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Akano's Blog



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Equation of the Day #14: Chaos

Posted by Akano , in Math/Physics Nov 08 2015 · 286 views
Chaos, deterministic, PRNG and 1 more...
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

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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

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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?

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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.

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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.

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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.

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Halloween approacheth

Posted by Akano , in Video Games, Life Oct 28 2015 · 241 views
Halloween, Autumn, Fall, Mario and 1 more...
'Sup, BZPeoples?

I've been away a while attending friends' weddings, visiting various peeps, rereading Harry Potter (still as magical as always), and working on research. If all goes well, this Friday will see me submitting a paper to the Journal of Chemical Physics. :) All in all, it's been a good few months. But now is the season of autumnal haunts, and I have grown very happy with the drop in temperature and the swap of palette.


Over my fall break, I visited KK in the land of cheese, Wisconsin, and he introduced me to the joys of Punch Out!! for Wii and Super Mario Maker. I now own both, and have been having loads of fun playing others' levels and designing levels of my own (mostly goofing off). When I've finalized a level I'll post it for you all to play if you like. ^_^ In particular I've been working on a Ghost House.

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It's been super fun. Also, RubberRoss' levels are utterly evil.

EDIT: Untimely Haunt v2: CD1D-0000-00C8-8D19

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A Brief History of Everything

Posted by Akano , in Math/Physics Jul 20 2015 · 195 views

feat. Neil deGrasse Tyson.



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I don't have a problem

Posted by Akano , in Video Games Jul 18 2015 · 225 views
Consoles, b/w, Zelda
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I don't know what you're talking about.

(P.S. Stable 3D is the best thing ever. Totally worth the upgrade.)

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Monthly update

Posted by Akano , in Life May 14 2015 · 252 views

Haldo, everypeoples!

I just wanted to check in (as I haven't posted in well over a month). The semester is wrapping up pretty well. I've started writing a paper to published this summer (!) and have been deep in spectrum assignment land. I also have some friends getting married at the end of the month and will be playing trumpet for their wedding with two of my friends. ^_^ After that happens, I'm going to a conference to give a talk on my research (a first!).

So, yeah, a lot of exciting stuff coming up. Basically, pretty busy up until mid-June.

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Bill Cipher AMA happened

Posted by Akano , in Gravity Falls Apr 02 2015 · 469 views

So, the Bill Cipher AMA happened on reddit yesterday, and there were lots of funny and interesting revelations. Things I found noteworthy:

Spoiler


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Source: Gravity Falls Topic

Discuss! 8D

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My morning

Posted by Akano , in Life Mar 23 2015 · 322 views

I'd like to describe to you all the most memorable scene from my morning walk into school.

I emerged from the tunnel under the train tracks and headed towards the last intersection on my walk on my way to school. In the bare bushes of one of the office buildings on the street twittered a bunch of small sparrows, not a care in the world. I watched them as they chattered, when all of a sudden a brown blur appeared in my periphery. A hawk (I believe a broad-winged hawk) was in a dive, ready to snatch up a meal from the group of sparrows. It swooped, then crashed into the bare branches of the bushes. I could tell immediately after it crashed that it was unsuccessful in its quest, and as it looked around during its shocked stupor, I could see in its face the expression of "Well, this is just great." The poor thing sat there for a moment and eventually freed itself, its ego more bruised than its body, as it flew atop the building and reconsidered what its next move was.


All in all, it was really cool, and I'm glad I got to witness it. :)

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From: Gravity Falls Topic

Posted by Akano , in Gravity Falls Feb 23 2015 · 236 views
Blatant ad, Gravity Falls, 8D

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I noticed that the old topic for this awesome show was past the revival limit, so I thought I'd try to reboot discussion. Monday was the most recent episode, Northwest Mansion Mystery, and the next episode, Not What He Seems, will air on March 9.

So, let the discussion begin!

As for author theories,

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Also, the Teaser for "Not What He Seems" looks very promising. 8D

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Source: Gravity Falls Topic

I made a Gravity Falls discussion topic in Completely Off Topic. Feel free to fuel the discussion over there if you're a fan of the awesome Disney show. :D

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Awesome Cartoons

Posted by Akano , in Life Feb 16 2015 · 319 views
OverTheGardenWall, Cartoons rule and 1 more...
As brothers tend to do, KK and I spent our Saturday evening watching a cartoon about brothers navigating an unfamiliar world simply trying to get home. Over the Garden Wall is a brilliant miniseries, and if you haven't seen it, you should. The story is captivating, mysterious, and makes you ask all sorts of questions to the very end. The animation is beautiful (similar but different to Gravity Falls), the characters are incredibly fun and interesting, and the plot doesn't necessarily progress the way you think it will, but at the same time has many familiar story elements. Also, the celebrity voice acting is done really well, which isn't always the case when you have big names in your cartoon credits.

Go watch it. It's a great ride.

Also, slightly related, new Gravity Falls tonight! 8D

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It's a super special day that comes around every year...

Posted by Akano , in Life Feb 14 2015 · 194 views

It's Tekulo's birthday! I made him a thing. Go wish him well over there, if you like. *poke* *prod*

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Oak Log Bans

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About Me

Akano Toa of Electricity
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Premier Members
Stone Champion Nuva
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1,500+ posts
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+2 for Premier Membership
+1 from Pohuaki for reporting various things in Artwork


Name: Akano
Real Name: Forever Shrouded in Mystery :P
Age: 27
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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Hieroglyphs And The Like

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