defined this new unit, i, we can now create a new set of numbers called the complex numbers, which take the form z = a + bi, where a and b are real numbers. We can now take the square root of any real number, e.g. the square root of -4 can be written as ±2i, and we can make complex numbers with real and imaginary parts, like 3 + 4i.
     
    How do we plot complex numbers? Well, complex numbers have a real part and an imaginary part, so the best way to do this is to create a graph where the abscissa (x-value) is the real part of the number and the ordinate (y-axis) is the imaginary part. This is known as the complex plane. For instance, 3 + 4i would have its coordinate be (3,4) in this coordinate system.
     
    What is the magnitude of this complex number? Well, it would be the square root of itself multiplied by its complex conjugate, or the square root of (3 + 4i)(3 - 4i) = 9 + 12i - 12i +16 = 25. The positive square root of 25 is 5, so the magnitude of 3 + 4i is 5.
     
    We can think of points on the complex plane being represented by a vector which points from the origin to the point in question. The magnitude of this vector is given by the absolute value of the point, which we can denote as r. The x-value of this vector is given by the magnitude multiplied by the cosine of the angle made by the vector with the positive part of the real axis. This angle we can denote as ϕ. The y-value of the vector is going to be the imaginary unit, i, multiplied by the magnitude of the vector times the sine of the angle ϕ. So, we get that our complex number, z, can be written as z = r*(cosϕ + isinϕ). The Swiss mathematician Leonhard Euler discovered a special identity relating to this equation, known now as Euler's Formula, that reads as follows:
     

    eiϕ = cosϕ + isinϕ


     
     
     
    Where e is the base of the natural logarithm. So, we can then write our complex number as z = reiϕ. What is the significance of this? Well, for one, you can derive one of the most beautiful equations in mathematics, known as Euler's Identity:
     

    eiπ + 1 = 0


     
    This equation contains the most important constants in mathematics: e, Euler's number, the base of the natural logarithm; i, the imaginary unit which I've spent this whole time blabbing about; π, the irrational ratio of a circle's circumference to its diameter which appears all over the place in trigonometry; 1, the real unit and multiplicative identity; and 0, the additive identity.
     
    So, what bearing does this have in real life? A lot. Imaginary and complex numbers are used in solving many differential equations that model real physical situations, such as waves propagating through a medium, wave functions in quantum mechanics, and fractals, which in and of themselves have a wide range of real life application, along with others that I haven't thought of.
     
    Long and short of it: math is awesome.
     

  10. Akano

    Hobbits...

    By Akano,

    Tomorrow my roommate (of three years), a fellow physics graduate student, and I are going to watch the three extended Lord of the Rings movies. In a row. In one day.
     
    My roommate has tried this once on another occasion, but we couldn't help but bleed into the next day. Here's hoping that we can manage it this time.
     
    Also, three day weekend! 8D
     

  14. Akano
    I like triangles. I like numbers. So what could possibly be better than having BOTH AT THE SAME TIME?! The answer is nothing! 8D
     
    The triangular numbers are the numbers of objects one can use to form an equilateral triangle.
     

    Anyone up for billiards? Or bowling? (Image: Wikimedia Commons)


     
    Pretty straightforward, right? To get the number, we just add up the total number of things, which is equal to adding up the number of objects in each row. For a triangle with n rows, this is equivalent to
     



     
    This means that the triangular numbers are just sums from 1 to some number n. This gives us a good definition, but is rather impractical for a quick calculation. How do we get a nice, shorthand formula? Well, let's first add sequential triangular numbers together. If we add the first two triangular numbers together, we get 1 + 3 = 4. The next two triangular numbers are 3 + 6 = 9. The next pair is 6 + 10 = 16. Do you see the pattern? These sums are all square numbers. We can see this visually using our triangles of objects.
     

    (Image: Wikimedia Commons)


     
    You can do this for any two sequential triangular numbers. This gives us the formula
     



     
    We also know that two sequential triangular numbers differ by a new row, or n. Using this information, we get that
     



     
    Now we finally have an equation to quickly calculate any triangular number. The far right of the final line is known as a binomial coefficient, read "n plus one choose two." It is defined as the number of ways to pick two objects out of a group of n + 1 objects.
     
    For example, what is the 100th triangular number? Well, we just plug in n = 100.
     

    T100 = (100)(101)/2 = 10100/2 = 5050


     
    We just summed up all the numbers from 1 to 100 without breaking a sweat. You may be thinking, "Well, that's cool and all, but are there any applications of this?" Well, yes, there are. The triangular numbers give us a way of figuring out how many elements are in each row of the periodic table. Each row is determined by what is called the principal quantum number, which is called n. This number can be any integer from 1 to infinity. The energy corresponding to n has n angular momentum values which the electron can possess, and each of these angular momentum quanta have 2n - 1 orbitals for an electron to inhabit, and two electrons can inhabit a given orbital. Summing up all the places an electron can be in for a given n involves summing up all these possible orbitals, which takes on the form of a triangular number.
     



     
    The end result of this calculation is that there are n2 orbitals for a given n, and two electrons can occupy each orbital; this leads to each row of the periodic table having 2⌈(n+1)/2⌉2elements in the nth row, where ⌈x⌉ is the ceiling function. They also crop up in quantum mechanics again in the quantization of angular momentum for a spherically symmetric potential (a potential that is determined only by the distance between two objects). The total angular momentum for such a particle is given by
     



     
    What I find fascinating is that this connection is almost never mentioned in physics courses on quantum mechanics, and I find that kind of sad. The mathematical significance of the triangular numbers in quantum mechanics is, at the very least, cute, and I wish it would just be mentioned in passing for those of us who enjoy these little hidden mathematical gems.
     
    There are more cool properties of triangular numbers, which I encourage you to read about, and other so-called "figurate numbers," like hexagonal numbers, tetrahedral numbers, pyramidal numbers, and so on, which have really cool properties as well.
     

  15. Akano
    Science is awesome. I am currently reading a journal article about how people are making the acoustic version of iridescence. For those who don't know, iridescence is what certain insects, jewels, soap bubbles, and CDs exhibit as that rainbow effect that changes color depending on what angle it's viewed. The sonic or acoustic version of this is creating something that varies in pitch depending on the angle at which you stand relative to it.
     
    Awesome.
     

  19. Akano
    Today I wanted to talk about one of my favorite equations in all of mathematics. However, I won’t do it justice without building up some framework that puts it into perspective. To start out, let’s talk about waves.

    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.
     
    Image: Wikipedia
    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:
    Animation: LucasVB on Wikipedia
    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
     

    where
     

    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
     

    Or, graphically,
     

    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!
  20. Akano

    January update

    By Akano,

    Haldo, BZPorples,
     
    I hope everyone had some awesome holiday funtimes! Mine were packed with traveling, visiting friends, gift giving and receiving, and all the food. All of it.
     
    I also saw The Force Awakens twice while I was home. It was pretty fantastic. The part where we find out that Chewbacca is Rey's father was quite the twist![/trololololol]
     
    Now I am back at school. Though classes don't start until next week, I'm in my lab typing this and sorta doing work. (I've been at a loss to find a certain physical quantity for the past week and have been trying to cope with this by watching various videos. Right now I'm watching Cosmos: A Spacetime Odyssey. When Knowledge Conquered Fear = WIN!)
     
    Otherwise, things are going the way they've always been going.
     

  21. Akano

    WIRELESS WIZARD!

    By Akano,

    Not sure what started this fad, but I'm okay with this.
     

    I Am A:
     
    True Neutral Human Wizard (3rd Level)


     
    Ability Scores:
    Strength- 9
    Dexterity- 12
    Constitution- 12
    Intelligence- 16
    Wisdom- 14
    Charisma- 12
     
    Alignment:
    True Neutral- A true neutral character does what seems to be a good idea. He doesn't feel strongly one way or the other when it comes to good vs. evil or law vs. chaos. Most true neutral characters exhibit a lack of conviction or bias rather than a commitment to neutrality. Such a character thinks of good as better than evil after all, he would rather have good neighbors and rulers than evil ones. Still, he's not personally committed to upholding good in any abstract or universal way. Some true neutral characters, on the other hand, commit themselves philosophically to neutrality. They see good, evil, law, and chaos as prejudices and dangerous extremes. They advocate the middle way of neutrality as the best, most balanced road in the long run. True neutral is the best alignment you can be because it means you act naturally, without prejudice or compulsion. However, true neutral can be a dangerous alignment when it represents apathy, indifference, and a lack of conviction.
     
    Race:
    Humans are the most adaptable of the common races. Short generations and a penchant for migration and conquest have made them physically diverse as well. Humans are often unorthodox in their dress, sporting unusual hairstyles, fanciful clothes, tattoos, and the like.
     
    Class:
    Wizards- Wizards are arcane spellcasters who depend on intensive study to create their magic. To wizards, magic is not a talent but a difficult, rewarding art. When they are prepared for battle, wizards can use their spells to devastating effect. When caught by surprise, they are vulnerable. The wizard's strength is her spells, everything else is secondary. She learns new spells as she experiments and grows in experience, and she can also learn them from other wizards. In addition, over time a wizard learns to manipulate her spells so they go farther, work better, or are improved in some other way. A wizard can call a familiar- a small, magical, animal companion that serves her. With a high Intelligence, wizards are capable of casting very high levels of spells.
     

  22. Akano

    Rv B Season 10

    By Akano,

    We're four episodes into the new season of Red vs. Blue, and I still haven't heard Elijah Wood's voice nor seen his character on screen.
     
    Just sayin'.
     

  24. Akano

    Pokédex 3D Pro

    By Akano,

    You look really cool, and I'm sure I'd enjoy your features and such, but you're $15. For a Pokédex.
     
    If you had 3D battling capabilities/games that would work in tandem with BW/B2W2, I'd probably get you, but you don't. Sorry.
     

  25. Akano

    Thesis Submitted

    By Akano,

    Six years, four papers, and 179 pages later, I have submitted my Ph.D. thesis.
     
    Now I have to defend it. But first, STRESS RELIEF VIA VIDEO GAMES!
     
     

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