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Akano

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Blog Entries posted by Akano

  3. 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.
     

  4. Akano

    *Sweeps up the snow*

    By Akano,

    Hello, BZPeople! I just wanted to drop in and let you know that I'm quite alive and well and have had a wonderful holiday season. I thought I'd give you all a quick rundown of my last month or so after I vanished without a trace.
    Went home and celebrated Christmas with my folks. Got AN AWESOME INSECT COLLECTION from Tekulo; a custom lightsaber, a LEGO coffee mug, and a Knights minifigure set from KK; a portable charger thing from my mom; an Elvis mug/cocoa mix from my aunt; and a gift box from my dad which had a Bazinga shirt and socks, TARDIS/Dalek ice tray (!!), and other miscellaneous nerdy things.
    Spent New Year's with some friends from college, one of whom is having a baby in the extremely near future. (I will soon be an uncle, which is kind of awesome.)
    Went to Walt Disney World for a week with my mom and Tekulo. Magic Kingdom is still my favorite park, Haunted Mansion is still my favorite ride, Space Mountain is still my favorite thrill ride, and Tower of Terror is still awesome. I didn't really enjoy Epcot all that much (though I had fun riding Spaceship Earth after virtually zero wait), and Universal Studios was fine except for the head-jostling the Rockin' Roller Coaster gave me. (I remember it being much more avoidable as a kid...)
    Now I'm back at school doing research. Also, thanks to all who supported Tekulo in his recent blog entry where he came out. It makes me very happy to see that here. Keep being awesome.
     

  5. 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
     

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



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



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



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



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

  7. Akano

    Going home for turkey

    By Akano,

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

  9. Akano
    KK got me sick. Jerk.
     
    In other news, MAJORA'S MASK 3D CONFIRMED XIKQADRVVGSJUPOLWTW!!!
     
    I'm pretty psyched about this development and am throwing my money at the screen to make it be mine now. It's not working.
     
    Now, please excuse me while I hack up my left lung.
     

  11. Akano

    On Vacation

    By Akano,

    Nearly two weeks ago, my brothers, parents, roommate (whom Tekulo has taken to call Friend: Toa of Friendship, so I will refer to him as such), and two family friends took a relaxing trip down to Hilton Head Island, SC. Our first day in the condo wasn't terribly eventful; we all went to the beach and enjoyed the sun and sea. Tekulo built a rockin' sand castle, with a moat, staircases, and other hidden gems, and our dad got pooed on by a crow/raven/corvid. Twice. Also, KK decided to take advantage of my skin condition and scratch "Hi" on my back, which wasn't too bad due to the cool ocean keeping the itchiness at bay.
     
    Day 2 was fairly laid back. The youngins (my brothers, Friend, and I) stayed at the condo for the most part and partook in various stay-in type activities whilst the older folks (parents and family friends) went out and did adult things. Those who weren't allergic to shellfish had a dinner of shrimp, scallops, and the like; the two of us who were allergic (including me) had steak, because that's ocean food.
     
    Day 3 we ended up eating dinner at a restaurant known for its awesome seafood, and while you would think that limited my choices (it did), I was able to have sashimi tuna with avocado and peanut sauce. It was by far the best meal I had there, and it still haunts me.
     
    On Day 4, the elders went to Savannah, GA to tour the city and see the sights, while we youngins did awesome stuff. Our morning featured parasailing thanks to Tekulo's reservation-making skills. Friend: Toa of Friendship went up with KK, while Tekulo and I paired up for our flight. It was a real thrill and surprisingly quiet when you get a few hundred feet above the boat. The motor was a faint hum in the distance, and the view of the bay was fantastic. I would definitely recommend it to those who have never done it before, even if you have a fear of heights (like me). After returning to shore, we found out the same company offered kayaking to a wildlife preserve marsh nearby, so we reserved that for the afternoon while checking out Harbour Town in the meantime. We climbed up the lighthouse, grabbed a bite to eat, and then met up for a kayaking adventure into marshland. It was a fun little tour, filled with the floating excretions of a manatee (I may have glimpsed its nostrils!), herons and egrets, hermit crabs, and dogs barking at us from nearby villas. The park was very quiet and serene, and it was nice to be near nature. Once summer rolls around in the north I'd like to do some more kayaking/canoeing activities. After we returned from our marshland adventure (in fairly windy, rocky waters), we had dinner at a local pub, then went on to play 18 holes of minigolf. Friend: Toa of Friendship won with just two over par, and I claimed second place with Tekulo and KK trailing behind us.
     
    Day 5 was the youngins' return to Ohio in which Tekulo read The Promise, KK played Fire Emblem, and my 3DS's battery died from playing A Link Between Worlds.
     
    All in all, the trip was lots of fun, but it was nice to finally get back home after two weekends of constant traveling.
     
    WLEI DPSFS ASFS WLLOQWDIFE WD DPS UIJTI UIKHLSZ. DPSY ASFS UIIL WJT MSHD DI DPSKESLBSE.
     

  12. Akano

    *Slow clap*

    By Akano,

    Well done, Alex Hirsch. Well done.
     
    The latest Gravity Falls episode was excellent. Totally squashed theories about the author of the journal and, while answering some questions, raised even more. Can't wait for the next episode...
     

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

  14. Akano
    I have been up to lots of stuffs recently. Mostly of the electronic gaming variety.
     
    First off, I played Mega Man X for the first time courtesy of the Wii U Virtual Console (prompted by a fantastic video by Egoraptor). Fantastic game; the more I play SNES games, the more I regret not owning a SNES in childhood.
     
    After playing Fire Emblem (Rekka no Ken) and Sacred Stones, I finally caved into my roommate's demands that I play Awakening; OMG SO MUCH AWESOME! Probably one of my favorite games of all time, and definitely looking forward to playing it again.
     
    Right after completing Awakening, I received Professor Layton vs. Phoenix Wright: Ace Attorney in the mail. Another fantastic game; Layton puzzles and story combined with Phoenix Wright courtroom shenanigans made for an awesome crossover. I just wish Maya hadn't been given a valley girl accent. :/
     
    And, finally, my roommate got some 3DS Smash Bros codes, one of which he shared with me. I'm loving Mega Man so far; will definitely be playing as him once the Wii U version comes out (not getting the 3DS version; I'd like to spare my buttons of a painful death). I really wish the demo included Robin as a playable character, though; of the new roster, I'm looking forward to playing as him the most.
     
    On a more academic note, this Thursday is my Ph.D. preliminary exam oral defense, so that'll be fun. I've already worked through the problems that I will be asked about, and I think I've solved all of them. Hopefully all will go well.
     

  15. Akano
    Haldo, BZPeople!
     
    As you may have noticed, this summer the blogs were highly accented with my extreme absence since I took a trip to the northern UK. The reason for this absence is simple: I was studying for, and subsequently started taking, my preliminary examinations for continuation to my Ph.D. in physics. Last week for me was filled with three four-hour exams: one on Monday (Quantum Mechanics), one on Wednesday (Electromagnetism), and one on Friday (Classical, Special Relativity, and Statistical Mechanics grab bag). I am happy to say that I survived the initial onslaught of my prelims and am now in phase two: a twelve-hour take-home exam to be done over seventy-two hours (10 a.m. today to 10 a.m. Thursday). I have glanced at the problems but not worked on them yet, but I feel pretty good about the quantum section.
     
    In other news, the new school year is set to start in a week, but I have a class starting this Thursday. That'll be quite fun. 8D
     

  16. 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.
     

  17. Akano

    I'm in Scotland

    By Akano,

    So, I'm currently in Edinburgh, UK for official Hogwarts business doing research with a collaborator at the local university. Things I have done include
    Vaguely losing consciousness on the plane ride over the Atlantic to adjust to new time zone. Would not qualify it as sleeping.
    Instead of checking into the hotel (which didn't allow checkins until 2 p.m. local time), climbing Arthur's Seat to the summit with my advisor like a boss.
    Enjoyed a Guinness. (Not Scottish, I know, but arguably fresher than those sold in the US.)
    Things I have not yet done include
    Tossing a caber.
    Wearing a kilt.
    Playing bagpipes.
    Trying haggis. (Will probably do this at breakfast tomorrow, though.)
    So far so fun. Also I've done research. Totally why I'm here.
     

  18. Akano
    It's been a while since I posted something, eh?
     
    Last week I went home to visit my family for a couple of weeks. It's been pretty low key so far; visited some friends over the weekend, mowed the lawn (so lovely), and tried to get things a bit cleaner around the house. I also ate way too much food at various gatherings.
     
    This weekend my dad and I pick up KK from grad school (he successfully defended his Master's!) and will enjoy two long, arduous, 10 hour-long trips. Then, I fly to Wisconsin for a conference, so that should be fun.
     
    I also got the Back to the Future DeLorean LEGO set, and I must say it's a very nice representation. Also, Doc Brown's "Great Scott!" face is perfect.
     
    I really want to make the purple skateboard in that set an actual hoverboard using magnets and a superconductor. One of these days...
     

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

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




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



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

  21. 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.
     

  22. 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.
     

  25. Akano

    Together, We Ride

    By Akano,

    So, my roommate finally coaxed me into playing the first Fire Emblem (actually the seventh, but the first one released in the US) for GBA, and I'm loving it. After finishing it, I'm definitely looking forward to playing Fire Emblem: Awakening, because I've heard nothing but good things about it.
     
    The music is great in that awesome nostalgic way, the characters are fun and memorable, the magic wielders are freakin' awesome, and the gameplay is fantastic – in order to keep all your troops alive, obtain everything worth getting in each level, recruiting all new troops, and beating the levels is a fun challenge.
     
    Other things I've done in the past week or so include:
    Finishing MetalBeard's Sea Cow (Awesome!)
    Learning how to derive the formula for the volume of an n-dimensional sphere (really clever trick!)
    Drinking lots of tea.


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