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Akano

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  1. Akano
    In continuing my trend of reviewing 90s LEGO System sets, I'm turning my criticism to one of the most beloved, nostalgic LEGO sets arguably of all of LEGO's existence.
     
    Presentation
    From the design of the box to the instruction manual, these are the first things you see before building the set.
     




     
    Ah, the desert. Something about it inspires adventure. Perhaps it's the survival aspect of it. Anyway, the box pulls off the desert background very nicely, and the action of a bunch of cavalry men attempting to catch bandits gives off the Wild West feel. Also, the back of the box has alternate models, which LEGO doesn't do anymore and is something that I personally miss a lot from the newer sets. The large set also has the flap on top which opens to reveal the interior of the box. The plastic/cardboard sheet inside was used to preview minifigs and/or exclusive/unique pieces, but seeing as mine was used, this is not the case here.
     
    Building
    Half the fun is had building the set. How fun is it to build and how easy or challenging is it?
     




     
    This set is rather ridiculous with the space its pieces take up, so I couldn't organize them completely. There are 668 pieces total, which makes for a mild build. I think I built it up within an hour, since the build isn't terribly difficult. Also, I couldn't help but hear Gerudo Valley in my head as I was building. I can't imagine why.
     


    These pieces and the 1×2 and 1×4 log pieces are extremely prominent, in case you couldn't tell.


     
    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.
     
    There are four 16×32 baseplates in this set (no, I'm not kidding), so the build is split up into four parts. First, let's look at the entrance. the entrance:
     


     

    A fort? That's easy! Feel free to follow along with my simple step-by-step instructions. I make building FUN!!


     
    The entrance is pretty...entrancey. The watchtowers are cool, even though (as seen from the back) they have no ladders to lead up to them, so one must wonder how a guard could get up there to look out from there. The doors are barricaded from the inside with a bar made from a 1×10 plate and two 1×2 hinge pieces, as are the other doors on this set. It's pretty cool how they designed that (it is a fort, after all).
     
    Now, the walls:
     




     
    The first wall is rather nondescript. Just a guard wall with a ladder leading up to the top.
     




     
    The second wall has a little more going on. There's a side entrance and a rock wall made with, which I recently found out is called, a Big Ugly Rock Piece (BURP). The cactus at the bottom of the wall hides a cache of coins. This wall also has an actual tower on it to place a sentry in, which is cool. We also have a flag like the one at the entrance adorning the tower.
     
    The final part of the set is headquarters:
     




     
    This is my favorite part of the set, as it actually has something going on other than being a wall. The headquarters reminds me of the Hydeout in the Borscht Bowl Club from the first case of Apollo Justice: Ace Attorney. It has its secrets, like the hiding place in the chimney behind the fireplace.
     


    I don't know what you're talking about; I'm not hiding anything.


     
    Or the spinning table that sends the bandit or cavalry member to the prison/floor below.
     


    You have a full house? I only have a two pair: a pair of ones, and another pair of ones.


     
     
     
    Here, you can see the Colonel and Dewey Cheatum (no, I did not make that name up) playing at the aforementioned table. Rotate the table counter-clockwise, and Dewey ends up doing time for the vast number of card-based crimes he's committed.
     


    Not pictured: The incarceration of whoever came up with the name "Dewey Cheatum."


     
    The room is accessed by its minifig regulars by a set of stairs that can also be found in the LEGO Studios Vampire Crypt, along with others.
     




     
    Now that we've covered the set itself, let's go over the minifigures:
     




     
    There are two factions. Let's first go over the bandits.
     




     
    From left to right, we have Black Bart, Flatfoot Thompson, and (*sigh*) Dewey Cheatum. Of all the figs, I think Dewey's is the coolest. His playing cards style vest and top hat clash a lot with his nasty gold-toothed scowl. It's as though class and hillbilly merged. Flatfoot Tom has a great western villain face, and Black Bart always struck me as the muscle of the gang.
     




     
    The cavalry contains three soldiers, two lieutenants, and the colonel (whose face somewhat resembles that of the LEGO Chess king). Since this set is the headquarters of the cavalry, it contains the largest number of their figures, which is definitely a plus for this set.
     


    I may have shot the sheriff, but I did not shoot this guy.


     
    This set does not contain the sheriff (but Sheriff's Lock-Up does, and I hope to find a good one soon...), but it does contain his deputy. He has a white hat and hair and a sweet cowboy vest. Not much else to say about him; however, he looks to me like he's always chill about everything, even if a gunfight were to start.
     
    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?
     




     
    Other than the doors opening and closing and the table trap that leads to the prison, there isn't really that much functionality with this set, which disappoints me a little bit. However, this set is a great setting for a western adventure, so while there isn't much to do with the set, there is a lot that someone could come up with at the set.
     
    Final Thoughts
    Once it's all said and done, how does the set stack up? Should I get it?
     


    Who shot first, Dewey or the Colonel?


     
    This set, being a very nostalgic set, is quite pricey. If you can find this set for less than $100 and you want this set, grab it. I understand why it's so loved, but I personally wouldn't place it as more important than, say, Pharaoh's Forbidden Ruins or Night Lord's Castle.
     
    Pros
    What's to like?
    Big
    Many minifigs
    Classic theme

    Cons
    What's not to like?
    Price
    Not much functionality

    I hope you enjoyed this one. For the gallery, go here when it becomes public.
     

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

    Fg = - G m1m2/r2


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

    Fg = m a = - G m ME/r2


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

    a = - G ME/r2


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

    E = m c2


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

  3. 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.
     

  4. Akano

    Me playing Pokémon Moon

    By Akano,

    Me: "Hau just handed me a person named Max Potion. I think that's human trafficking."
     
    KK: *bursts into laughter*
     
    Me: *joins in laughter because he didn't expect that joke to land*
     
    Another typical Friday evening.
     

  5. 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.
     

  6. Akano
    While days 2-4 were relatively uneventful (mostly collaborating days, doing research) my last three days in Edinburgh were pretty cool.
     
    Day 5: I climbed Arthur's Seat for a second time; it was sunny, so the view was even better than on Sunday when my advisor and I climbed it together. I also got proof That I made it to the summit. 8D
     
    Day 6: I went to Edinburgh Castle and walked around the former residence of the Scottish royal family. There were a few museums that were mainly dedicated to the history of Scottish military. I also saw the Scottish crown jewels, but was not allowed to take photos (and didn't realize I wasn't allowed until a lady yelled at me for taking out my phone).
     
    Day 7: I journeyed to Pencaitland, which is about 12 miles outside of Edinburgh, for a tour of the Glenkinchie distillery. However, I was late for my booked tour, as the bus didn't take me the whole way to the distillery, and it was a 2 mile walk to the distillery from the bus stop. I literally was walking, in the middle of Scottish farmland, following signs that I hoped were telling me the truth and leading me in the right direction. I eventually got there, however, unharmed but late. The lady at the front desk was very accommodating, though, and fit me into the next tour. It was quite lovely, and we got to see all the steps of brewing whiskey followed by a tasting session. After tasting, I picked up a bottle for my dad and a smaller bottle for me (if they sold 50 cl bottles, they would have been the same size, but, alas, they only had them in 20 and 70 cl sizes). I tried to get on the distillery's shuttle back to Edinburgh, but unfortunately I had asked to join it after it had left; they told me not to worry, though, as there was a local barman who would come and pick me up with two other women who also took the bus and take us to his pub, which was a few seconds' walk from the bus stop. The three of us had a drink at his pub (his only request in exchange for picking us up, which was completely fair in my opinion) and we got to talking. The one lady was around my age and still in college, while the older woman was her aunt and was retired. We got to talking about math and science, since the aunt had studied nutrition science for her job and enjoyed talking about science. We also discussed beer, the tour, and previous and future travels we were planning to take. All in all, it was a lovely afternoon, and we sat together on the bus back to Edinburgh and talked some more. Once we were back in the city, we said our goodbyes and were very glad to have met each other.
     
    Then, this morning, I boarded my flight back to the states, and now I am quite tired, as it is (as I'm typing this) around 1:30 a.m. back in Edinburgh, and my body wants to be very well asleep. I am happy to say, though, that it was a fruitful trip, both for research and fun, as now my code that I've been working on for nearly a year finally works and has reproduced the results of the paper we modeled it off of! Now, we get to push it into new parameter space to aid us in our spectroscopic analysis.
     
    Well, that's all for now. I'm going to go eat some cookies with my friends.
     

  7. Akano

    Sorting

    By Akano,

    So, I decided to take the Pottermore Sorting quiz with all questions available on some third party quiz site, because while I love the Thunderbird, the house's traits do not really fit me (I can be adventurous, but let's face it: I'm a schooler). So, I took the full Pottermore quiz, and my results were somewhat predictable.
     

     
    Indeed, Horned Serpent is my Ilvermorny house by a decent margin, with my second place house being Pukwudgie, then Wampus, and Thunderbird being my least compatible house. (lolololololol)
     

     
    My Hogwarts house, however, is not Ravenclaw, despite my scholarly ways. I'm a Gryffindor. Ravenclaw was indeed my next most compatible, followed by Hufflepuff and finally, by a large margin, Slytherin.
     
    These results make much more sense to me. Remember, kids, larger sample sizes are better.
     

  8. 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.
     

  9. 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!
  10. Akano
    I made it in GIMP. A cardioid is the envelope formed by a set of circles whose centers lie on a circle and which pass through one common point in space. This image shows the circle on which the centers of the circles in the above image lie. A cardioid is also the path traced by a point on a circle which is rolling along the surface of another circle when both circles have the same radius (here is a cool animation of that).
     
    What is the cardioid's significance? Well, it looks like a heart, which is kind of cool. It's also the (2D) pickup pattern of certain microphones (I have a cardioid microphone). If a sound is produced at a given point in space, the pickup pattern shows an equal intensity curve. So, if I place a microphone at the intersection point of all those circles, the outside boundary is where a speaker producing, say, a 440 Hz tone would have to be to be heard at a given intensity. So, the best place to put it would be on the side where the curve is most round (the bottom in this picture) without being too far away from the microphone.
     
    Another interesting fact about the cardioid is that it is the reflection of a parabola through the unit circle (r = 1. Here's what I mean). In polar coordinates, the equation of the above cardioid is given by
     

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

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

  13. Akano

    *Grabs Popcorn*

    By Akano,

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

  14. Akano

    Physics Jokes!

    By Akano,

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

  15. 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.
     

  16. 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.
     

  17. Akano
    I've felt rather unproductive recently on BZP, so I decided to do another set review. This time, we look at the new Monster Fighters theme's largest set (for now), Vampyre Castle!
     
    Presentation
    From the design of the box to the instruction manual, these are the first things you see before building the set.
     

    The trunk dented the upper left corner of the box. Sad face.


     
    The titular castle sits here in front of a sinister, spooky background with bats trailing from the moon and a ghostly green glow emanating from the castle, its inhabitants, and the moon. The back of the box shows the secrets and interior of the castle, allowing the potential buyer to see all the features of the set.
     
    Building
    Half the fun is had building the set. How fun is it to build and how easy or challenging is it?
     

     



     
    This monstrous set (HA! Pun) has seven bags of pieces. For 949 pieces to the set, it's not too surprising that there are so many bags, but still, I think this is the largest set I've built to date (and probably the most expensive one I've ever bought). All the pieces from the seven bags are shown above, and I can tell you that it took a great chunk of time to build. Expect to set aside 2-3 hours to build this (or more, if you're not a seasoned System-building veteran. I took a bit longer, but I was taking pictures as I went. Also, I did take a couple breaks here and there to spend some time with friends ).
     



     
    Although it is a long build, it's really cool to watch as the whole thing comes together, which is one of my favorite experiences in building a LEGO set. While time consuming, I didn't find the build too difficult, but that may be due to having past experience with System sets.
     
    To be honest, the only annoyance for me was the stickers. While they do allow for LEGO to not have to print their designs on the pieces themselves (which honestly probably saves them some money), I am quite particular when applying stickers, and it's annoying when I mess up the first time. The stickers do glow in the dark though, along with many other pieces, so that adds a bit to the cool factor if you like glow-in-the-dark.
     
    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.
     
    Let's start with our heroes and their car:
     



     
    The car is a pretty slick ride. For those of you who like sand green, this is your car. I'm quite the fan of the early 20th century vibe the technology in these sets gives off, and considering that Jack and Rodney's prosthetic limbs are steam-powered, the steampunk vibe is good, too. The car is a one-seater and can accommodate either Jack or Rodney, since the right leg sits on a 2×1 grate, so there are no studs to prevent Rodney from driving it. The most notable feature of the car is the net launcher, which sits in the back, and has a sort of piston that, when pushed, makes the net fly out.
     

    It's like Spiderman's web power, only with a machine and the ability to catch bat people.


     



     
    The heroes included in this set are Jack McHammer and the leader of the Monster Fighters, Dr. Rodney Rathbone. Rodney sports a very dapper vest (with pocket watch) and bowler hat, a steampunk leg, a rapier, and a revolver. Jack has a steampunk arm, a green flannel shirt with a leather vest, a knit cap, and Mjolnir.
     

    I wouldn't doubt it if he took it right out of Thor's hand.


     
    Now, let's dive into the castle and its villainous vampire owners.
     



     
    The castle is centered on the main tower where the Moon-Eclipse-Whatchamajig stands.
     



     
    As you can see in the picture, a hinge with pin pieces is what connects the parts of the castle to each other. The main tower, though, only has connections on one side. The castle is sort of connected in a sideways U or S shape, where the main area is the top branch, the bend is the drawing room/spiky death pit tower, the middle is the gate, and the final bend/line is the prison tower. Only the main/drawing room connection is hinged, though, which allows the castle to have its layout.
     



     
    The inside of the main tower has five rooms; the attic/organ room, the solar eclipse machine room (which is more like a clock tower chamber than a room, like where Basil of Baker Street and Ratigan duel), the drawing room, the bedroom/potions room, and the coffin in the basement. My personal favorite is the potion/bedroom, as the bed is really neat and the potion room contains some cool pieces. I like how in the basement the curtains around the coffin are portrayed by bricks rather than fabric; I think that if they used fabric it wouldn't have the same feel.
     



     
    Also, the moonstone device is really neat to behold. It has a cool mechanism for revealing the moon wheel thing behind the moonstones; the moonstones are all on 90° axle joints all connected by Y-shaped axle holders. When the moon wheel is pushed out, the axles that the moonstones aren't connected to end up between two large cart wheels behind the moon disk, and so when the disk is retracted, the moonstones return to covering the disk, which is pretty cool.
     



     
    The secondary turret is cool due to the cute little map/drawing room that looks comfy-cozy until the floor falls out from under you and you land in a spiky pit of doom. However, if you survive and manage to climb into the attic, you get some treasure! I'm guessing this is the guest room of the castle.
     



     
    The next portion is the gate, which features creepy old school cast iron doors and, if that doesn't prevent you from entering, a secondary gate which has fangs as the blocking mechanism.
     

    Enter if you dare!


     
    You know, to attract the neighbors.
     



     
    The final portion of the set is the prison, where a poor, unfortunate soul has been wasting away for a while, since he's only a skeleton now.
     

    Poor guy, I just want to give him a hug.


     
    However, if either of your heroes get trapped here, you can always bust them out with the nifty lever on the outside wall of the prison.
     

    Behold, my finger's debut on BZP! Greet him with kindness, please.


     
    And now, Lord Vampyre and his wife (does she have a name?).
     



     
    As you can probably tell, LEGO seems to be enjoying making minifigures with printing on the back and front of their bodies, and it seems to be quite prevalent in this theme (as with the latest series minifigures). They also have flip faces, another trend becoming more popular (which debuted in the Studios sets that were released a decade ago. I feel so old...).
     



     
    Also, note that we get another appearance of the Series 4 Mad Scientist's flask in this set, so the piece is becoming more and more widespread. We also get a translucent green wine goblet, several gems, a toad, a mushroom, and six of the seven the moonstones (zombie moonstone not included).
     

    Note: these probably won't evolve your Clefairy, Jigglypuff, Munna, Nidorino, Nidorina, or Skitty.


     
    And last, but certainly not least, the Manbats!
     

    Not to be confused with Batman.


     
    These minifigures are definitely a first for LEGO with their arms that actually spread outward. However, they don't actually move inward/outward, so the poseability of their arms is quite limited without looking rather goofy. They have ears similar to the Elf and Werewolf Series Minifigures, but are a lot bigger and more...batlike.
     
    Also, to give you a sense of how massively huge this set is, I compared its size to the largest Fright Knights set released, Night Lord's Castle, which is the tallest castle set to date that LEGO has released.
     

    I sense a war between Vampyre and Bat Lord brewing.


     
    It's pretty incredible how far LEGO has come as far as detailed sets go, since you can really see the difference between the present day and the early 90s. But notice that Night Lord's Castle just barely keeps its title; in fact, the only reason it's taller than the Vampyre Castle is due to Willa's telescope at the top. Also note that Night Lord's Castle contains only 598 pieces in comparison to the Vampyre Castle's 949; a lot of those additional pieces go into the detail.
     
    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?
     

    Have at you!


     
    There are definitely a lot of possibilities with this set. It has several rooms which make good vignettes/settings for spooky interactions between your heroes and villains, and the appearance is right out of Halloweentown (of Nightmare Before Christmas). Along with the prison wall lever, the prisoner can be moved up and down with an axle and the moonstone device can be engaged with a similar sort of piston that the car's net launcher has. Also, the trapdoor to the spiky pit makes for some good fun.
     
    Final Thoughts
    Once it's all said and done, how does the set stack up? Should I get it?
     

    Let me play you the song of my people.


     
    This set is pretty darn cool. It gives off a great spooky aura and is perfect for anyone who is a fan of Halloween, castles, and/or the Monster Fighters theme. It supplies us with a good cast of minifigs and is definitely one of the best sets released this year. If you can afford the $99.99 price tag, pick this up; you'll be glad you did.
     
    Pros
    What's to like?

    Hugely gigantic castle
    Cool minifigures
    Cool car
    Awesomely spooky
    Some neat pieces to grab

    Cons
    What's not to like?

    The price tag

    I hope you all enjoyed my take on this set. If there are any pictures you want me to take or any aspect of the review/set you want my to clarify, just ask.
     

  18. Akano

    This Day

    By Akano,

    I've decided that This Day Aria is my favorite song from A Canterlot Wedding. Its juxtaposition of Cadance singing about her love for Shining Armor with Chrysalis singing about her callous indifference and plot to use him is amazing.
     
    Also, it's (at least partially) a villain song, and those tend to be awesome.
     

  20. Akano

    Professor Akano

    By Akano,

    Over the past couple of months I visited and interviewed at a college interested in hiring someone to teach one semester of courses for their physics major series. My name was given to them through the grapevine and, after giving a mock lecture, presenting my research over lunch, and meeting with the department, I got the job. Starting later next month, I will officially be Professor Akano.
     
    I'm excited.
     

  21. 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.
     

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

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