Andromeda, Topology, and SMBC

A few things. First off, I saw Andromeda for the first time ever last weekend, on the 10th! I was at Big Pine Key and… ahh, it was so beautiful! Two and a half million year old light… it’s rather sexy, actually. And now that I know exactly where to look and what to look for, I’m able to see it from my house! Oh, it’s so beautiful. It’s a whole other galaxy. Looking at hundreds of millions of stars in about one square degree of sky… all that light traveling 2,538,000 light years, only to end that ages long journey on my retina… it’s really sexy to think about. Alright, I might be just the tiniest bit turned on by space. And by “tiniest bit” I mean quite a bit, as in, a lot.

On a random note, if I ever get the chance to name a chocolate bar, I’m going to call it Andromeda for one simple reason. There’s already a “Milky Way” chocolate. There must be an Andromeda. “Large Magellanic Cloud” doesn’t sound delicious… well, to me it does, but it doesn’t really sound like a chocolate bar. It’d be appropriate for cotton candy though.Oh great, now I’m thinking about galaxy names and what kind of candy it would be. “Arp 87” doesn’t really sound like any sort of candy. xD Now to look at the Hubble Deep Field again because it’s so beautiful… and then there’s that picture of Andromeda… mmm…

Second semester is going to start… today, really. I gotta really work on my grades now. Like, really work on them. But now I don’t have college apps to stress me out. It’s also time to put this lighting to the test. Hopefully all should work out well and hopefully my dysfunctional family doesn’t get in the way again. My dad’s a little verbally abusive… but he’s away now, and hopefully he’ll stay away for a long time now so that I can get a little peace of mind…

This reminds me. No one gave me the room number of my Government teacher. I have to go to her class today, and apparently the school never gave the students their schedule changes. Ugh, yet another reason why my high school is… unfavorable… I can’t wait until I’m in a more organized college environment.

I’ve also taken up teaching myself topology. A professor friend of mine (he won’t tell me what college he’s a professor at, but I know he’s from Boston–not MIT though, I don’t think) has given me a copy of Armstrong (1983) Basic Topology, and it’s surprisingly readable. I’m working my way through it. Of course, it being a third-year topology textbook, and having taught myself only the basics of Set Theory, the book is not easy, so to speak, but it is definitely manageable. After getting through some of it, I’ll try and watch some lectures on YouTube. I’m sure there’s an OpenCourse one, but I want to see the other ones too and see which one I like the best. Virtual teacher quality matters, too!

On a side note, if you combine xkcd and SMBC together, that’s 95% me. There’s even a post that describes my “talk mathy to me” thing so accurately: http://www.smbc-comics.com/?id=2934#comic

Ahhh…

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Andromeda, Topology, and SMBC

All Infinities Are Big, But Some Infinities Are Bigger Than Others

Someone on EP asked the following question:

If you combine two infinite continuums to make an infinite plane, would the number of points be infinity squared?

Of course, the site’s math nerd HAD to answer this question! And my response got rather long.  Not infinitely long, though!


Continue reading “All Infinities Are Big, But Some Infinities Are Bigger Than Others”

All Infinities Are Big, But Some Infinities Are Bigger Than Others

Optimization Problems [Paper Box Folding]

As ExperienceProject’s math tutor, if it’s Algebra 1 and beyond, and if I’m able to understand it, or if it’s in my capability to learn what’s being asked of me, I shall teach it. Although, as much as I understand these types of problems, infallibly I do poorly on the tests. It’s not because I forget what to do… I usually run out of time because I’m very OCD about the details.

Someone asked me to help them with optimization problems earlier. At the time, I didn’t know how to do them. Turns out a few days later, that’s what was taught in my class. I’m assuming he’s in AP Calc AB too. He might have been a few days ahead of me in the curriculum. I guess the only good thing about Common Core is that everyone’s on the same page… literally, in a way, give or take a few days. The rest of this Common Core stuff is just Complete Crap. But a rant about how much I hate Common Core is for another post. For now, I’ll just post my explanation about optimization problems here, for anyone that needs help in them.

As said to him, they’re awfully easy once you get the hang of them. Really, there’s a higher chance that any error you make is algebraic rather than in the calculus itself.

Here’s how to do optimization problems:
Continue reading “Optimization Problems [Paper Box Folding]”

Optimization Problems [Paper Box Folding]

AS3 and Math Competitions

I finally have the means to learn ActionScript. I seem to be picking it up very quickly. I will soon have an image render of my now-theoretical phi fractal to compile a mathematical thesis paper with, as well as of course the flash program itself, in which you’ll be able to input an iteration depth and degree of accuracy to which the function is calculated to.

Also, as this is a blog that’s generally about the more academic side of me, I should mention that Mu Alpha Theta has announced that we will be doing two math competitions this year. One of them–the upcoming one–I forget the name of, and the later one is AMC. Finally! I’ve been wanting to do this for ages and ages! Ever since MathCounts in middle school!

AS3 and Math Competitions

Think

This post is very recent.
All I can think of saying is that I just let things go in this one, and just went on and on and on, because it was, after all, a stream of consciousness to try and explain all that goes on in my head (and what better way to do that than stream of consciousness).

Original Post Date: October 11, 2014
Original Post: http://bit.ly/1vjqEAz


Sometimes I’ll get a reply to a post, which I reply back with what starts out as a long comment, then ends up being too long to post as a comment, so I decide to post it as an experience, and allow it to be as long as it turns out to be.

When I do that, I’m essentially typing up a stream-of-consciousness, where I type whatever comes to my mind at the time, even if it seems like I’m jumping from topic to topic, and then eventually reach a conclusion that sums up everything I’ve mentioned through all the verbosity and redundancy and inconsistency of the text just to make it appear consistent while having definite deliberation as does the rest of the passage.

This is perhaps the longest passage I’ve ever written for EP, and I’d particularly like you to pay attention to the very last paragraph at the least, if this is TL;DR for you.
Continue reading “Think”

Think

Nullum Magnum Ingenium Sine Mixtura Dementiae Fuit

Because why not?

There is no great genius without some touch of madness!

I fail at pulling off the Einstein face.
Very soon you’ll find out what and all is scrawled onto the board behind me. It has to do with the Fibonacci iterator and the Euclidean Plane. 😀

E/c²=M. √-1=I. PV/nR=T. It's funny how I was talking about MIT to a classmate of mine today, and she interrupts me mid-sentence, asking me what my shirt meant. It took her 2 minutes to realize what it meant...
I tried to do the Einstein look. I have the hair. I have the tongue. I have the math. But I still can’t pull it off. On a side note, my mathematical theory is written on the board behind me.

My gawd I have a Miley Cyrus tongue…

Image

The Beauty of Chaos and Mathematics

Original Post Date: Jun 20, 2014
Original Post: http://bit.ly/1DjhRCZ

I don’t know what else to add. This is one of my posts where I pretty much have my own ideas. Well, I guess I should have mentioned the Three-Body-Problem, and more about Godel’s Incompleteness Theorem…


The Beauty of Chaos and Mathematics

Mathematics is not itself about numbers and symbols. Those numbers and symbols are a part of the grand metaphor for the absolute truth of what is mathematics.

Given two particles of identical mass that only but gravitationally interact with each other, it seems obvious how they’d mathematically interact. The force that draws them together is simply Newton’s gravitational constant times both of their masses divided by the square of the distance between them. It’s nothing more than hardly high-school physics. An infinitesimal of time ticks by and the particles move accordingly. They have a new position and a new trajectory. The function is applied again, and the particles move accordingly, again, with a new trajectory. This process goes on and on.

It still appears to be a simple system. After all, with two particles, it seems that complexity is completely nonexistent in the system. It might seem complicated to a layman, but it is not complex, by definition—nor is it complicated to any high school student with a basic understanding of physics and math. In fact, this system is a far less complex system than reality; reality would include far more forces than just the gravitational force, and far, FAR more particles. Either way, this particle system seems rather simple.

Will it follow a predictable pattern? It might seem like it does at first glance. That is the beauty of mathematical chaos. This one simple pattern follows a seemingly unpredictable nature, even though it is defined by one simple function. This Lorenz Attractor system is a good example of mathematical chaos. If we change any of the parameters by just a tiny bit, no matter how small, given an enough amount of time, the systems appear to have no similarity in their trajectories. It’s simply beautiful.
Throw in a third particle. How about a hundred particles? Maybe we’ll throw in the few googols or so of subatomic particles that we know of into the mix, and all of the strange ways they behave, just to add to the complexity. The beauty is that these systems are theoretically governed by a set of a few mathematical rules. Whether we have discovered these patterns or not yet, these patterns do exist.

Gödel’s Incompleteness Theorem states that there might be problems that we will never be able to solve. From a philosophical sense, this is true. For all we know, solipsism could be the truth, and everything is but a mere illusory projection from an observer’s mind. The thing is… math is absolute truth. Everything can be mathematically defined in some way. The Pythagorean Theorem will always hold for any right triangle on a perfectly flat surface, and methods of topology and the Law of Cosines can compensate for other distortions of triangles and other angles. It’s truly beautiful.
The traditional definition of chaos—where things run uncontrollably and rampant—is the antithesis of beauty. Mathematical chaos, on the other hand, is the epitome of beauty. Take the Mandelbrot Set, for example. It’s nothing more than taking a point on the Complex Plane, putting it into a function where one term is squared and added to the initial term, then fed back into the squared term, and looping—iterating it—for enough times that the point either goes out of bounds or doesn’t. Yet this simple generation does not generate a simple shape. In fact, it generates one of the most complex structures in all of mathematics—a fractal.

Due to the principles of Chaos Theory, with “time” being the number of iterations done to the point, a point that is a billionth of a billionth of a billionth of a billionth of a pixel away from the one right next to it might be in the Mandelbrot Set, while the neighboring point isn’t. The Mandelbrot Set is almost unanimously agreed to be the most be the most beautiful object in all of mathematics, as it is an extremely elegant demonstration of Chaos Theory.

Now think about Conway’s Game of Life. Imagine an infinitely huge sheet of graph paper. Each cell can either be shaded in or not shaded in—that is to say, alive or dead… and no, there are no Schrodinger cells. Each cell has eight neighbors. Let’s say we have a living cell. If there are one or no neighbors, the cell dies, as if of loneliness. If there are two or three cells, the cell lives on to the next generation. If there are any more neighbors, the cell dies, as if by overcrowding. If three neighbors surround a dead cell, the cell comes to life. From these simple rules, we can again use the principles of Chaos Theory to create a game board that essentially looks… living, hence its name.

To a layman, this may not seem all too fascinating. To a mathematician and mathematical biologist, these patterns take us one step closer to describing our own universe. Could we not be the byproduct of a very large array of an n-dimensional version of the Game of Life? Who is to say we aren’t? While this theory lies forever in the Conjecture Limbo of Gödel’s Incompleteness Theorem, it is certainly a sobering thought to think that our universe may be governed by a set of rules that an elementary school kid could understand.

Most people think of math as the horrible way schools teach it. I do not like school mathematics. I do not like it one bit. Why don’t I like school math? All of the classes—even Geometry—manages to squeeze out all of its elegance and beauty, and show us a framework that looks—let’s face it—ugly. It’s horrible—it’s like teaching that art is all about painting different types of fences. Math is an art, and its beautiful constituents—ESPECIALLY when it comes to geometry—are ripped out of the subject and presented to us in school. No wonder why kids do not like math. They don’t know what math is.

Math is an absolutely beautiful topic. Chaos Theory and fractal geometry are some of the many elegant, absolutely lovely mathematical topics. If I were to invent a math class, I’d teach all about these amazing concepts of math—what real math is, mixed in with some geometry. In an art class, people learn about the life of the artist. Why shan’t we say the same for math classes—let’s learn about, at least in part, the life of the mathematician. After all, math is none other than an art that uses symbols as metaphors to represent reality.

I may be only 17, but one does not have to be old to be wise. It simply takes a little elucidation to what really is to be captured by its allure, drawn towards not a Siren Song, but the calling of absolute truth and reality. Mathematics is not the greatest human invention. Mathematics has always existed and always will exist as absolute truth. Mathematics is the greatest human discovery. It may not seem like it at first glance, but it’s like a chocolate éclair—looks are deceiving, and all you need to do is to take one bite and you’ll have wondered why you ever hesitated to dig in.


I don’t know what else to add. It’s 6AM. I should get ready for school.

The Beauty of Chaos and Mathematics