Category: Math/Programming

Well, last semester was quite hard. But I’m back home, adjusted to living in Florida again, and ready to relax.

Maybe I’ll finally blog more now. Ugh.

Silly joke aside, how does one react to having one’s home utterly destroyed. What is the point of capturing this shell of a city after ISIS has destroyed all essence of human life in it. While a victory, it’s more a symbolic one from the humanist point of view.

Hopefully the people will come back, and it’ll flower under the Arabian sun once again. If not, then I wish the families the best of luck.

This series really emphasized change. Every character in the show transitioned from lofty goals and dreams to dead eyed stares. All because of one person who grasped the ability to change in his hands: Walter.

I’m not advocating myself to become a drug dealer. I’m not in the empire business.

Nor the money business.

Hell, I don’t know what my business is. Is it “make-the-world-a-better-place” business? I certainly hope so, but I don’t know what I should be doing to attain that while still living at a standard of living I like.

But what Walter taught me is to not wait until a stage 3 cancer derails my life. Take charge of the roaring bronco that is change.

It’s fairly obvious that a square is a regular lattice polygon, but it’s fair from obvious that it’s the only one. In the past semester, we actually proved that it’s the only regular polygon on the lattice without invoking any complicated mathematics.

Sketch of Proof:
We first show that regular triangles and hexagons can’t exist. The easiest way is to apply the irrationality of sine to either the area or a rotational perspective. Regular hexagons cannot exist because it’s composed of equilateral triangles.

For any other polygons, we apply infinite descent. Let P1P2…PN be the vertices of the polygons, then consider the vector P2P3 applied to P1. Since it’s a lattice vector applied to a lattice point, we will end up with another lattice point. Applying this procedure to all of the lattice points will result in a smaller lattice polygon! This concludes the proof.

Now what are the implications, and some other problems to consider?

Well for one, what other lattices will give us more regular polygons? For example a triangular lattice will give a triangle and a hexagon as the regular polygons. Is there any others? Does there contain a maximal lattice with such numbers?

Furthermore, does this imply the irrationality of sines and cosines of certain angles? If a regular polygon in the lattice exist, then the trig functions are rational. Is the converse true? I’m pretty certain it is, but haven’t worked it out yet.

Back from a 5 day cruise from the Bahamas, stopping in Half Moon Cay and Nassau. There were some beautiful places there, but more importantly it was utterly relaxing. The inevitable explosion of memories from BSM was also delayed a little as I was constantly distracted with other fun stuff.

Now back to some regular postings,

1. I posted my review for the BSM algebra exam in the hopes that it would help people. It is available here or under the “About Me” page.
2. I really want to do a Clash of Clans math review sometimes this winter break…
3. An investigation into the irrationality of trig functions from a lattice point of view? We’ll see if I have the time.

 

As my time at Budapest comes to a close, I can’t help but be reminded of the times during honor band in high school.

For three days, sixty other musicians and you gel together and make beautiful music for the sake of music. Friendships are made, and is inevitably short-lived with the nature of gathering students from all over the state. During the last rehearsal of one such honor band, the conductor said something quite profound to me. He told me to look around and know that this is the last time this group of people will be assembled. Consequently, that was also the last time that particular sound associated with the ensemble was made in history.

After the concert, the sound of the ensemble is forever lost to history.

I guess BSM, or any other camp for that matter,  is something like that. That group of people will never be together again, and that synergy of relationships will be gone after the program. But as the saying goes, it is better to have loved than not loved at all.

And so, another happy chapter comes to a close in my life.