(8,9,7)

(streak)

I always tell myself, okay, I will actually just draw something facetiously and get it over with, nobody comes to this blog to admire my GIMP mouse doodles, but then perfectionist tendencies kick in and I get carried away and it ends up taking more than an hour or so.

linear

Okay. I hope I didn’t make any mistakes. 38-hour reprieve. Let it be.

edit: In hindsight, (6,10,6) is likely better. Sorry.

edit 2: None of the words are “multiply” or “multiplies”. Don’t take this too seriously; it sucks.

Advertisements

Φllotaxy

This is beautiful. Why do they have to make it sound all mysterious and difficult? That’s (the reciprocal of) the golden ratio, by the way.

[Textbook]

Transcript since the resolution is far from awesome: “Most angiosperms have alternate phyllotaxy, with leaves arranged in an ascending spiral around the stem, each successive leaf emerging 137.5° from the site of the previous one. Why 137.5°? Mathematical analyses suggest that this angle minimizes shading of the lower leaves by those above.”

\frac{137.5^\circ}{360^\circ} = 0.3819\ldots = 1 - 0.6180\ldots = 1 - \frac{1}{\phi} = \phi^{-2}

edit: Go figure, the angle itself is called the golden angle.

Matrix Intuition

Stopped by a friend’s house a few days ago to do homework, which somehow devolved into me analyzing what programming language I should try to learn next in a corner, which is completely irrelevant to the rest of this post. Oops.

Anyway, in normal-math-curriculum-land, my classmates are now learning about matrices. How to add them, how to multiply them, how to calculate the determinant and stuff. Being a nice person, and feeling somewhat guilty for my grade stability despite the number of study hours I siphoned off to puzzles and the like, I was eager to help confront the monster. Said classmate basically asked me what they were for.

Well, what a hard question. But of course given the curriculum it’s the only interesting problem I think could be asked.

When I was hurrying through the high-school curriculum I remember having to learn the same thing and not having any idea what the heck was happening. Matrices appeared in that section as a messy, burdensome way to solve equations and never again, at least not in an interesting enough way to make me remember. I don’t have my precalc textbook, but a supplementary precalc book completely confirms my impressions and “matrix” doesn’t even appear in my calculus textbook index. They virtually failed to show up in olympiad training too. I learned that Po-Shen Loh knew how to kill a bunch of combinatorics problems with them (PDF), but not in the slightest how to do that myself.

Somewhere else, during what I’m guessing was random independent exploration, I happened upon the signed-permutation-rule (a.k.a. Leibniz formula) for evaluating determinants, which made a lot more sense for me and looked more beautiful and symmetric

\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^n A_{i,\sigma_i}

and I was annoyed when both of my linear algebra textbooks defined it first with cofactor expansion. Even though they quickly proved you could expand along any row or column, and one also followed up with the permutation formula a few sections later, it still felt uglier to me. Yes, it’s impossible to understand that equation without knowledge of permutations and their signs, but I’m very much a permutations kind of guy. Sue me.

Continue reading

Selection Camp

Hey look, a post!

[Whiteboard completely filled with math scribbling]

Amidst the two times five days of math camp and all the accompanying extra schoolwork (plus midterms (plus compound interest produced by procrastination)), I haven’t been writing a lot. So this post has been accumulating for perhaps a month, and there are still a few interesting bits left out because they’re relevant to what might be the next post.

I could try to record everything that happened during the two camps of math immersion I’ve been going to in the last couple of weeks, but that would be boring to read and to write. The interesting bits simply aren’t common enough to justify all that. Still, I should at least write something about exactly what I skipped six days of school plus the normal midterm date, for everybody who is curious, because I do not know the cardinality of that set, er, how many people that is.

But first a completely irrelevant aside: I put “apmo problem release date” into Google and got my last text-based post as the fifth hit. I tried again in an incognito Chrome window to make sure Google wasn’t bubbling based on my account, and then through a random proxy in France just to be safe. What conclusion am I supposed to draw? Maybe, nobody is blogging about this stuff? A negative correlation between math competition and rambling indiscriminately online? No idea.

*drum roll*

A few words of warning: firstly since people don’t actually use English this is all loosely translated, a rather lossy process in terms of atmosphere. But I’m too lazy to write everything down in two languages, so this is all there is. Also, if you don’t know enough math then some of the humor will go right over your head. But the idea that there are significantly many readers of my blog in that category is probably just wishful thinking.
Continue reading