# What Aspect(s) of Math Do You Find Beautiful?

1. Jul 29, 2010

### murmillo

I find this question quite interesting, because there is a reason why some people are so engaged in mathematics. So I was wondering what people in this forum find so enthralling about math.

When I was in elementary school, I absolutely loved math (and dinosaurs and maps). I spent my free time drawing curious shapes and trying to find interesting properties of those shapes and patterns in numbers. Back then I was certain that whatever job I would do in the future, had to involve math. I simply could not imagine a day going by in my life without thinking of math. I loved finding things out.

Now I'm a junior in college and a math major in the Honors Program of a well-known liberal arts college (well, well-known for some; unknown for most people, as most liberal arts colleges are). I still enjoy reading math for fun, and doing math, but maybe not as much as before. Perhaps it's because I'm not in a classroom setting with professors and students to collaborate with. I'm sure that has something to do with it, but I'm finding that I'm getting a little bored. Perhaps the math I'm doing is still at an elementary level and all the truly beautiful theorems will come later?

I've taken the really "basic" courses: calculus and linear algebra. I've also taken a semester of real analysis. This fall I'll be taking algebra (I've already read the chapter on groups) and elementary differential geometry. I've looked at some of the basic nitty-gritty definitions and theorems, and none of them seem beautiful. I skipped to near the end and found a theorem saying, in my own words with complete lack of rigor, that if you take the globe and make a 2-D map of it you're bound to distort the distances of at least two places. That to me sounds beautiful because it's obvious but sort of not obvious at all at the same time.

Right now I'm trying to grapple with a topology text because I'm trying to find some branch of mathematics that will be so interesting that it can be a passion. I think I need something that I can think about in my head while in bed.

So, what parts of math do you find beautiful?

I'm putting this thread in Academic Guidance because I am considering going for a Ph.D. in math or statistics because I want to teach college students, and this thread should help people who need help deciding.

Last edited: Jul 29, 2010
2. Aug 1, 2010

### david.aloha

Personally, I try to visualize everything I do because I find it very fascinating - sort of like when you're a kid and you ask why something works the way it does because you just need to know. I see it unfolding in my mind, and I try to develop sort of an intuitive grasp of it that is beautiful and meaningful. I personally find many algebraic constructs to be stiff and hard to visualize, although sometimes I can break them down and understand their relations (I have yet to take a course on analysis, but I hope that it will delve into this realm). Currently, I'm particularly interested in complex numbers, quaternions, and fractals - and I'm looking forward to taking a course in complex numbers/analysis.

As someone who is also a programmer, I'm also interested in computational modeling. I've taught myself to program out of an additional interest in game programming, but through that I've learned the satisfaction of rendering images to a screen and manipulating the interactions and physics behind them.

That's what I find beautiful. I'd also like to take at least one upper level course in geometry. Also, I have memories of being fascinated by the way the water flowed in the wake of my grandpa's boat, as well as the way that flames flicker in a fire pit. I'd love to take some courses on fluid dynamics since I find these things beautiful and fascinating in real life.

3. Aug 1, 2010

### gretun

I think the relationship between Calculus and Physics is what I find most beautiful

4. Aug 1, 2010

### physics girl phd

Symmetries. I remember once a professor returning from a meeting about graduate school finances (money taken in by departments during admissions and divied out (differently) to the departments for uses in recruitment and 1st year expenses) and I remember looking at them briefly, and seeing enough symmetry in the functions that I could quickly analyze things and say, "at least there's conservation of money."

I'll also agree that complex numbers are cool. I just did a segment on AC power and "phasor diagrams" with my EM class, and plotting rotating vectors on the real/imaginary axes for getting something useful in a quick way is wonderful.

I also just like different coordinate systems. Did you know there are many other possible 3-D systems other than cartesian, cylindrical, and spherical (although plotting and working on those with calculus is still pretty fun too). Of course you do... you've taken higher math. There are of course fun 2-D systems too...like parabolic ones. These 2-D systems are pretty neat to younger (MS and HS) students.

5. Aug 1, 2010

### mg0stisha

I love how you can learn many foreign languages and still not being able to communicate with a big part of the world, but you can learn math and discover the language of the universe. Math is spoken (almost) uniformly by every intelligent being we know of. Beautiful.

6. Aug 1, 2010

### david.aloha

That's a good point physics girl phd about parabolas. It was 2D non-linear functions and their transformations that really renewed my interest in mathematics in high school. Here were these methods for making sense of what was previously a garble of "do it this way" identities and formulas. I started learning these basic ways to analyze things without the use of some arbitrary identity or formula that was already there for me. In that same math course I tackled conics, permutations and combinations, logarithms and geometric series, and made extensive use of graphical approaches to problems. It just got better when I did calculus. The only downside is, for some reason, the stream I was in for math (pure) in high school didn't include vectors, fractals, or matrices and the "lower" (applied) stream did. Later, aside from what sometimes seems like the sometimes arbitrary use of matrices, I found these topics to be interesting and meaningful.

And Gretun, that's a big part of where I fell in love with calculus. I looked at what I was doing in physics and my calculus course and went, this just makes so much sense. I saw the rate at which something changed and that it was summed up by this relatively simple and intuitive concept (once I worked it out visually) and the way you could track the accumulation of something and derive meaning from it opposite to the derivative.

I really like this .gif on wikipedia's page for torque:

It's so simple, but it seems to me like such an intuitive way of looking at things once I wrapped my head around it the first time.

Last edited: Aug 1, 2010
7. Aug 2, 2010

### ekrim

I find it beautiful when someone is able to express the same concept with completely different mathematics. The connections and reoccurring themes that pop up when you study a variety of fields is truly amazing.

8. Aug 2, 2010

This

9. Aug 3, 2010

### donnylee

For me it's simple.

Using topology to prove the fundamental theorem of algebra.

Now that's beautiful.

10. Aug 4, 2010

### Theorem.

That it isn't a very good liar

11. Aug 4, 2010

### jf22901

I don't find any of it beautiful! It's just something I have to know, and be good at, to be able to do what I want in life.

However, that's not to say I don't enjoy it. There's something incredibly satisfying about using maths to solve problems, particularly ones you've been working on for weeks. However, other days I look at a page of equations and my heart sinks!