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Akano

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Blog Entries posted by Akano

  1. Akano

    Ponies Premiere

    By Akano,

    These villains are getting less and less backstory development as we go, huh? In terms of development:
     

    Sombra < Chrysalis < Discord / Nightmare Moon


     
    That said, though, I enjoyed the episode quite a bit. The songs were okay, but I've only listened to them once, really. However, it was nice for Spike to get a role as Twilight's supporting vocals. Cathy Weseluck amazes me.
     
    I did think Rainbow Dash was a bit over the top, but I enjoyed everyone else.
     
    También, sombra = shadow en español, por tu información.
     

  2. Akano

    Snow Days

    By Akano,

    So, today was the second day this week where classes have been cancelled for snow/winter weather. Today the cancellation occurred due to power outages and falling frozen tree branches.
     
    So, I'm now at home doing work since there's actually power here. So that's been fun.
     
    Outside looks especially pretty, though.
     

  5. Akano

    Mwee And Stuff

    By Akano,

    Hey, guys. Just wanted to let you all know that I've made a new topic. Basically, I'm rewriting Inventor of Metru Nui and I've made a new topic of it which you can find here. I'd like feedback on it, since I'm trying to revamp the old story with whatever skills I've gained since writing the original.
     
    In other news, tomorrow is Thanksgiving, and hopefully I'll not be lazy and deliver you a comic by then.
     
    With that, I leave you another lovely Physics equation. ^^
     
    ΣF=ma
     

  6. Akano

    Wow...

    By Akano,

    It's been a while since something like this happened. I was shocked and amazed Friday night.
     
    Our band got a standing ovation.
     
    It was amazing how much better our dance and our spirit was compared to that of last week. The crowd got pumped, they cheered, and we danced. It was amazing. I haven't felt that pumped in a while.
     
    It was the best show for our homecoming game.
     
    Oh, and we won.
     
    Rock on.
     

  7. Akano
    Today, the yellow Toa of Electricity you all know and love takes a step towards a journey he has not encountered before and leaves an old journey behind to be remembered for the rest of his life. I exit the decade of teenage, and enter my twenties.
     
    Today is my twentieth birthday!
     
    And yet, I don't feel too different. I think the change from one age to the next doesn't immediately happen on someone's birthday, but over time takes effect. Or I'll just stay the random jokester I've always been. 8D
     
    Also, today is George Washington's birthday. Our first president deserves more attention than I do, but then again I assume that's what President's Day's all about. Either way, happy birthday, George!
     

  10. 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
     

  11. Akano

    Spirit of Justice

    By Akano,

    Finally, we got an Ace Attorney game with fun Apollo Justice cases! What was once thought impossible has been achieved!
     
    Seriously, though, Spirit of Justice ruled. It definitely made up for Dual Destinies' lack of awesome. Also, the puns were taken up to eleven. No complaints here.
     
    Although Case 4 was VERY out of place. Basically, the only thing worthwhile there was Blackquill's appearance. I didn't really like how they portrayed Athena in that one.
     

  12. Akano
    So, I be back from the untamed lands of the north, and my sleep schedule is messed up. And I have work tomorrow. Fun times.
     
    On the other hand, I saw Legend of Korra yesterday, and OH MY GOSH SEASON FINALE MUST SEE FINAL SHOWDOWN (?) BETWEEN AMON AND KORRA AHH!!!
     
    Also, I got the Mines of Moria LEGO set and have finished building half of it (the cool half with Balin's tomb, the doors, and the well). The rest will wait until tomorrow. I hope to review it (after I review the Vampyre Castle and Gandalf Arrives).
     

  14. Akano

    Origami Shapes!

    By Akano,

    I made a Post-It note dodecahedron:


    It was fun. 8D
     
    For the record: each face is made up of five Post-It notes, each a different color, so that, per face, no color is repeated.
     

  15. 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.
     

  16. Akano

    Welcome To The Herd!

    By Akano,

    I am now officially a brony thanks to KK and Tekulo. Lauren Faust can really make a cartoon series that multiple groups can enjoy while still being targeted at girls.
     
    The fact that Timmy Turner's voice is in the series doesn't hurt either.
     
    My favorite is Fluttershy.
     

  18. 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. 

  20. Akano
    I've decided to post this review here, as I'm not sure this fulfills the requirements to actually post this in the LEGO Sets forum.
     
    Presentation
    From the design of the box to the instruction manual, these are the first things you see before building the set.
     

    ]

    It's a polybag. The front features the set's picture and a picture letting you know that the eagle character you get is Ewar. The back is full of legalese, as per usual, and has a cutout option for a free child ticket to LEGOLAND.
     
    Building
    Half the fun is had building the set. How fun is it to build and how easy or challenging is it?
     



    At only 33 pieces, this set does not take long to build.
     



    Set Design
    Now that the set is complete, we can critique how it looks from every angle. New or interesting pieces can also be examined here.
     



    I would first like to point out that this set came with about 10 extra trans-light-blue round tile pieces, which I thought was ridiculous, but I'm not complaining. The vessel is pretty simple in design, but still rather cool in my opinion. It's like someone combined a snowmobile and a jet ski...and made it fly. The really cool thing about this set, though, is Ewar.
     



    First off, Ewar is an anthropomorphic eagle, which ranks up there in awesomeness with anthropomorphic echidnas. He also has a really simple but clever wing-mount that reminds me of the armor of the old Hydronaut figures. This also allows his wings (and those of the other wingèd Chima races) to be poseable, unlike the winged mummy figures from Pharaoh's Quest. I like Ewar's flip face as well; one side has goggles and the other is just his bare face, which has nice detail on it. The design on his torso reminds me a lot of ancient Egypt, and his helmet (head piece?) is really cool as well.
     
    Playability
    The other half of the fun is in playing with the set. How well does the set function and is it enjoyable to play with?
     

    Woohoo! Cowabunga, dude!


    While it is a small set and, thus, is kinda limited on playability by itself, I'm sure that it makes a nice addition to any larger set if you're building some kind of Chima army or need a new high-tech hover vehicle for your utopian LEGO world. Ewar himself is a cool minifig that also allows for some fun play time.
     

    Hey, Einstein, I'm on your side!


    Final Thoughts
    Once it's all said and done, how does the set stack up? Should I get it?



    Alas, jetski-snowmobile, you are the only one who listens to me.


    This set was originally released as a free promotional with a $75 set order from LEGO S@H. It's now on BrickLink for about $5 USD. I don't know if I would personally spend that much money just to get him, but I'm glad I was able to get him through LEGO's promotional. If you're a fan of Chima, go for it.
     
    Pros
    What's to like?
    Ewar is cool
    Lots of extra pieces
    Jetski-snowmobile
    Cons
    What's not to like?
    Small
    It's a cute little set. If you want to see the gallery, go here when public.
     

  22. Akano

    Sick Again...

    By Akano,

    I swear, this is ridiculous. Yesterday, I felt great, now I'm sitting at home wrapped in a blanket dealing with sore muscles, sore throat, and a headache.
     
    And I'm really tired.
     
    Ooh, but listening to Glenn Beck heals it a bit. ^^
     

  24. 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.
     

  25. 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.
     

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