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Akano

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  1. Akano

    Concert And Comics

    By Akano,

    Okay, I just came out with a new comic and no one has seemed to notice, so I'm going to redirect people to my comics topic to check it out. It incorporates that new idea I said I had earlier. The new idea was HAND DRAWN COMICNESS!
     
    My CGing skills haven't developed to the point that I could CG in two days and have it out, so for now the comic is just a sketch. I'll try to improve that aspect, though. Hope you all enjoy it!
     
    Also, I had a band concert on Monday that was with both band and choir as a Christmas concert. I must say, it was quite amazing. No, not the choir part, our part. Out of all the groups — Concert Choir, Symphonic Chorale, Show Choir, and Wind Ensemble — the Wind Ensemble was the only group to get a standing ovation. Boo-yah! I even had a solo at the end of Sleigh Ride for my trumpet — I made a horse-whinney sound. ^^;
     
    The crowd seemed to love it, though, and they loved when we played Trans-Siberian Orchestra's "Christmas Eve/Sarajevo 12/24." That earned us our standing ovation.
     
    The Concert Choir was terrible, though. And the choirs got to do more songs than the band. But, we're still the crowd favorite. ^^
     
    And now, as usual, a physics equation. I know you've all waited for this.
     
    For uniform circular motion, ΣF = mv²/r, where ΣF is the sum of the forces acting on the plane of the circle, m is mass of the object in motion, v is its velocity or speed, and r is the length of the radius.
     
    Of course, you could also say that v²/r = a, where a is centripital acceleration, bringing us back to ΣF = ma. ^^
     

  3. Akano

    Dear Chairman...

    By Akano,

    I love watching series over and over again to see if there are any subtleties the writers threw in that I never noticed during my first viewing. However, I rarely find a series that, when I watch it, I get the same feeling of suspense, the same feeling of revelation, as I do when I watch Red vs. Blue Season 6: Reconstruction.
     
    If you haven't seen Reconstruction yet, be warned that there will be spoilers in this entry.
     
    What is it that's so great about this particular season of a very comedic and ridiculous take on the Halo universe? Well, let's start with the premise of Red vs. Blue. We have two teams of ridiculously inept soldiers who are "at war" with each other for what they believe is the fate of the universe. Being the ineffective soldiers that they are, their battles usually end in whacky hijinks and the exchange of insults, and they very much keep those personalities in Season 6. A brief summary of the characters and their personalities:
     
    Red Team
     
    Sarge - Gruff and regimented leader of the Red Team, older, comes up with convoluted plans and ridiculous tactics. His hate for the Blues is only surmounted by his hatred of Grif.
     
    Grif - Lazy comic relief who is smarter than he looks, just unmotivated. Has a sort of love/hate relationship with Simmons.
     
    Simmons - The nerd of the Red Team, enjoys math, sucks up to Sarge every chance he gets.
     
    Lopez - The Red team's robot who can only speak (poorly translated) Spanish. Deadpan snarker.
     
    Donut - (absent from Reconstruction) Guy who wears pink armor and is rather effeminate.
    Blue Team
     
    Church - Self-appointed leader of the Blue Team
     
    Tucker - Lazy member of Blue Team who only thinks about picking up chicks.
     
    Caboose - The token cool dude of Blue Team. Probably the most popular character on the show for his ridiculous lines.
     
    Tex - A special ops soldier and Church's ex-girlfriend. The only soldier who can actually do something.
    Now, mix these characters with Agent Washington, a completely serious special ops soldier (like Tex) with no tolerance for humor. Surprisingly, this works extremely well (considering the number of ways they could have screwed this relationship up). Together, they are all trying to find a new threat known as the Meta who is killing off Freelancer agents (like Tex and Wash) to obtain their armor abilities and AI, which help them in battle.
     
    Then there is the overarching banter between the Director of Project Freelancer and the Oversight Sub-Committee Chairman. These conversations open every episode in the form of audio letter and alternate between the two, and they illustrate one of the most awesome passive-agressive power struggles I've ever witnessed in any series (and they are never on screen throughout the entire season!). While brief at the beginning of each episode, the subject of the dialogue, while at first seems unrelated, is actually intertwined with the entire motivation of the events of the season.
     
    And if that didn't seem to make things come full circle, the big reveal in the season further seals the deal. When I watch this one moment when Washington fully reveals why the Reds and the Blues were stuck in the middle of a boxed-in canyon in the middle of nowhere, why these Freelancer AI have plagued them and caused all their problems from the get go, and why he needs to put a stop to what Project Freelancer has done and bring them to justice, I am stunned. I always watch the scene and marvel at how perfectly everything is drawn together. I get the same goosebumps during each subsequent viewing of that scene that I got the first time I watched it. The reveal is always fresh; it always keeps me on the edge of my seat; it never gets stale, and that is why I consider this the crowning moment of the entire Red vs. Blue series.
     
    And I can't think of any other series that does that to me.
     

  5. Akano
    Today, my brother KopakaKurahk and I were walking up and down a nearby cul-de-sac and, while he was observing some lovely Virginia creeper and poison ivy decorating a peaceful evergreen, I noticed, with the help of my insect nerdiness, a spider that I originally thought was sitting within its web waiting for prey. Turns out, the spider had already obtained its evening meal: a small, black wasp that was no longer than a centimeter. Oh, the little stinging terror tried and tried to escape its inevitable fate of being a tasty liquified meal, but to no avail. The spider had latched on to the back of its thorax and was biting hard, not letting go despite the wasps squirming and attempts of stinging.
     
    For an amateur entomology enthusiast who despises most wasps, the feeling I obtained from this scene was true bliss.
     
    For those of you out there who don't care for my insect ranting (a.k.a., if your username is not Atako ) I have another news update on the life of this Bionicle-obsessed college student. The Toa of Electricity has achieved a new first: I have a real job now! *cue the dramatic chipmunk*
     
    It's not bad. It's not thrilling either, but what part-time job is? Well, I can think of one, and it's a job one of my friends has at his college (and it totally involves insect studies ). I currently reside in a mail room mainly sending out t-shirts. I keep that job until they're all sent out, then I need to find another place to work. But, still, that's not too bad.
     
    Wow, I haven't had this long a blog entry in quite some time. Whee! 8D
     

  6. Akano
    So, I just saw the new Sherlock Holmes movie Friday night. (I know, I'm behind the times) I have to say, IT WAS EPIC! I love Sherlock Holmes mysteries, and this movie was no exception.
     
    Also, for those of you comic lovers out there, NEW COMIC! I know, can you believe it?
     
    Naruto: Believe it!
     
    Now I'm off to research fractals. 8D
     

  8. Akano
    Since it is now the end of the semester at my school, professors are winding down the semester by ramping up the students' workloads. Isn't that just dandy?!
     
    Thus, real life takes over and Akano's free time goes from some to nearly none. On the bright side though, my brother KK and I recently had a recent acquisition of a Legend of Zelda DS Lite! (Do do do doo!) Thus, super fun happy fourth-gen Pokémon games are going to be played by us (KK has already gotten Platinum. I personally am waiting for HeartGold to come out in the US).
     
    That's really all I've got right now. As a parting gift, here's a cute little Rotom to wreak havoc on your appliances.

     

  9. Akano
    This is one that I didn't really know much about until recently, so I thought I'd share it. Today's equation is known as the Virial theorem,
     



     
    or, in component form,
     



     
    The word "virial" comes from the Latin vis, which means "force" or "energy," and looking at the equation, it makes sense why it's called that. Here the big Σ means sum, the "k" index denotes the kth particle of a system of N particles, V is the potential energy function affecting the kth particle, T is the potential energy of all the particles in the system, and rk is the position of the kth particle. This essentially relates the kinetic energy of all the particles to the positions and forces exerted on each particle (since -grad V is the force when energy is conserved, which is an assumption we are making). The brackets 〈 〉 denote that we're taking an average, so 〈T〉 is the average kinetic energy, etc.
     
    Now, you may be thinking, "okay, that's a cute equation, I guess, but I don't see how it's particularly useful." Okay, here's where the usefulness comes in. Let's say I want to know the mass of some distant galaxy, but I don't have a good galaxy-weighing device on hand. We know that the gravitational potential energy of an object is given by
     



     
    where m is the mass of the star, M is the mass of the center of the galaxy, and r is the distance from the center of the galaxy. Taking the distance r and multiplying by the gradient of the potential yields...the potential again, with a negative sign out front. So, for gravity,
     



     
    Plugging this into the Virial theorem above and noting that 2T = mv^2 (where v is speed), we get that, for an object in the gravitational pull of an object of mass M,
     



     
    Thus, we have at our disposal a way of measuring the mass of something like a galaxy by measuring only the speeds of stars and their distance away from the center. That's pretty incredible.
     
    This actually is one of the ways scientists support the idea that there is dark matter in the universe; the Virial theorem gives an average of what speeds the stars in our galaxy should have based on their distance away from the center of the Milky Way, but what we actually observe is startlingly different. Thus, we can conclude that something is wrong with our knowledge of how gravity within a galaxy works. Based on this and other observations, the idea that there's extra stuff that can't be seen that adds to the gravitational force of a galaxy seems to be a reasonable idea.
     
    In my research on diatomic hydrogen (H2), the Virial theorem is used in a different capacity. When figuring out the potential energy of an electron (or two) around the two positively charged protons, the virial has the Coulomb force term (which is just -V, just like gravity) and an additional term that pops up from assuming that the electrons are keeping the protons at equilibrium. I won't go too much into the physics, but the final product is
     



     
    where E, T, and V are the total energy, kinetic energy, and potential energy of the electron(s), respectively, and R is the distance between the nuclei. This tells us something useful about the energy of the electrons; more specifically, it tells us about how the energy changes as you move the nuclei farther apart or closer together. In other words, since E = T + V,
     
     



     
    which is very useful when constructing potential energy curves for hydrogen.
     
    On a slightly related note, our lab's paper got published! Akano is now a for reals, published scientist! 8D
     

  10. Akano

    Equation of the Day #1

    By Akano,

    So, I've decided to do one of those daily-like blog entries, though I can't guarantee that I'll be able to do this every day (being a busy grad student and all). I figured that, being a physics grad student, math might be one of my stronger suits (next to reviewing LEGO sets), so I'm going to try and share an equation with you and see if I can explain it well enough for people to understand. 8D
     
    Tonight's equation: The wave equation.
     




     
    This says that the sum of the change in the change in the function, ψ, with respect to the coordinates used to represent it is equal to the inverse square of the speed of the wave,c, modeled by ψ times the change in the change of ψ with respect to time.
     
    This equation is the governing equation for all wave phenomena in our world. Sound waves, light waves, water waves, earthquakes, etc. are governed by this mathematical equation. In one dimension, the wave equation simplifies to
     




    which has the lovely solutions
     




     
    where A and B are determined by appropriate boundary conditions, and ω/k = c. This equation governs things like vibrations of a string, sound made by an air column in a pipe (like that of an organ, trumpet, or didgeridoo), or even waves created by playing with a slinky. It also governs the resonances of certain optical cavities, such as a laser or Fabry-Perot cavity.
     
    Since waves are one of my favorite physical phenomena, I find it very appropriate to start with this one.
     

  11. 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.
     

  12. Akano
    You've probably heard of the Uncertainty Principle before. In words, it says "you cannot simultaneously measure the position and the momentum of a particle to arbitrary precision." In equation form, it looks like this:
     



     
    What this says is that the product of the uncertainty of a measurement of a particle's position multiplied by the uncertainty of a measurement of a particle's momentum has to be greater than a constant (given by the reduced Planck constant, h over τ = 2π). This has nothing to do with the tools with which we measure particle; this is a fundamental statement about the way our universe behaves. Fortunately, this uncertainty product is very small, since ħ is around 1.05457 × 10-34 J s. The real question to ask is, "Why do particles have this uncertainty associated with them in the first place? Where does it come from?" Interestingly, it comes from wave theory.
     




     
     




     
     
    Take the two waves above. The one on top is very localized, meaning its position is well-defined. But what is its wavelength? For photons, wavelength determines momentum, so here we see a localized wave doesn't really have a well-defined wavelength, thus an ill-defined momentum. In fact, the wavelength of this pulse is smeared over a continuous spectrum of momenta (much like how the "color" of white light is smeared over the colors of the rainbow). The second wave has a pretty well-defined wavelength, but where is it? It's not really localized, so you could say it lies smeared over a set of points, but it isn't really in one place. This is the heart of the uncertainty principle. Because waves exhibit this phenomenon – and quantum particles behave like waves – quantum particles also have an uncertainty principle associated with them.
     
    However, this is arguably not the most bizarre thing about the uncertainty principle. There is another facet of the uncertainty principle that says that the shorter the lifetime of a particle (how long the particle exists before it decays), the less you can know about its energy. Since mass and energy are equivalent via Einstein's E = mc2, this means that particles that "live" for very short times don't have a well-defined mass. It also means that, if you pulse a laser over a short enough time, the light that comes out will not have a well-defined energy, which means that it will have a spread of colors (our eyes can't see this spread, of course, but it means a big deal when you want to use very precise wavelengths of light in your experiment and short pulses at the same time). In my lab, we use this so-called "energy-time" uncertainty to determine whether certain configurations of the hydrogen molecule, H2, are long-lived or short lived; the longer-lived states have thinner spectral lines, and the short-lived states have wider spectral lines.
     
    So while we can't simultaneously measure the position and momentum of a particle to arbitrary certainty, we can definitely still use it to glean information about the world of the very, very small.
     

  13. Akano
    Today I want to talk about mass. Sometimes you'll hear it defined loosely as "the amount of stuff in an object." There are, however, two separate definitions of mass in classical physics. The first definition comes from Newton's second law.




     
    This mass is known as the inertial mass. The larger an object's inertial mass, the more it resists being accelerated by a given force. The second definition of mass also comes from Newton, but it is instead determined by his law of gravitation.
     



     
    The mass here determines how much two massive objects attract one another; this is known as the gravitational mass. But here's the interesting thing about these two masses: there is no law of physics that says these masses are one and the same. Such a notion is known in physics as the equivalence principle. The weak equivalence principle was discovered by Galileo; he noticed that objects with different masses fall at the same rate. Einstein came up with the strong equivalence principle, which discusses how a uniform force and a gravitational field are indistinguishable when you look at a small enough portion of spacetime. The only reason we believe these two masses are equivalent is because experiments show that they are equal to within the precision of the instruments with which we measure them, and there are ongoing experiments trying to narrow down that precision to determine if there is any difference between the two.
     

  14. Akano
    The room you are currently sitting in is probably around 20°C, or 68°F (within reasonable error, since different people like their rooms warmer or colder or have no control over the temperature of the room they're reading this entry in). But what does it mean to be at a certain temperature? Well, we often define temperature as an average of the movement of an ensemble of constituent particles – usually atoms or molecules. For instance, the temperature of a gas in a room is given as a relation to the gas' rms molecular speed:
     



     
    Where T is the absolute temperature (e.g. Kelvin scale), m is the mass of the particles making up the gas, and k is Boltzmann's constant. But this is a specific case. In general, we need a more encompassing definition. In thermodynamics, there is a quantity known as entropy, which basically quantifies the disorder of a system. It is related to the number of ways to arrange the elements of a system without changing the energy.
     
    For instance, there are a lot of ways of having a messy room. You can have clothes on the floor, you can track mud into it, you can leave dishes and food everywhere. But there are very few ways to have an immaculately clean room, where everything is tidy and put in its proper place. Thus, the messy room has a larger entropy, while the clean room has very low entropy. It is this quantity that helps to define temperature generally. Denoting entropy as S, we have that
     



     
    Or, in words, temperature is defined as the change in energy divided by the change in entropy of something when its volume remains fixed, which is equivalent to the change in enthalpy (heat) divided by the change in entropy at constant pressure. Thus, if you increase the energy of an object and find that it becomes more disordered, the temperature is positive. This is what we are used to. When you heat up air, it becomes more disorderly because the particles making it up are moving faster and more randomly, so it makes sense that the temperature must be positive. If you cool air, the particles making it up slow down and it tends to become more orderly, so the temperature is still positive, but decreasing. What happens when you can't pull any more energy out of the air? Well, that means that the temperature has gone to zero, and movement has stopped. Since the movement has stopped, the gas must be in a very ordered state, and the entropy isn't changing. When the speed of the gas particles is zero, we call its temperature absolute zero, when all motion has stopped.
     
    It is impossible to reach absolute zero temperature, but it isn't intuitive as to why at first. The main reason is due to quantum mechanics. If all atomic motion of an object stopped, its momentum would be known exactly, and this violates the Uncertainty Principle. But there is also another reason. In thermodynamics, there is a quantity related to temperature that is defined as
     



     
    Since k is just a constant, β can be thought of as inverse temperature. This sends absolute zero to β being infinity! Now, this makes much more sense as to why achieving absolute zero is impossible – it means we have to make a quantity go to infinity! It turns out that β is the more fundamental quantity to deal with in thermodynamics because of this role (and others).
     
    Now, you're probably thinking, "Akano, that's all well and good, but, are you saying that this means that you can get to infinite temperature?" In actuality, you can, but you need a special system to be able to do it. To get temperature to infinity, you need β to go to zero. How do we do that? Well, once you cross zero, you end up with a negative quantity, so if we could somehow get a negative temperature, then we would have to cross β equals zero. But how do we get a negative temperature, and what would that be like? Well, we would need entropy to decrease when energy is added to our system.
     



     
    It turns out that an ensemble of magnets in an external magnetic field would do the trick. See, when a compass is placed in a magnetic field, it wants to align with the field (call that direction north). But if I put some energy into the system (i.e. I push the needle), I can get the needle of the compass to point in the opposite direction (south). When less than half of the compasses are pointing opposite the external field, each time I flip a compass needle I'm increasing entropy (since the perfect order of all the compasses pointing north has been tampered with). But once more than half of those compasses are pointing south, I am decreasing the disorder of the system when I flip another magnet south! This means that the temperature must be negative! In practice, the compasses are actually molecules with an electric dipole moment or electrons with a certain spin (which act like magnets), but the same principles apply. So, β equals zero is when exactly half of the compasses are pointing north and the other half are pointing south, and β equals zero is when T is infinite, and it is at this infinity that the sign on T swaps.
     
    It's interesting to note that negative temperatures are actually hotter than any positive temperature, since you have to add energy to get to negative temperature. One could define a quantity as –β, so that plotting it on a line would be a more intuitive way to see that the smaller the quantity, the colder the object is, while preserving the infinities of absolute zero and "absolute hot."
     

  15. 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.
     

  16. Akano
    The definition of a planet has been under scrutiny several times, and with New Horizon's recent visit to Pluto, the discussion of Pluto's demotion was on everyone's minds (at least, back in July). But I'm not going to talk about Pluto's demotion (though I think it was totally appropriate from a scientific perspective). Instead, I'm going to talk about the Moon.
     
    Should the Earth-Moon system be considered a binary planet? This sounds outlandish at first, since the Moon is a moon, obviously. It orbits the Earth as a natural satellite, just as the Galilean moons (Ganymede, Callisto, Io, and Europa) orbit Jupiter, Titan orbits Saturn, Triton orbits Neptune, and so on, right?
     
    The definition of a moon is vague, and thus there are multiple ways of determining whether or not a planet-moon system is really a binary planet. One way of drawing the line between the two descriptions is by finding the barycenter (or center-of-mass) of the system. The center of mass of a collection of N masses is given by
     



     
    where M is the total mass of the system, and mi and ri are the mass and position of the ith object, respectively. If the center of mass of a two-body system lies outside the larger object in that system, call it a binary planet. This makes sense, right? This means that the smaller body doesn't orbit the larger body, but instead they both orbit some point in space. For instance, the barycenter of the Pluto-Charon system lies outside Pluto (0.83 Pluto radii above Pluto's surface), the larger of the two bodies, while the Earth-Moon barycenter lies within the Earth (just under 3/4 of an Earth radius from the planet's center). By this definition, the Pluto-Charon system is a binary (dwarf) planet system, while the Earth-Moon system is is a planet-moon system. (Although, we are slowly losing our moon due to tidal acceleration. In a few billion years, the Moon will have drifted far enough away that the barycenter of the Earth-Moon system will leave the interior of our planet.) However, when you plug in values for the Sun-Jupiter system, you find that the center of mass lies outside the Sun! Indeed, Jupiter is the only natural satellite of the Sun for which this is true. (Does this mean Jupiter should have a different classification from the rest of the planets? Not really; the Sun is around 1000 times more massive than Jupiter, so the reason for this is that Jupiter is very distant from the Sun.)
     
    Maybe a different definition is needed to distinguish planet-moons from binary planets, then, since the Sun-Jupiter system is not a binary star (Jupiter is slightly too small to generate nuclear fusion). Another proposition is to look at the so-called tug-of-war value of a body. The tug-of-war value of a moon determines which Solar System object has a stronger gravitational hold, the Sun or the moon's "primary" (the Earth is the Moon's primary). Using Newton's law of gravitation
     



     
    we can take a ratio of the Sun's pull on a satellite to the primary's pull. The result is the tug-of-war value, proposed by Isaac Asimov.
     



     
    Here the subscripts s and p refer to the Sun and the primary, respectively; m is the mass of the body referred to by the subscript; and d is the distance between the moon and the body referred to by the subscript. If the tug-of-war value is larger than 1, then the primary has a larger hold on the moon than the Sun, whereas if it's less than 1, the Sun's gravity dominates. For the Earth-Moon system, it turns out this number is 0.46, which means that the Sun pulls on the Moon with more than twice the force of Earth's pull. This is an oddity among moons, but is not unique. It does mean, though, that the Moon, when viewed from the Sun, never undergoes retrograde motion; it moves across the solar sky without changing direction. Another way to put this is that the Moon is always falling toward the Sun (like the planets), and never in its orbit does it fall away from the Sun (unlike most moons). If you look at the orbits of the Earth and Moon from the point of view of the Sun, they dance around each other in careful step, which is unlike most other moons in the Solar System. For Asimov, this was reason enough to consider the Earth and Moon as a binary planet system.
     
    This tug-of-war value does not, however, classify Pluto and Charon as a binary dwarf planet system (they're too far from the Sun for their tug-of-war value to be less than 1). Perhaps the definition of a binary planet is a difficult one to pin down.
     
    Should the Moon be promoted to planet, just as Pluto was renamed as a dwarf planet? I don't know, but it gives us something to think about as we look up at the starry night, watching the dance of all the chunks of rock and gas hurtling through space in our sky, to music written by nature and heard through science.
     

  17. Akano
    The above image is known as the Pentagram of Venus; it is the shape of Venus' orbit as viewed from a geocentric perspective. This animation shows the orbit unfold, while this one shows the same process from a heliocentric perspective. There are five places in Venus' orbit where it comes closest to the Earth (known as perigee), and this is due to the coincidence that
     



     
    When two orbital periods can be expressed as a ratio of integers it is known as an orbital resonance (similar to how a string has resonances equal to integer multiples of its fundamental frequency). The reason that there are five lobes in Venus' geocentric orbit is that 13–8=5. Coincidentally, these numbers are all part of the Fibonacci sequence, and as a result many people associate the Earth-Venus resonance with the golden ratio. (Indeed, pentagrams themselves harbor the golden ratio in spades.) However, Venus and Earth do not exhibit a true resonance, as the ratio of their orbital periods is about 0.032% off of the nice fraction 8/13. This causes the above pattern to precess, or drift in alignment. Using the slightly more accurate fraction of orbital periods, 243/395, we can see this precession.
     



     
    This is the precession after five cycles (40 Earth years). As you can see, the pattern slowly slides around without the curve closing itself, but the original 13:8 resonance pattern is still visible. If we assume that 243/395 is indeed the perfect relationship between Venus and Earth's orbital periods (it's not; it precesses 0.8° per cycle), the resulting pattern after one full cycle (1944 years) is
     

     
    Click for hi-res image.


     
    Which is beautiful. The parametric formulas I used to plot these beauties are
     



     
    Where t is time in years, r is the ratio of orbital periods (less than one), and τ = 2π is the circle constant.
     

  18. Akano
    Hydrogen is the simplest and most common neutral atom in the universe. It consists of two particles – a positively charged proton and a negatively charged electron. The equation that describes the hydrogen atom (or any one-electron atom) in the nonrelativistic regime is the Schrödinger equation, specifically
     



     
    where ħ is the reduced Planck constant, μ is the reduced mass of the electron-nucleus system, Z is the number of positive charges in the nucleus that the electron is orbiting, e is the charge of a proton, τ is the circle constant, ε0 is the vacuum permittivity, and ψ is the wavefunction. Solving this equation (which is nontrivial and is usually done after a semester of Advanced Quantum Mechanics) yields a surprisingly simple formula for the energies of the atom,
     

    ,

     
    where h is Planck's constant, c is the speed of light, me is the rest mass of the electron, and n is any integer larger than or equal to 1. The constant R∞ is known as the Rydberg constant, named after Swedish physicist Johannes Rydberg, the scientist who discovered a formula to predict the specific colors of light hydrogen (or any hydrogen-like atom) would absorb or emit. Indeed, the formula I gave, En/hc, is equivalent to the inverse wavelength, or spatial frequency, of light that it takes for the atom in its nth energy state to free the electron of its atomic bond. Indeed, this was a puzzle in the early 20th century. Why was it that hydrogen (and other atoms) only absorbed and emitted specific colors of light? White light, as Isaac Newton showed, is comprised of all visible colors of light, and when you split up that light using a prism or similar device, you get a continuous rainbow. This was not the case for light emitted or absorbed by atoms.
     
    The equation above was first derived by Niels Bohr, who approached solving this problem not from using the Schrödinger equation, but from looking at the electron's angular momentum. If electrons could be considered wavelike, as quantum mechanics treats them, then he figured that the orbits of the electron must be such that an integer number of electron wavelengths fit along the orbit.
     

     
    Left: Allowed orbit. Right: Disallowed orbit. Image: Wikimedia commons


     
    This condition requires that
     



     
    The wavelength of the electron is inversely related to its momentum, p = mv, via Planck's constant, λ = h/p. The other relation we need is from the physics of circular motion, which says that the centripetal force on an object moving in a circular path of radius r is mv2/r. Equating this to the Coulomb force holding the proton and electron together, we get
     



     
    Plugging this into the quantization condition, along with some algebra, yields the energy equation.
     
    What's incredible is that hydrogen's energy spectrum has a closed-form solution, since most problems in physics can't be solved to produce such solutions, and while this equation only works exactly for one-electron atoms, it can be modified to work for so-called Rydberg atoms and molecules, where a single electron is highly excited (large n) and orbits a positive core, which need not be a nucleus, but a non-pointlike structure. In my lab, we consider two types of Rydberg molecules.
     



     
    The example on the left is an electronic Rydberg molecule, while the one on the right is called an ion-pair Rydberg state, where a negative ion acts as a "heavy electron" co-orbiting a positive ion. To model the energies of these kinds of states, we use a modified energy equation.
     



     
    where I.P. represents the ionization energy of the electron, and the new quantity δ is known as the quantum defect. It's a number that, for electronic Rydberg states, has a magnitude that's usually less than 1, while for ion-pair states can be quite large (around –60 or so in some cases); it in some sense contains information of how the core ion, e.g. H2+, is oriented, how the electron is spread over space, how its polarized, and so on. It's a vessel into which we funnel our ignorance in using the approximation that the molecule is behaving in a hydrogen-like manner, and it is surprisingly useful in predicting experiments. Currently my research involves studying electronic Rydberg states of molecular nitrogen, N2, and looking at heavy Rydberg states of the hydrogen molecule, H2 to gain a better understanding of the physics of certain states that have been experimentally observed in both systems.
     

  19. Akano

    Equation of the Day #18: 12

    By Akano,

    Yesterday I stumbled across this image (which I recreated and cleaned up a bit). It's a beautiful image. Arranged around the edge is the circle of fifths, which in music is a geometric representation of the twelve tones of the Western scale arranged so the next note is seven semitones up (going clockwise in this figure). The notes are all connected in six different ways to the other notes in the "circle," known as intervals, which are color-coded at the bottom. I thought, "Wow, this is a really cool way to represent this geometrically. How neat!" However, I found the original website that the image came from, and it's a pseudoscience site that talks about the fractal holographic nature of the universe. While fractals do show up in Nature a lot, and there are legitimate theories proposing that the Universe may indeed be a hologram, what their site is proposing is, to put it lightly, utter nonsense. But instead of tearing their website apart (which would be rather cathartic), I instead want to point out the cool math going on here, because that sounds more fun!
     
    Looking at the bottom of the graphic, you'll notice six figures. The first (in red) is a regular dodecagon, a polygon with twelve equal sides and angles. This shape is what forms the circle of fifths. The rest of the shapes in the sequence are dodecagrams, or twelve-pointed stars. The first three are stars made up of simpler regular polygons; the orange star is made up of two hexagons, the yellow is made up of three squares, and the green one is made up of four triangles. The final dodecagram (in purple) can be thought of as made up of six straight-sided digons, or line segments. These shapes point to the fact that twelve is divisible by five unique factors (not including itself): one set of twelve, two sets of six, three sets of four, four sets of three, and six sets of two! You could say that the vertices of the dodecagon finalize the set as twelve sets of one, but they're not illustrated in this image. So really, this image has less to do with musical intervals and more to do with the number 12, which is a rather special number. It is a superior highly composite number, which makes it a good choice as a number base (a reason why feet are divided into twelve inches, for instance, or why our clocks have twelve hours on their faces).
     
    The final dodecagram in cyan is not made up of any simpler regular polygons because the number 12 is not divisible by five. If you pick a note in the circle of fifths to start on, you'll notice that the two cyan lines that emanate from it connect to notes that are five places away on the "circle," hence the connection to the number 5. In fact, it would be far more appropriate to redraw this figure with a clock face.
     
     
    This new image should shed some more light on what's really going on. The dodecagrams each indicate a different map from one number to another, modulo 12. The only reason this is connected to music at all is due to the fact that a Western scale has twelve tones in it! If we used a different scale, such as a pentatonic scale (with five tones, as the name would suggest), we'd get a pentagon enclosing a pentagram. Really, this diagram can be used to connect any two elements in a set of twelve. The total number of connecting lines in this diagram, then, are
     
     
    where the notation in parentheses is "n choose 2," and Tn is a triangular number. This figure is known in math as K12, the complete graph with twelve nodes. And it's gorgeous.
     
    So while this doesn't really have anything to do with music or some pseudoscientific argument for some fancy-sounding, but ultimately meaningless, view on the universe, it does exemplify the beauty of the number 12, and has a cool application to the circle of fifths.
     

  20. Akano
    Ah, the pentagram, a shape associated with a variety of different ideas, some holy, some less savory. But to me, it's a golden figure, and not just because of how I chose to render it here. The pentagram has a connection with the golden ratio, which is defined as
     



     
    This number is tied to the Fibonacci sequence and the Lucas numbers and seems to crop up a lot in nature (although how much it crops up is disputed). It turns out that the various line segments present in the pentagram are in golden ratio with one another.
     



     
    In the image above, the ratio of red:green = green:blue = blue:black is the golden ratio. The reason for this is not immediately obvious and requires a bit of digging, but the proof is fairly straightforward and boils down to a simple statement.
     
    First, let's consider the pentagon at the center of the pentagram. What is the angle at each corner of a pentagon? There's a clever way to deduce this. It's not quite clear what the interior angle is (that is, the angle on the inside of the shape at an individual corner), but it's quite easy to get the exterior angle.
     



     
    The exterior angle of the pentagon (which is the angle of the base of the triangles that form the points of the pentagram) is equal to 1/5 of a complete revolution around the circle, or 72°. For the moment, let's call this angle 2θ. To get the angle that forms the points of the pentagram, we need to invoke the fact that the sum of all angles in a triangle must equal 180°. Thus, the angle at the top is 180° – 72° – 72° = 36°. This angle I will call θ. While I'm at it, I'm going to label the sides of the triangle x and s (the blue and black line segments from earlier, respectively).
     



     
    We're nearly there! We just have one more angle to determine, and that's the first angle I mentioned – the interior angle of the pentagon. Well, we know that the interior angle added to the exterior angle must be 180°, since the angles both lie on a straight line, so the interior angle is 180° – 72° = 108° = 3θ. Combining the pentagon and the triangle, we obtain the following picture.
     



     
    Now you can probably tell why I labeled the angles the way I did; they are all multiples of 36°. What we want to show is that the ratio x/s is the golden ratio. By invoking the Law of sines on the two isosceles triangles in the image above, we can show that
     



     
    This equation just simplifies to sin 2θ = sin 3θ. With some useful trigonometric identities, we get a quadratic equation which we can solve for cos θ.
     



     
    Solving this quadratic equation yields
     



     
    which, when taken together with the equation for x/s, shows that x/s is indeed the golden ratio! Huzzah!
     
    The reason the pentagram and pentagon are so closely tied to the golden ratio has to do with the fact that the angles they contain are multiples of the same angle, 36°, or one-tenth of a full rotation of the circle. Additionally, since the regular dodecahedron (d12) and regular icosahedron (d20) contain pentagons, the golden ratio is abound in them as well.
     
    As a fun bonus fact, the two isosceles triangles are known as the golden triangle (all acute angles) and the golden gnomon (obtuse triangle), and are the two unique isosceles triangles whose sides are in golden ratio with one another.
     



     
    So the next time you see the star on a Christmas tree, the rank of a military officer, or the geocentric orbit of Venus, think of the number that lurks within those five-pointed shapes.
     

  21. Akano

    Equation of the Day #2

    By Akano,

    I had a wonderful time today; I got to see old friends from my undergrad today and went exploring a corn maze; it was a lot of fun.
     
    Also, equation of the day: Newton's Second Law
     



     
    where F is the net force, a is acceleration, and m is the mass of the object in question. It's such a simple-looking equation, but it contains so much physics. Want to know the path of a free-falling object subject only to the force of gravity? You use this equation. Want to know the attractive force and classical orbit of planets/atoms? You use this equation. Want to know the physics of a car skidding on pavement? You get the idea.
     
    This equation is a staple of physics and is used extensively in intro and classical physics. Newton, you clever devil, you.
     

  22. 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!
  24. Akano

    Equation of the Day #3

    By Akano,

    Just so everyone knows, I obtained the awesome LEGO Haunted House over Fall Break and have pictures that will work wonderfully in a review. You can probably expect that next week at some point (I hope). For now, let's go over another fun physics equation! This one is probably very familiar to you, though you may not have any idea what it means. I give you mass-energy equivalence:
     



     
    Where E is energy, m is mass, and c is the speed of light. It's a very simple-looking equation with only three parameters, but what does it mean? Well, it means that anything with mass – you, your cat, your house, the Earth – has latent energy stored in it, and the amount of mass determines that latent energy. For an object at rest, this correlates to the rest mass of the object. If an object is moving really fast (near the speed of light) its kinetic energy causes it to actually get heavier, since the object can never actually reach the speed of light (only objects with no rest mass move at the speed of light).
     
    So, if we have an object sitting and doing nothing, and it suddenly glows for a split second, then stops, where did the light come from? Well, light has energy, as we know, so we could calculate the energy of the light that escapes our object. If the light emanates in all directions, then the net kinetic energy of the object is unchanged. But conservation of energy says that energy can neither be created nor destroyed! Have we violated the laws of physics with our weird glowing object? Well, no, because if you were somehow able to weigh the object pre- and post-glow, you would find that the mass of this object is actually slightly less after the light is given off.
     
    But wait! Doesn't conservation of mass say that matter can neither be created nor destroyed? Well, yes, it does say that. So the only way for this to make sense is if the mass is converted into the energy that was emitted. We know that energy can be converted into different forms (electric, mechanical, thermal, etc.), so this must mean that mass is another form of energy that can be converted to and from! Pretty neat, huh?
     
    Minutephysics has a cool video on this with a bit more technicality and pretty pictures of radioactive cats, but this is my text-based explanation simplified.
     
    Another thing that may cross your mind is that this looks very similar to Newton's second law:
     



     
    So, does Newton's second law equate force with acceleration? Well, no, because in the mass-energy equation, the constant of proportionality, c2, is a universal constant; it is the same for any and all objects in the universe. The mass of an object, however, varies from object to object, and is thus not a fundamental, universal constant, so while these equations are similar and relate two seemingly different entities, they do not conceptually perform the same task.
     

  25. Akano
    If you're building something and want to tell other people how to build it, it's useful to show the dimensions of said something (how big it is) relative to other things that people are familiar with. However, there are very few things in this world that are exactly the same size as other similar things (e.g. not all apples weigh the same or have the same volume). So, some smart people once upon a time decided to make standards of measurement for various properties of matter (which I think we can all agree was a smart decision). I wanted to talk about one of these today: the meter.
     
    The word meter (or metre for those who live across the pond/in Canada) comes from the word for "measure" in Greek/Latin (e.g. speedometers measure speed, pedometers measure steps, &c.), but the meter I'm talking about is the International System (SI) unit of distance. The original definition of the meter was one ten-millionth of the distance from the Earth's equator to the North Pole at sea level (not through the Earth). The first person to measure the circumference of the Earth was the Greek mathematician/astronomer/geographer Eratosthenes (and he was accurate to within 2% of today's known value) circa 240 B.C., so this value was readily calculable in 1791 when this standard was accepted.
     
    In 1668, an alternative standard for the meter was suggested. The meter was suggested to be the length a pendulum needed to be to have a half-period of one second; in other words, the time it took for the pendulum to sweep its full arc from one side to the other had to be one second. The full period of a pendulum is
     



     
    So, when L = 1 m and T = 2 sec, we get what the acceleration due to gravity, g, should be in meters per second per second (according to this standard of the meter). It turns out that g = pi2 meters per second per second, which is about 9.8696 m/s2. This is very close to the current value, g = 9.80665 m/s2 which are both fairly close to 10. In fact, for quick approximations, physicists will use a g value of ten to get a close guess as to the order of magnitude of some situation.
     
    So, you may be wondering, why is it different nowadays? Well, among a few other changes in the standard meter including using a platinum-iridium alloy bar, we have a new definition of the meter: the speed of light. Since the speed of light in a vacuum is a universal constant (meaning it is the same no matter where you are in the universe, unlike the acceleration due to gravity at a point in space), they decided to make the distance light travels in one second a set number of meters and adjust the meter accordingly. Since the speed of light is 299,792,458 meters per second exactly, this means that we have defined the meter as the distance light travels in 1/299,792,458th of a second.
     
    This is all nice, but it's not a very intuitive number to work with. After all, we humans like multiples of ten (due to having ten fingers and ten toes), so why not make a length measurement of the distance light travels in one billionth (1/1,000,000,000th) of a second (a.k.a. nanosecond)? That seems a bit more intuitive, don't you think? It turns out that a light-nanosecond is about 11.8 inches, or about 1.6% off of the current definition of a foot. In fact, one physicist, David Mermin, suggests redefining the foot to the "phoot," or one light-nanosecond, since it's based off of a universal constant while the current foot is based off the meter by some odd, nonsensical ratio.
     

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