The first day of classes went pretty well. It looks like I'll be waking up early a lot this quarter, but that does mean I'll be done with most time-sensitive things by 4ish or earlier each day, meaning I will have evenings to watch Orioles games and do homework. In other news, I'm teaching Physics 7C again, so magnets and birefringence ahoy!

But the best news is that none of the Sushi crew teach on Monday nights! Hence, tonight will be delicious.

What? Mathematics? Yes, yes. Here it is. The cosine and sine of an angle theta can be defined respectively as the x and y components of a vector (oriented an angle theta above the horizontal) from the origin to the unit circle centered at the origin. That might sound complicated, but you might recall that it's as simple as drawing a triangle and finding the blue and red distances in the diagram below.

Note that the total distance (in yellow) from the origin to the circle is always 1 since the radius of the unit circle is one by definition. This, when combined with the pythagorean theorem, gives rise to the famous trigonometric identity cos^2 + sin^2 = 1.

This simple definition led me to think: why a circle? My next thought was, "why can't I define functions in the same way with different shapes," and that is when the madness began. For you see, the cosine and sine functions are so nice due to the beautiful uniformity of the circle. My functions will have none of this niceness, and will require several additional stipulations in their definitions. 

I now define the Poly_n and Gon_n functions of theta as the x and y components respectively of the vector (with an angle theta above the x axis) going from the origin to the regular n-gon centered at the origin whose side length is such that the distance to any vertex from the origin is one. Another caveat to this definition is that I mandate that all polygons be oriented such that below the x-axis a side of the polygon be parallel to that same axis. To make this more clear, let's check out what Poly₃ and Gon₃ look like:

Since a regular 3-gon is more commonly referred to as an equilateral triangle, you'll note that creating this diagram from my definition was pretty simple; Poly₃ has taken the place of cosine and Gon₃ has taken the place of sine. Note however that we lose the ability to label the yellow distance as one; in fact, it will only be one at the vertices. In addition, the Poly and Gon functions will be piecewise because of the sharp edges. Let's quickly take a look at the the Poly₄ and Gon₄ functions:

Though my simple colored lines look similar, you'll note that these functions will have a completely different form to them than those before. In fact, I hesitate to do more at the moment because they're just so darn ugly. Plotting the functions themselves would be terrible thanks to the jagged shape of polygons, in fact anything I do with them would have to start from basic geometry or with the niceness of sine and cosine. Possible future directions for this include finding the yellow distance and naming it something, finding the form of a few of the piecewise functions using geometry and writing them as a Fourier series of sines and cosines and ultimately finding the power series expansions for them. This could lead to finding some "useful" identities and maybe even derivative relations. Sadly, I can think of absolutely no use for this formulation. The one nice thing I can say about the Poly_n and Gon_n functions is that in the limit as n goes to infinity, they blessedly become cosine and sine! That's it for this adventure into strange maths: join us next time when I discuss approximations in all their vague glory.

EarthBound Segment #1: I'm Onett!
My playthrough of EarthBound has begun! I had planned on posting some of my favorite quotes from each segment, however after eagerly cataloguing each of the game's first dozen or so lines, I realized the futility involved. EarthBound's script is simply one of the most beautiful ever crafted. Each sentence uttered, each sign read delights the mind with its marvelous oddity. It really must be experienced directly.
Anyway, let's dive right in! After learning of Giygas's threat to the universe from Buzz Buzz (who was quickly and tragically smote by the fiendish Larda Minch), Ness realized that he needs to gather the Earth's Eight Melodies, as well as three comrades in order to save the world! Based in Onett, he's told that the first melody is nearby, at the location known only as Giant Step. After a new day dawned Ness's first steps were to:
-Explore Onett and its penchant for roadblocks (they're going for the world record!), mayoral propaganda (by the esteemed B.H. Pirkle) and social revolutions (the Fresh Breeze Movement wants to clean up the town!)
-Talk to local billboardsmen, treasure hunter and vagabond Liar X. Aggerate who survived a meteor impact because he eats lots of garlic. Apparently he has something he wants to show Ness later...but only when he's alone...
-Purchase some handy equipment and grab the Mr. Baseball hat from the secret hideout
-Get his first smash attack and revel in the bone-crushing 16-bit sound effect
-Be ambushed by an eccentric, but mysteriously wise photographer in front of his house

I stopped playing today after defeating EarthBound's first real villain (Starman Jr....well, he just doesn't quite cut it as far as villainy goes), Frank Fly.

First of all you'll need his battle music. Got that funky beat playing? Alright, now you're grovin' to Frank's style. Frank is the leader of the Sharks, Onett's local ruffian gang. Ness finds that if he can defeat Frank, thus quelling the hoodlums on the streets, the mayor of Onett will give him the access to Giant Step that he desires. Frank fights dirty: he uses knives against a kid (then again, Ness does have a baseball bat), talks trash and retreats to his robot, Frankenstien Mark II when things are looking dangerous. Once conquered though, Frank acknowledges Ness's strength, and will support his struggles for the rest of the game! Frank Fly, as it turns out, is one cool guy.