carlo angiuli (blog)

As I may have mentioned before, I’m in Douglas Hofstadter’s course titled “Group Theory and Galois Theory Visualized.” We’ve been talking a lot about the beauty of mathematics — a slightly strange concept which mathematicians universally acknowledge but cannot quite quantify.

I’ve touched on this subject before — what is beauty (or perhaps, elegance) in mathematics? Last time I addressed it, I concluded that elegance, to me, is a high results to complexity ratio. I gave the example of Euler’s formula, which I find incredibly simple but deep. Beauty is about the insightfulness and depth of results, not just the usefulness.

(Example from group theory: the classification of finite simple groups, as a single corpus, is neither insightful nor deep nor useful! And it’s certainly one of the ugliest — and longest — proofs out there! Not that there aren’t good moments. It’s just kind of silly to represent the whole thing as a “proof.”)

But I’m mostly commenting again on mathematical beauty because of a comment Hofstadter made about beauty in general — what is found beautiful and by whom? (After all, most people don’t find math beautiful at all!)

He noted that, while people claim “beauty is in the eye of the beholder,” there’s clearly some sort of standard for beauty. After all, if there weren’t, why would we have art museums filled with the most “beautiful” art? It does, however, take a special sort of person to enjoy the sort of beauty that math offers. Though, in that regard, math isn’t unlike most other activities — after all, why do marathoners want to run 26 miles for pleasure? It’s just that, as Hofstadter points out, enjoying mathematics requires somebody who wants instead to… “run a matharon,” so to speak.

Have I mentioned that I love puns?

(This is the act that I performed at the first annual IU math department talent show last night. The preceding act was a bass/recorder duet.)

Wow, there are some great acts here. In particular, I think the basis we just heard was great. His music spanned our three-space quite nicely. Anyway, I was going to bring some predatory birds here, but then I realized it wasn’t a talon show.

Okay, I’d like to make a request of you before I start my act. Please laugh very loudly at everything I say, because nobody might actually find it funny.

So, math comedy. When I told my friends I was going to do a math stand-up act, one of them replied, “Chuck Norris knows the tangent of pi over two!” Well…okay. I’m not sure how to respond to that.

Math comedy is certainly a niche audience, though. Even among mathematicians. If you ask a statistician if they’ve heard a joke before, they say “Probably.”

Anyway, there are a lot of oldies-but-goodies. There’s the joke about the mathematician who gives a talk about 13-dimensional space. Afterwards, an engineer comes up to him and says, “Wow, how could you possibly visualize 13-dimensional space?” The mathematician responds simply, “That’s easy, I just visualize n-dimensional space, and set n equal to 13.”

Of course, many sub-disciplines come with their own occupational hazards. They say topologists can’t tell the difference between a doughnut and a coffee mug, them being homeomorphic and all. I’m not sure if that’s true; I’ll ask Kent after the show.

And then physicists get their own brand of flak from mathematicians. Physicists, you see, use a special brand of mathematics. The really fuzzy type…that’s usually wrong, but somehow comes up with the right answers all the time. I think one thing in particular illustrates physicist math. Those of you who know some physics may know that electric and magnetic waves propagate as orthogonal sinusoidal waves. The direction in which they are pointing, the vector representing the energy flux of the wave, that’s called the Poynting vector. I don’t know about you, but I never make distinctions about which of my vectors are pointing. They all are!

Anyway, the other day I was going to a geometry conference, and I was speaking on constructible diagrams. I was flying out of the airport, but I was stopped at security because of my straightedge and compass. They found my weapons of math construction. I ended up missing my plane. But it’s okay; luckily I had three points in my pocket, so I defined my own plane and got there on time.

You know, we mathematicians are always trying to prove to everyone that there’s math everywhere. In particular, there’s a lot of math in the Bible; did you know that? For example, a lost story from the gospels. One day, Jesus said, “The kingdom of heaven is like x squared plus 3x plus 5!” Somebody went up to Matthew and asked him, “What is Jesus talking about?” “Don’t worry,” responded Matthew, “that’s just another one of his parabolas.”

Then there’s also the story in Genesis with Noah’s Ark. After the ark landed, Noah told all the animals to go forth and repopulate the world. Two snakes stayed behind, and told him, “We can’t do that until you build us a wooden desk.” So, whatever, he built it, and lo and behold, they started to reproduce. He asked them what the problem was, and they said, “Well, we’re adders. We need log tables to multiply.”

The other day I was proving a theorem. It was a long theorem, with a lot of significant intermediate stages. I got to one of those stages, and I said to myself, “Do I have to finish? Lemma stop here.”

Medicine has made great strides recently. When right triangles get old, they sometimes start to sag, their right angle turns into 89 degrees, 88 degrees… Anyway, they made this injection, you can just apply it to the triangle, and the angle will snap back up to a right angle. It’s called Pythagorean serum.

The other day I was at the concession stand. I wanted a medium order of Fibonachos, and my friend wanted a small order. But then I realized that a small plus a medium cost the same as a large.

I usually eat more healthily. I found a grape that could commute, it’s called an abelian grape.

I thought up a great anagram for Banach-Tarski. Ready? It’s… “Banach-Tarski Banach-Tarski.”

Some people have wondered why Newton didn’t contribute to group theory. It’s because he wasn’t Abel.

Have you heard? A former vice president recently released some rap tapes to teach computer science. It’s called “Al Gore Rhythms.”

Even mermaids like math. They wear algae bras.

Okay, just one more and I’ll leave you guys alone. So, as you know, lately, the military has been having issues with how its officers are perceived. Some kernels have expressed concern at their rather zero-dimensional images.

Hey people, I know I haven’t written anything here in a long time. I find myself very busy, or when I’m not busy, very occupied being not busy. Blogging is busy-ness for non-busy times, and unfortunately at those times I find myself doing other things, like playing Portal (best. game. ever.) or just hanging out with people. I will, however, start writing in my blog more frequently! I promise. It probably won’t be in the next two weeks, though, because I have papers and finals.

However, I had a question for everyone. Right now, the Writings section of my website is kind of bare. I have written some papers I am pretty proud of, and I was wondering whether people would actually be interested in reading them. Of course, I would only post interesting papers on interesting subjects. For example, I have written some pretty nifty papers this year on luminiferous aether and on special relativity with which I am rather pleased. If you are somewhat interested in the history of science, I think you would find them good discussions of fascinating topics. I could also dig into the Carlo vault and post older papers, for example, my McCarthyism paper that demonstrated I am in fact competent with history, just not at remembering all its minutiae.

Are people interested? I know proposing that some people might like to read papers in their spare time is a pretty crazy thing to say, but some weirdos actually like that sort of thing.

I passed out of Calc III. Judging from the final, I will have to get used to tests consisting of fewer, larger problems than they did in high school.

I am taking a Calc III final tomorrow morning. Luckily, Calc III appears to be things I know well (partials, iterated integrals), not things I need to review heavily–it doesn’t even have line integrals! Win. I hope.

Apparently when you learn things poorly the first time, you don’t have great retention six months later. Time to study!

Things like the Koide formula make me kinda happy. It’s exciting that we have already developed frameworks to understand so many things, but that so much still remains. We clearly know very little about the universe.

A show of brute computation doesn’t interests me nearly as much as an elegant result. If I had to define elegance in this context, I think I would say that a subject with a high results to complexity ratio is very elegant, especially for math. I think first-year calculus (I and II) is very elegant, as it makes many very difficult problems easy, and makes insoluble problems possible.

To me, the most elegant result obtained by first-year calculus is Euler’s formula, or in its most famous special case, that e^(i*pi) = -1. The fact that this can be conclusively proved to or even discovered by high-schoolers with knowledge of Taylor series is, I think, a testament of how elegant calculus is. Of course, there are other ways to prove it, but I think Taylor series are the most elegant way–the clearest, most straightforward, least contrived; no clever logic is required at all.

Of course, Euler’s many formulae remain the gold standard of elegant results in mathematics. While he didn’t always convincingly prove his results, I think it was more important that he simply generated them. Sometimes simpler logic goes further than more bulletproof logic. I certainly don’t believe I should be mentioned in the same paragraph as Euler, but as to myself, I was quite happy with my intuitive proof using Gram-Schmidt that, for vector space V and subspace W, that dim(W) + dim(W perp) = dim(V). The idea is too conceptual to forge into any sort of formal proof, but it makes perfect sense; I would be interested in seeing a more rigorous proof, but I haven’t gone out of my way to look for one. I am, for the most part, satisfied with rationality.

Andrew Wiles’s proof of Fermat’s last theorem was certainly a tour de force in the mathematical world, and certainly, as I understand, contains many interesting ideas (which no, I am not myself qualified to appreciate). It’s certainly nice that the theorem–and subsequently, the full Shimura-Taniyama theorem–have been proven, but the proofs are long, complex, involved, and *gasp* even, in some places, significantly case-oriented. It’s a true achievement, but to me isn’t as awesome (in the pre-slang sense of the word) as much shorter results. The epsilon conjecture hinted at great possibilities, but to me, V - E + F = 2 has for centuries been quite profound enough.

I don’t mean to say that I don’t consider the above to be true accomplishments; they certainly are. I suppose it’s simply that I am more interested in the concept than the execution itself. This is why I doubt I will be entering math itself as a field, though I fully expect to be using math in whatever I end up doing.

And it’s also, I suppose, why I have always been so happy to help somebody with math who seems actually interested in the ideas themselves, not just how to do well on the next test. It’s easy to tell the students who truly want to hear the concepts and understand them for themselves apart from the students who simply want an algorithm to memorize. And it’s perfectly understandable why math could be so hated by students who view it as simply a set of algorithms. Because then, as John once sagely pointed out, math is no more than a “demented mind-game.”

So the other day I was making a cup of tea, and asked my sister whether she would like some as well. Since I didn’t mind leaving the teabag in my own cup while I drank it, I realized that the easiest way to make two cups of tea of approximately the same strength would be to first put the teabag in her cup, and then to move it to my cup for the remainder of the time. This is because the teabag lets flavor out the fastest at the beginning, and gets increasingly slow at it as time progresses. Of course the tea flavor always increases; there’s no point at which the teabag starts reclaiming flavor for itself.

With all these words like “increases” and “increasingly slow” this sounds like a good opportunity for some calculus.

The graph above represents f(t), the derivative of tea flavor with respect to time. Note that df/dt (which will now be referred to as Tea/Time) is always positive, but decreases in value approaching zero as time approaches infinity. The strength of the tea–denoted F(t)–is the definite integral of this graph over some time interval.

Let’s make some assumptions to make this problem easier. Since I occasionally forget about my tea as I let it cool down before drinking, let’s assume that I don’t start drinking my tea until time t=infinity. Thus, the strength of her tea is equal to the integral from zero to whatever time I remove the teabag (called t=a), and the strength of my tea is equal to the tail integral from a to infinity.

We can of course solve for a algebraically:

Of course, this simply verifies our original observation that I want to remove the teabag at the point at which her tea is half as strong as the teabag would be if I waited until time infinity before removing the teabag from a single cup. We are, however, happy to see our integrals come out to a statement we already knew to be true.

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I guess what I’m asking is, “Is it a bad thing that I started forming integrals in my head when my sister said she wanted tea?”

Yeah, I don’t really feel like writing a lot about orientation. I had a good time, so call me a bad blogger or something, but I’m not sure what there is to say. Very quick summary: Randomly ran into CJ on Monday. Hung out with Wells Scholars much of the time. Had a good meeting with my adviser on Wednesday morning, then a very nice talk with the math department about what math courses to take, and then a decent talk with the chem department about taking organic chem first semester.

My first semester courses are:

Mathematics S343 - Honors Differential Equations I
Chemistry C341 - Organic Chemistry I
Honors H205 - Origins and History of the Universe
History and Philosophy of Science X102 - Science Revolutions: From Plato to Nato
Telecommunications T160 - Videogames: Historical & Social Impact

Yeah, you read that last one right.

In terms of my schedule:

Monday: 11:15 CHEM, 2:30 CHEM
Tuesday: 9:30 HPSC, 11:15 MATH, 4:00 HON, 5:45 TEL
Wednesday: 11:15 CHEM
Thursday: 9:30 HPSC, 11:15 MATH, 4:00 HON, 5:45 TEL
Friday: 11:15 CHEM

Yes, my days are kinda lopsided… >_<

So John Wiltshire-Gordon and I realized we don’t actually have any idea how matrices work. Rather, why they work.

We figured out how Gaussian elimination works generally. It looks like if you have an augmented matrix [A | B] and you Gaussian-eliminate it into [C | D], that AD = BC, always. That’s why it works to find the multiplicative inverse as well as the solution to a linear system. We also traced the determinant throughout, and saw that each time you multiply the pivot, you change the determinant. (Well, we knew that already.)

The real question is, what really is a determinant?

I’m sure this will make some great Monday Mathematics columns later.