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Akano

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Blog Entries posted by Akano

  1. Akano
    Today, we take a journey to Middle Earth and take a look at the LEGO reimagining of the Mines of Moria.
     
    Presentation
    From the design of the box to the instruction manual, these are the first things you see before building the set.
     



     
    I really like the LEGO LOTR logo with the one ring on Sauron's finger in the upper right corner along with the map of Middle Earth in the background. The back highlights various parts of the set. I do like their cave-like choice of background for the front picture.
     
    Building
    Half the fun is had building the set. How fun is it to build and how easy or challenging is it?
     



     
    This set comes with four bags of pieces which build the well & Balin's tomb, the wooden doors, the center of the wall, and the outsides of the wall. The build took me about 2-3 hours or so (not counting distractions). I must admit that at times I jumped the gun with some of the smaller pieces and thought I had run out of pieces, meaning that I was frantically searching my pile of pieces for I piece I simply misplaced during the build, so follow instructions carefully.
     



     
    Again, this set does come with stickers (grr!) for some of the detail, such as the dwarves' record and Balin's tomb.
     
    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 totally forgot the cave troll in this pic, sorry.


     
    The set consists of three main parts: the wall, the door, and the tomb and well.
     



     
    The door has some pretty good detail, but some of that is provided by stickers with the wood plank patterns. They also included the axe that locks the doors shut for about 10 seconds from the movie, and this axe has a metallic copper blade, which I thought was cool. I also like the use of spyglass pieces as torches. Looking from the back, the wall on the right has a lever that makes it fall.
     



     
    I like the detail of the cave troll; however, he has absolutely awful balance while holding his hammer, so the number of poses he can have while wielding it are limited. He has a tendency of falling forward, since his mold is rather forward-heavy, even when not wielding the hammer. If you have studs beneath his feet, then he should be fine, but he's rather unstable just standing on a table.
     



     
    The wall consists of bookshelves with some of the oldest books stickers can convey. There is also some of the dwarves' treasure underneath the floor in its own little nook. For some reason the treasure chest moves in and out.
     



     
    The sides of the wall have towers that can be toppled using levers underneath them. The box shows one of these landing on the troll.
     



     
    These are my favorite parts of this set. I don't know if it's the fact that they're small and simple or if it's because these are the parts most focused on in the movie, but I love the well and the tomb. The tomb actually opens to reveal Balin's skeletal remains.
     

    Shouldn't Balin's skeleton be dwarf-sized?


     
    My only qualm with the well would be that the skeleton does not have any armor/helmet on it unlike the movie. Otherwise, I love this part of the set.
     
    Now, onto the minifigures:
     



     
    The heroes are very true to their movie selves save probably for Pippin; I don't really see Billy Boyd when I look at him; in fact, Pippin's face isn't too different from Shire Frodo's face. My favorite is probably Gimli,since the detail in his face, beard, and helmet are amazing. Boromir's appearance is probably second in terms of movie accuracy.
     



     
    the goblins are pretty creepy looking, and while I'd say their faces are pretty good, they seem to have too much hair (but maybe I'm remembering the movie wrong). I do like the shield that the one carries, as it's very jagged and looks goblin-made.
     

     



     
    All the characters have back printing on their torsos, which is more impressive on some than others. They also all have flip faces, which is amusing for Gimli, since all that changes when his beard is on is that his eyebrows go from neutral to angry. Also, they tried to make Legolas look intimidating, which just never happens in the movies.
     



     
    The hair/helmets of the characters all seem to be new molds to match the movies. I particularly like Gimli's helmet, which again has a lot of detail put into it (including printing).
     



     
    As far as interesting pieces go, there are quite a few jewels in this set. You also get a set of weapons that came originally in the Heroica games which I found interesting. They're mainly there to litter the ground of the tomb, probably to serve as the weapons of the dwarves who once populated the halls. You also get a bone and a rum bottle that originated the Pirates of the Carribean sets.
     
    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?
     

    For the Shire!


     
    This set has quite a bit of play to get out of it, since you can basically reenact the battle in the tomb with it. It's also makes for good scenery for any type of underground adventure you'd like your characters to embark on, so there's a good deal of potential. Any Lord of the Rings fan, of course, will just want to spend hours having our heroes take out that cave troll.
     
    Final Thoughts
    Once it's all said and done, how does the set stack up? Should I get it?
     



     
    For any fan of the movies this set is a good pick, since it is, I think, one of the more accurate depictions of one of the movie scenes. All the minifigures in this set, save for the skeletons and Gimli, are exclusives, so it's a must-have for those who want, well, at least half of the Fellowship.
     
    Pros
    What's to like?

    Great detail
    Very true to the film
    The minifigures
    Cool pieces

    Cons
    What's not to like?

    The price tag
    Cave troll is a bit unbalanced

    I hope all the LOTR fans who are reading enjoy this review. For the gallery, go here when it becomes public.
     

  2. 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.
     

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



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

  4. Akano

    WIRELESS WIZARD!

    By Akano,

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

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


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

  5. Akano
    KK and I had an awesome time. I'll have pics posted later, so for now I'll just use a few words to describe my experience:
     
    There were so many AMAZING creations, from a 7 foot long Serenity to some awesome Egyptian temple structures to Erebor from The Hobbit, and that doesn't even scratch the surface of the tip of the iceberg. The BIONICLE creations were really neat, and I enjoyed stopping by those tables. Speaking of, I have now officially met the following BZPeople (yes, that's what I call us/you/whatever) IN PERSON:
    Sumiki a.k.a. Cap'n Subnuki a.k.a. Hat Man
    Black Six a.k.a. Andrew a.k.a. you all know who he is
    The Architect a.k.a. Blog Lurker
    Xaeraz a.k.a. Xaeraz
    Aanchir a.k.a. He who draws Andrew Francis characters
    BZPeople whom I saw but did not actually meet (sorry, I'll try harder next time not to be overwhelmed with ALL THE AWESOME and stop to talk to people more).
    Makaru
    Avohkah Tamer
    Bionicle Raptor (related: I also forgot to buy a BZPower shirt. )
    Possibly others whose appearances I don't know and thus would not recognize them.
    All in all, it was an awesome afternoon well spent. I'll have some pictures up some time later this week, including my haul (not too substantial, but pretty cool).
     

  6. Akano
    This week marks the last week of the first half of the semester, which means that my half-semester Optics course is coming to a close. It's been busy (hence my lack of presence), but I almost have one full course under my belt. So far there have been two lessons I gave that could have been better presented, but the students have been fairly engaged and seem more or less okay with the material. I guess we'll see how it goes once the final happens.
     
    I did try writing a new Equation of the Day, but my schedule has been quite packed. I would like to get that rolling again after the end of the semester, since I greatly enjoy writing those entries.
     

  7. 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.
     

  8. Akano
    For those of you who aren't Glenn Beck fans, the title is actually a term of endearment.
     
    The reason for the Glenn Beck inside joke? I was at his Insiders Convention in Akron, Ohio this past Saturday, and I had a great time. I got to shake Glenn's hand, got a picture with him, and he talked with my mom and spoke to my dad on the phone due to some very humorous circumstances. Unfortunately, I did not inherit my mother's socializing genes, so I didn't say much. >>;
     
    Also, it appears my poor Lappy's hard disk drive has kicked the bucket. Fortunately, I backed up all my necessary files onto my desktop computer and I'll be getting a hard drive soon when my dad returns home. Until then, it's me and my desktop.
     
    Also, I am going to finally make a new comic. I have the idea and I'm ready to get started on it right now.
     
    Until next time, my fellow BZP brothers, Awwwwww Yeeeeeeeaaaaaaah….
     

  13. 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.
     

  14. Akano

    Complex Numbers

    By Akano,

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

    eiϕ = cosϕ + isinϕ


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

    eiπ + 1 = 0


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

  15. 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.
     

  16. Akano

    Recent updates

    By Akano,

    Due to the combination of the servers being taken offline and me being in school, it's been a while since my last entry. I managed to find most of my entries from Google's cache (Aug 26 - now), so they've been posted as though I had posted them on their original entry days, but the comments have disappeared into the aether.
     
    On a more jubilant note, I've been playing Pokémon Y and Ace Attorney: Dual Destinies recently (this is in no way like Tekulo's recent entry). I'm on the first part of case three of the latter, and I have to say that case two was very enjoyable. The twist was really good, and all the crazy weirdness of the people in that case now makes sense (it kind of reminds me of Turnabout Big Top from Justice for All).
     
    In Pokémon Y, I have gotten through Victory Road and am now ready to face the Pokémon League:
     







    Also, I think it's fantastic that I can customize my character with a yellow hoodie and well-matching fedora.
     



     
    My party currently consists of Keaton (Delphox), Anubis (Lucario), Lapras, Sylveon, Snorlax, and my HM user (a linear combination of Machoke, Talonflame, Pansage in near equal probability). I absolutely love the fact that the Fennekin family is based on mages, and Sylveon is absolutely fantastic. Since evolving it, I don't think I've had it faint once.
     

  17. 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. 

  18. Akano
    I am currently unable to upload photos to Brickshelf, and I'm not sure why. I'm trying to upload a .zip file of pics I want to use for a review, but after a long time of "uploading," the folder I attempt to upload to is empty. I tried doing one photo at a time, but that's failing as well.
     
    Anyone else experiencing this?
     

  20. Akano

    Halloween approacheth

    By Akano,

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

      

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

  21. 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.
     

  22. Akano

    Allons-y!

    By Akano,

    So, I may have just finished watching all six series of Doctor Who available on Netflix.
     
    And I'm quite glad I did. Awesome show. David Tennant is best pony Doctor.
     
    I want a sonic screwdriver.
     

  23. Akano

    A True, True Friend

    By Akano,

    Confound these ponies!
     
    They drive me to sing.
     
    I liked the finale, but I kinda wish it were a two-parter. It felt slightly rushed as one episode.
     
    Also, SONG OVERLOAD.
     

  24. Akano

    Golden Sun

    By Akano,

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

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