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Akano

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Blog Entries posted by Akano

  1. Akano
    I made it in GIMP. A cardioid is the envelope formed by a set of circles whose centers lie on a circle and which pass through one common point in space. This image shows the circle on which the centers of the circles in the above image lie. A cardioid is also the path traced by a point on a circle which is rolling along the surface of another circle when both circles have the same radius (here is a cool animation of that).
     
    What is the cardioid's significance? Well, it looks like a heart, which is kind of cool. It's also the (2D) pickup pattern of certain microphones (I have a cardioid microphone). If a sound is produced at a given point in space, the pickup pattern shows an equal intensity curve. So, if I place a microphone at the intersection point of all those circles, the outside boundary is where a speaker producing, say, a 440 Hz tone would have to be to be heard at a given intensity. So, the best place to put it would be on the side where the curve is most round (the bottom in this picture) without being too far away from the microphone.
     
    Another interesting fact about the cardioid is that it is the reflection of a parabola through the unit circle (r = 1. Here's what I mean). In polar coordinates, the equation of the above cardioid is given by
     

     
    where a is a scaling factor, and theta is the angle relative to the positive x-axis. The origin is at the intersection of the circles. The equation of a parabola opening upwards and whose focus is at the origin in polar coordinates is just
     

     
    which is an inversion of the cardioid equation through r = 1, or the unit circle.
     
    ♥
     

  2. Akano
    You may have learned once that classical mechanics all stems from Newton's laws of motion, and while that is true, it is not necessarily the best way to solve a given physical problem. Often when we look at a physical system, we take note of certain physical parameters: energy, momentum, and position. However, these can be more generalized to fit the physical situation in question better. This is where Lagrange comes in; he thought of a new way to formulate mechanics. Instead of looking at the total energy of a system, which is the potential energy plus the kinetic energy, he instead investigated the difference in those two quantities,
     




     
    where T is the kinetic energy and V is the potential energy. Since the kinetic and potential energy, in general, depend on the coordinate position and velocity of the particle in question, as well as time, so too does the Lagrangian. You're probably thinking, "okay, what makes that so great?" Well, if we were to plot the Lagrangian and calculate the area under the curve with respect to time, we get a quantity known as the action of the particle.
     




     
    where t1 and t2 are the starting and ending times of interest. Usually if the motion is periodic, the difference between these times is one period. Now, it turns out that for classical motion, the action is minimized with respect to a change in the path along which the particle moves for the physical path along which the particle actually moves. This sounds bizarre, but what it means is that there is only one path along which the particle can move while keeping the action minimized. Physicists call this the Principle of Least Action; I like to call it "the universe is inherently lazy" rule. When you do the math out, you can calculate an equation related to the Lagrangian for which the action is minimized. We call these the Euler-Lagrange Equations.
     




     
    These are the equations of motion a particle with Lagrangian L in generalized coordinates qi with velocity components denoted by qi with a dot above the q (the dot denotes taking a time derivative, and the time derivative of a coordinate is the velocity in that coordinate's direction). This is one of the advantages of the Lagrangian formulation of mechanics; you can pick any coordinate system that is best-suited for the physical situation. If you have a spherically symmetric problem, you can use spherical coordinates (altitude, longitude, colatitude). If your problem works best on a rectangular grid, use Cartesian coordinates. You don't have to worry about sticking only with Cartesian (rectilinear) coordinates and then converting to something that makes more sense; you can just start out in the right coordinate system from the get go! Now, there are a couple of special attributes to point out here. First, the quantity within the time derivative is a familiar physical quantity, known as the conjugate momenta.
     




     
    Note that these do not have to have units of linear momentum of [Force × time]. For instance, in spherical coordinates, the conjugate momentum of longitude is the angular momentum in the vertical direction, which has units of action, [Energy × time]. The Euler-Lagrange equations tell us to take the total time derivative of these momenta, i.e. figure out how they change in time. This gives us a sort of conjugate force, since Newton's second law reads that the change in momentum over time is force. The other quantity gives special significance when it equals zero,
     




     
    This is just fancy math language for saying that if one of our generalized coordinates, qi, doesn't appear at all in our Lagrangian, then that quantity's conjugate momentum is conserved, and the coordinate is called "cyclic." In calculating the Kepler problem – the physical situation of two particles orbiting each other (like the Earth around the Sun) – the Lagrangian is
     




     
    Note that the only coordinate that doesn't appear in the Lagrangian is ϕ, the longitude in spherical coordinates. Thus, the conjugate momentum of ϕ, which is the angular momentum pointing from the North pole vertically upwards, is a conserved quantity. This reveals a symmetry in the problem that would not be seen if we used the Lagrangian for the same problem in Cartesian coordinates:
     




     
    That just looks ugly. Note that all three coordinates are present, so there are no cyclic coordinates in this system. In spherical coordinates, however, we see that there is a symmetry to the problem; the symmetry is that the situation is rotationally invariant under rotations about an axis perpendicular to the plane of orbit. No matter what angle you rotate the physical situation by about that axis, the physical situation remains unchanged.
     

  3. Akano
    My very first Equation of the Day was about the wave equation, a differential equation that governs wave behavior. It doesn't matter whether you have linear waves (sine and cosine functions), cylindrical waves, or spherical waves, the wave equation governs them. Today I will focus on the second, the so-called cylindrical harmonics, or Bessel functions.
     
    A harmonic function is defined as one that satisfies Laplace's equation,
     



     
    For cylindrical symmetry, the Laplacian (the operator represented by the top-heavy triangle squared) takes the following form:
     



     
    This is where a neat trick is used. We make an assumption that the amplitude of the wave, denoted here by ψ, can be represented as a product of three separate functions which each only depend on one coordinate. To be more explicit,
     



     
    This technique is known as "separation of variables." We claim that the function, ψ, can be separated into a product of functions each with their own unique variable. The results of this mathematical magic are astounding, since it greatly simplifies the problem at hand. When you go through the rigamarole of plugging this separated function back in, you get three simpler equations, each with its own variable.
     



     
    Notice that the partial derivatives have become total derivatives, since these functions only depend on one variable. These are well-known differential equations in the mathematical world; the Φ function is a linear combination of sin(nϕ) and cos(nϕ) (this azimuthal angle, ϕ, goes from 0 to 2π and cycles, so this isn't terribly surprising) with n being an integer, and the Z function is a linear combination of cosh(kz) and sinh(kz), which are the hyperbolic functions. These equations are not what I want to focus on; what we've really been working so hard to get is the radial equation:
     



     
    This is Bessel's differential equation. The solutions to this equation are transcendental (meaning that you can't write them as a finite sum of polynomials; the sine and cosine functions are also transcendental). We write them as
     



     
    The Jn are finite at the origin (J0 is 1 at the origin, all other Jn are 0), and the Yn are singular (undefined) at the origin. They look something like this:
     







     
    The Jn are much more common to work with because they don't have infinities going on, but the Yn are used when the origin is inaccessible (like a drum head that has a hole cut in the middle). These harmonic functions are used to model (but are not limited to)
    Vibrational resonances of a circular drum head
    Radial wave functions for potentials with cylindrical symmetry in quantum mechanics
    Heat conduction in a cylindrical object
    Light traveling in a cylindrical waveguide

    Note that, while they kinda look sinusoidal, they don't have a set period, so the places where they cross the x-axis are have different intervals and are irrational; thus, they must be computed. This results in some weird harmonic series for instruments like xylophones, drums, timpani, and so on. I got into them because I'm a trumpet player, and the resonances of the surface of the bell of a trumpet are related to the Bessel functions.
     
    There are some
    (this one has a strobe effect during it) showing them in action. There are also some cool Mathematica Demonstrations related to them as well. There are also orthogonality relationships with them, but I'll save that for another day. 

  4. 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.
     

  6. Akano

    Fall And Winter

    By Akano,

    I was on a walk the other day after lunch and a beautiful aroma hit my nose. It was faint, almost fleeting, until it got slightly more intense, then faded away. It is the best smell in the world
     
    It is the smell of a fire burning in the fireplace of a distant house, which means one glorious thing: Winter is on its way.
     

  7. Akano
    So, this past week was my Fall Break, which allowed me to enjoy the comforts of home and gave me the opportunity to say hello to people who are still at my undergrad.
     
    I also received and played intensely Pokémon White 2, hence my lack of blog posts. I just beat the 8th Gym today (Marlon is weird...) and now have to prepare lecture for Wednesday. :\
     

  8. Akano

    Family vacation

    By Akano,

    Going to spend some time with KK, Tekulo, my mom and my roommate for the next week. I'll have BZP access, but I'm not sure if I'll actually be on during that time, so if I'm quiet for the next week, you now know why.
     

  9. Akano

    Fanboy Moment

    By Akano,

    Rob Paulsen responded to my question I tweeted earlier on whether his voice was in the LEGO City: Undercover trailer.
     

     
    While not a definitive answer, it still has made my day.
     

  10. Akano
    It's finals week at my grad school, but since I didn't take any classes this semester, I have no exams to study for (except my prelims, which are at the end of the summer ). I am, however, holding office hours for my students before they take their exams, so I'm not without stuff to do.
     
    I also went to Philly BrickFest two weekends ago, but I was only there for a couple hours since I had to leave for choir rehearsal. I did snap some pics, which will hopefully end up on my Brickshelf at some point. Maybe.
     
    In other news, I'm still obsessed with quantum mechanics and have been playing around with various mathematical things associated with it, like deriving the ladder operators and matrix elements for the quantum harmonic oscillator, deriving formulas for coherent states, and trying to find out what a true Hufflepuff is, anyway deriving coordinate transformations to the center of mass frame of two particles. Fun fun.
     
    (This is the part where you all look at me like I'm mad, and I reply with an expression like this: 8D)
     
    So, not too much going on with me right now, but I can't complain.
     

  11. Akano
    So, I'm working on a computer project for my Electrodynamics course. I'm using a computer method called the Relaxation or Finite Difference Method. It basically takes a physical scenario, divides the space of interest into a grid, and assigns voltages for each grid intersection. Then, using a computer language of choice (I'm using FORTRAN, like a boss), I make a program that essentially takes a weighted average of all the points whose voltages aren't fixed until the program doesn't change those voltages anymore. This gives a surprisingly good approximation for a physical system.
     
    I'm basically modeling a system with two conducting cylindrical shells of equal radius separated by some height and which are at voltages +V0 and -V0. The problem is that my output graphs do not look physical; the voltage just drops to near zero rapidly for points outside and between the cylinders, whereas I expect that the graph should gradually drop.
     
    Curse you, technology!
     

  14. Akano
    Yesterday, I obtained my 4-year…Norik head? Exactly what am I supposed to do with it? ~tries putting it on his face like a mask~
     
    You know, after one year, you can control time, after two years you get power over light, three years you get invisibility, but all I have now is a Rahkshi head?
     
    The fourth year stinks. XP
     

  15. Akano
    When I first came to this site, I was very much a lurker. My brothers and I had gotten into BIONICLE from its beginning in 2001, and we eventually stumbled on this site in the early, awkward days of the World Wide Web. It was here that I saw the creative output of the fandom – sprite comics, artwork, MOCs, comedies, epics, and much more.
     
    Fourteen years ago, I became a part of that creative output with the premiere of Akano's Comics.
     
    It's been six years since I made a comic (sorry, everyone, grad school happened; I do want to make more, though!), but my creative outputs didn't stop. I made set reviews and Equation of the Day (which I'm still working on, I promise!).
     
    Thank you, everyone, for being my first audience. Now that I'm a professor, I'll have many, many more, but you were the first to truly see the crazy inner workings of my mind. Thanks for sticking with me, and thanks for helping me improve. You guys helped someone who stumbled to craft paragraphs become an author. You helped someone who enjoyed comedy but couldn't quite formulate a joke become a half-decent comedian.
     
    Here's to more creative output.
     

  17. Akano

    Gen 2

    By Akano,

    Guys, Gold and Silver were released almost sixteen years ago, why are you all—
     
    OH! You mean BIONICLE...
     
    ...
     
    There was a gen 2?!
     

  18. Akano

    Going home for turkey

    By Akano,

    This evening, KK and I will embark on an epic quest to return home via Greyhound bus. Neither of us has ever partaken in such an adventure, so it will probably be full of intrigue, drool, and snorts due to the fact that we leave after midnight. 'Twill be interesting indeed.
     
    In other news, scattering theory is still something that eludes me. This is funny, because one of my research projects sort of uses scattering theory as an approximation technique. I get it there, but not when it comes to General Relativity.
     
    Also, dat Detective's Office set. 8L
     

  19. Akano

    Golden Sun

    By Akano,

    Just finished replaying the first game in the series, and it's still awesome. The world, the music, the characters, the story – all are so wonderful and charming. The soundtrack has to be some of the best music in video game history.
     
    After the long, gruelling process that is inputting a Gold level password from my Clear Data (I don't have a GBA link cable with me), I'm now starting The Lost Age. I will miss Isaac and Co. for now.
     

  20. Akano

    Goodbye, Spring Break

    By Akano,

    Alas, my Spring Break has come to a close. I did have fun, though. I got Sonic & Knuckles for the SEGA Genesis and enjoyed time with friends and family.
     
    This week I get a midterm for my grad level E&M class. We have over two weeks to do it, though, which is quite nice.
     
    Also, Happy St. Patrick's Day, all! If you are old enough to drink, please do so responsibly.
     

  23. Akano
    Yet again, my brothers suggested I do something that they've done. This time it was "watch Gravity Falls," a show which I knew of but didn't know much about. I have now finished the first season, and I must say it's a brilliant, funny show. In no particular order, what makes it stand out is
    The humor - By and far an important aspect of any comedy, the humor of Gravity Falls resonates very well, from the lamest pun to the brilliant stuff they get past the censors (and, wow, do they get a lot past the censors. This is a Disney show, right?). Expertly crafted and leaving me wanting more in the best way possible.
    The story - while not the most story-heavy series (a lot of the episodes are very standalone and can be watched without missing much of previous episodes), the story that is ongoing is very engaging. Gravity Falls, OR is a place where weird, paranormal stuff happens. Our main characters want to know why, thus we want to know why, and their curiosity becomes ours in a genuine, unforced way.
    The relationships are believable - Dipper and Mabel, the two main protagonists, are twin siblings who are sent to their great uncle (or, you guessed it, Grunkle) Stan's tourist trap, the Mystery Shack, for the summer. And they have a relationship that is completely believable (and as a twin, I can fully attest to it). Even when they have a scuffle or conflict, at the end of they day they can hug it out and not hate each other, which is very refreshing in a kids show. Also, the characters are not just defined by single character traits; for instance, Mabel has a fantastically overactive imagination and looks at the world from a very different angle than most of the other characters, but she's never called stupid or foolish by the others. Soos, the Mystery Shack's general repair and groundskeeper guy, who is overweight and sometimes dull-witted, is not defined by these traits, nor is he mocked for them; everyone treats him as they do everyone else, which is also really refreshing to see in a kids show. As for romantic relationships,
     
    This show has provided me with a lot of laughs and a fun world of mystery, and I look forward to what else it has in store.
     

  24. Akano
    "The proportionality factor σ (not to be confused with surface charge) is an empirical constant that varies from one material to another; it's called the conductivity of the medium. Actually, the handbooks usually list the reciprocal of σ, called the resistivity: ρ = 1/σ (not to be confused with charge density—I'm sorry, but we're running out of Greek letters, and this is the standard notation)."
     
    This is the man who allegedly can teach physics to gerbils. I wish I could take a class with him simply for his wit and skillz at teaching.
     

  25. Akano

    Halloween approacheth

    By Akano,

    'Sup, BZPeoples?
     
    I've been away a while attending friends' weddings, visiting various peeps, rereading Harry Potter (still as magical as always), and working on research. If all goes well, this Friday will see me submitting a paper to the Journal of Chemical Physics. All in all, it's been a good few months. But now is the season of autumnal haunts, and I have grown very happy with the drop in temperature and the swap of palette.
     
    Over my fall break, I visited KK in the land of cheese, Wisconsin, and he introduced me to the joys of Punch Out!! for Wii and Super Mario Maker. I now own both, and have been having loads of fun playing others' levels and designing levels of my own (mostly goofing off). When I've finalized a level I'll post it for you all to play if you like. In particular I've been working on a Ghost House.
     

      

     
    It's been super fun. Also, RubberRoss' levels are utterly evil.
     
    EDIT: Untimely Haunt v2: CD1D-0000-00C8-8D19
     
    Happy Halloween!
     

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