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Akano's Blog



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Happy Half-Tau Day!

Posted by Akano Toa of Electricity , in Math/Physics Mar 14 2014 · 136 views

Read here to find out why π is wrong! 8D

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My Weekend

Posted by Akano Toa of Electricity , in Life Mar 11 2014 · 96 views
Boston, playin all the bars and 1 more...
So this weekend two of my friends and I journeyed to the New England city known as Boston. We didn't actually do anything downtown, but we did get to see some history in Concord, such as the Old North Bridge (site of the fabled "shot heard 'round the world") and walked past Sleepy Hollow Cemetery (we didn't have time to walk through it, unfortunately), and went to a birthday party for a friend of a friend. We also saw one of my grad school professors who left my school due to his wife and him getting a job somewhere that didn't involve one of them taking a nearly three hour commute to work, and there was Indian food.

All in all, a good weekend with good company. Now I'm just relaxing and recovering from the drive, and will soon get to go home and build a pirate ship.

My body is ready.

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So, I just ordered Metal Beard's Sea Cow...

Posted by Akano Toa of Electricity , in LEGO Mar 06 2014 · 184 views
There goes my, life savings
Worth it!

My wallet may be in a bit of pain, though...

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Equation of the Day #9: Time Dilation and Length Contraction

Posted by Akano Toa of Electricity , in Math/Physics Mar 04 2014 · 145 views
Special Relativity, Einstein
Today I want to talk about something awesome: Special Relativity. It's a theory that was developed by this guy you may have heard of, Albert Einstein, and it's from this theory that arguably the most famous equation in physics, E = mc2, comes from. I'm not going to talk about E = mc2 today (in fact, I've already talked about it, but it's not the whole story!), but I wanted to talk about two other cool consequences of Special Relativity (SR), time dilation and length contraction.

First and foremost, the main fact from which the rest of SR falls out is the fact that the speed of light is the same for all observers moving with constant velocity, regardless of what those velocities may be. Running at 5 m/s? You see light traveling at the same speed as someone traveling 99 % the speed of light.

Wait, how can that be? This idea originally came from Maxwell's equations, which govern electromagnetism. When you solve these equations, you can put them into a form that results in a wave equation, and the speed of those waves is equal to that of light. This finding brought on the realization that light is an electromagnetic wave! But here's the interesting thing: Maxwell's equations do not assume any particular frame of reference, so the speed of the waves governed by Maxwell's equations have the same speed in all reference frames. Thus, it makes sense from an electromagnetic point of view that the speed of light shouldn't depend on how fast someone is traveling!

Now, we're still in a bit of a pickle; if all observers see light traveling at the same speed, how do things other than light move? Think about it. If you're driving down the highway at 60 mph and the car next to you is driving 65 mph, they appear to be moving 5 mph faster than you, don't they? So why doesn't this work with light? If I'm traveling 5 mph, shouldn't I see light moving 5 mph slower than normal? No; the problem here isn't that the speed of light is the same for all observers, but the fact that we think relative velocities add up normally. In fact, this relative velocity addition is simply a very good approximation for objects that are much, much slower than light, but it is not complete.

The answer to this conundrum is that length and time are different for observers traveling at different velocities. These two principles are governed by the equations

 

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

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WIRELESS WIZARD!

Posted by Akano Toa of Electricity , in D&D Feb 25 2014 · 156 views
Fad, Wizard, Lightning Bolt!
Not sure what started this fad, but I'm okay with this.


I Am A:

True Neutral Human Wizard (3rd Level)

 
Ability Scores:
Strength- 9
Dexterity- 12
Constitution- 12
Intelligence- 16
Wisdom- 14
Charisma- 12

Alignment:
True Neutral- A true neutral character does what seems to be a good idea. He doesn't feel strongly one way or the other when it comes to good vs. evil or law vs. chaos. Most true neutral characters exhibit a lack of conviction or bias rather than a commitment to neutrality. Such a character thinks of good as better than evil after all, he would rather have good neighbors and rulers than evil ones. Still, he's not personally committed to upholding good in any abstract or universal way. Some true neutral characters, on the other hand, commit themselves philosophically to neutrality. They see good, evil, law, and chaos as prejudices and dangerous extremes. They advocate the middle way of neutrality as the best, most balanced road in the long run. True neutral is the best alignment you can be because it means you act naturally, without prejudice or compulsion. However, true neutral can be a dangerous alignment when it represents apathy, indifference, and a lack of conviction.

Race:
Humans are the most adaptable of the common races. Short generations and a penchant for migration and conquest have made them physically diverse as well. Humans are often unorthodox in their dress, sporting unusual hairstyles, fanciful clothes, tattoos, and the like.

Class:
Wizards- Wizards are arcane spellcasters who depend on intensive study to create their magic. To wizards, magic is not a talent but a difficult, rewarding art. When they are prepared for battle, wizards can use their spells to devastating effect. When caught by surprise, they are vulnerable. The wizard's strength is her spells, everything else is secondary. She learns new spells as she experiments and grows in experience, and she can also learn them from other wizards. In addition, over time a wizard learns to manipulate her spells so they go farther, work better, or are improved in some other way. A wizard can call a familiar- a small, magical, animal companion that serves her. With a high Intelligence, wizards are capable of casting very high levels of spells.

 
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Physics Conference

Posted by Akano Toa of Electricity , in Math/Physics Feb 24 2014 · 98 views
Physics
As the fates would have it, the day after my birthday I hopped on a plane and went to a physics conference. I'm now sitting here in my lovely hotel room waiting for today's poster session at which I am presenting a poster on my research thus far. It involves stuff from my first published paper and some current "in the works" calculations that I'm doing to help our analysis along.

The talks up to this point have completely left me in the dust, so I'm hoping there will be discussion during the poster session that's more to my level of understanding on the various topics I've been exposed to.

Also the food is quite good.

 

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A quarter century

Posted by Akano Toa of Electricity , in Life Feb 22 2014 · 153 views

That's how old I am now.

Wat.

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Sleep

Posted by Akano Toa of Electricity , in Music Feb 19 2014 · 95 views

When I was in college I had the privilege of performing many beautiful pieces in both choir and band. While I got to sing Eric Whitacre's "Hope, Faith, Life, Love" and play trumpet in his instrumental piece "October," I never did get to sing this beautiful piece.

I absolutely love his suspensions and cluster chords. They give it a real ethereal quality, and it's beautiful.

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Love this theme

Posted by Akano Toa of Electricity , in Music Feb 17 2014 · 104 views
Doctor Who




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Equation of the Day #8: Bessel functions

Posted by Akano Toa of Electricity , in Math/Physics Feb 15 2014 · 159 views
Equation of the Day
My very first Equation of the Day was about the wave equation, a differential equation that governs wave behavior. It doesn't matter whether you have linear waves (sine and cosine functions), cylindrical waves, or spherical waves, the wave equation governs them. Today I will focus on the second, the so-called cylindrical harmonics, or Bessel functions.

A harmonic function is defined as one that satisfies Laplace's equation,

 

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For cylindrical symmetry, the Laplacian (the operator represented by the top-heavy triangle squared) takes the following form:
 

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This is where a neat trick is used. We make an assumption that the amplitude of the wave, denoted here by ψ, can be represented as a product of three separate functions which each only depend on one coordinate. To be more explicit,
 

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This technique is known as "separation of variables."  We claim that the function, ψ, can be separated into a product of functions each with their own unique variable.  The results of this mathematical magic are astounding, since it greatly simplifies the problem at hand.  When you go through the rigamarole of plugging this separated function back in, you get three simpler equations, each with its own variable.
 

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Notice that the partial derivatives have become total derivatives, since these functions only depend on one variable.  These are well-known differential equations in the mathematical world; the Φ function is a linear combination of sin() and cos() (this azimuthal angle, ϕ, goes from 0 to 2π and cycles, so this isn't terribly surprising) with n being an integer, and the Z function is a linear combination of cosh(kz) and sinh(kz), which are the hyperbolic functions.  These equations are not what I want to focus on; what we've really been working so hard to get is the radial equation:
 

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This is Bessel's differential equation. The solutions to this equation are transcendental (meaning that you can't write them as a finite sum of polynomials; the sine and cosine functions are also transcendental). We write them as
 

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The Jn are finite at the origin (J0 is 1 at the origin, all other Jn are 0), and the Yn are singular (undefined) at the origin. They look something like this:
 

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The Jn are much more common to work with because they don't have infinities going on, but the Yn are used when the origin is inaccessible (like a drum head that has a hole cut in the middle). These harmonic functions are used to model (but are not limited to)
  • Vibrational resonances of a circular drum head
  • Radial wave functions for potentials with cylindrical symmetry in quantum mechanics
  • Heat conduction in a cylindrical object
  • Light traveling in a cylindrical waveguide
Note that, while they kinda look sinusoidal, they don't have a set period, so the places where they cross the x-axis are have different intervals and are irrational; thus, they must be computed. This results in some weird harmonic series for instruments like xylophones, drums, timpani, and so on. I got into them because I'm a trumpet player, and the resonances of the surface of the bell of a trumpet are related to the Bessel functions.

There are some cool videos (this one has a strobe effect during it) showing them in action.  There are also some cool Mathematica Demonstrations related to them as well.  There are also orthogonality relationships with them, but I'll save that for another day.

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About Me

Akano Toa of Electricity
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Premier Members
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1,500+ posts
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Name: Akano
Real Name: Forever Shrouded in Mystery :P
Age: 25
Gender: Male
Likes: Science, Math, LEGO, Bionicle, Ponies, Comics, Yellow, Voice Acting
Notable Facts: One of the few Comic Veterans still around
Has been a LEGO fan since ~1996
Bionicle fan from the beginning
Misses the 90's. A lot.
Twitter: @akanotoe

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