Author: Marshall Jiang

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Lattice Polygons and Squares

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.

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From the Bahamas

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.

 

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Lost to History

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.

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Is Music Inevitable

Many discoveries in physics and mathematics seemed to occur concurrently, with a few people uncovering (or almost finding) the important concept. The standard example is how Gauss and Newton simultaneously developed calculus, albeit in a different notation. I have also read that if Einstein never existed, contemporaries of him would have found the same relationships in physic, albeit a few years later.

So does this apply to music? Is Beethoven’s 5th inevitable in a sense? Is the development of Romantic music automatic? What about contemporary music?

In a sense, yes. There are a finite number of melodies to be played. That finite number is quite large though. One can argue eventually a composer, let’s call him Waspthoven, would stumble upon the sequence of intervals and rhythms which is the melody, but that is not all of a symphony. Chords and “transitions” play an equally important role in any form of music. Waspthoven would then have to piece together the correct sequence of silence and sound, dissonance and consonance.

But then, would the period be ripe for the publication? Would the public have accepted Waspthoven’s composition? After all, Stravinsky’s Rite of Spring was met with such horror that riots broke out. Besides the public opinion, the matter of logistics also plagues our hypothetical composer. Will there be enough high-quality instruments to play what Waspthoven wanted? Would there be a concert hall to play it in?

Certainly some music are highly period dependent. Bach’s come to mind, when writing for religious purposes is the sole motive; a symphony as abrasive as the 5th would have never been performed then. That means if a genius like Stravinsky was born in the wrong time period, his or her work would be forever lost. It’s kind of poetic to think of unappreciated genius, but also sombering to think of where we could be now culturally, and scientifically.

I guess a correct way to phrase what I’m asking is that is music primarily driven by “Great Man” or is development primarily from the social period? Maybe some other theory of history?

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The Best Layover

Ha, what kind of layover is even good?

My itinerary for going back to the US:

  • Flight from Budapest at 6pm on the 22nd, ~3 hours.
  • Flight from London at 9am on the 23rd, ~10 hours.
  • Flight from Miami at 5pm on the 23rd, ~1 hour.

I’ll be in London for a random 12 hours. UGH.

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Long-Form and Attention Span

I’ve been getting into long-form journalism recently, and trying to read more in-depth materials rather than have superfluous posts from reddit. It’s quite weird that I can work on Putnam problems for hours on end, yet struggle through reading a 2000 word article. Maybe it’s just my motivation level?

I find also that my attention span has been getting shorter, and I am a more impatient person than in the past; this is especially true when I’m doing homeworks in pure-math classes. There are times when I just want a solution now, and not work at a problem for a bit. It seems to be a by-product of switching over to more computational stuff, where results can be obtained instantly and results can be almost “generated.”

Hopefully reading more will slow me down, because sometimes the mundane parts of life are worth slowing down for.