Mathematical art V2

Post Reply New Topic

I'm going through the prettiest equations of my life; definite integrals!

It is beautiful, even if it's quite abstract; for people with little math culture, it seems like magic, but the beauty of math resides in one's comprehension of math.

I agree with you!! I've been finding that a lot lately myself. I've been drawing geometrical shapes and basing some of my stuff off of the number pi, even trying to memorize it to use it in artwork (3.1415926535 so far)

The Fundamental Theorem of Calculus is a wondrous thing.

closest I ever get to that is fractals. There's a program called Chaos Rei (or possibly Kaos Rei) that's free, and you can enter equations for fractal designs (for that matter, go by Sourceforge and look for fractal programs)

If you're into topology, there's another free program called Topmod, also free. I use it to generate unusual shapes for other art programs, but you can do all sorts of topological shapes. Google 'em up

Aside from that, I am abysmally bad at math...

hippies...

Ah, beautiful mathematics. What is an aesthete like myself doing in such a major?

When I learned Calculus, we learned the Riemann Integral. However, the Darboux Integral is logically equivalent to the Riemann Integral (indeed, I found the duplicitous thing masquerading as a Riemann integral in my advanced calculus course), and, shall I say it? It makes the Riemann Integral look like a cantankerous old woman. The Darboux Integral accomplishes the same thing as the Riemann integral, only it does so entirely without reference to limits or infinity. Instead, there are Upper Sums, and there are Lower Sums, and Darboux Integrability occurs exactly when the infimum of the upper sums equals the supremum of the lower sums, and this is defined to equal the integral. How delicious!

The proof of the fundamental theorem of calculus likewise is also much more elegant.
Consider F differentiable on [a,b] s.t. F' = f is Darboux Integrable.
Then for any partition of [a,b] {a = x(0) <x(1)<...x(n) = b} we have that for any [x(k-1),x(k)] there is some there is a c(k) in this interval such that f(c(k))*(x(k)-x(k-1)) = (F(x(k)-F(x(k-1)), because of the Mean Value Theorem. If we set up a given Riemann sum, S, consisting of only such f(c(k))*(x(k)-x(k-1)) in that partition, then we come to a fixed value for S, namely, F(b) - F(a), because after converting the sum to the antiderivative we have a telescoping sum! But since by hypothesis f is Darboux integrable, we know that that the infimum of the upper sums equals the supremum of the lower sums and any Riemann sum is related to each by an inequality, so voila! The integral of f = F(b) - F(a)!

The Mean Value Theorem is such a beguiling thing. I think I must review it sometime.

A350XWB: How you shall enjoy your major if you can take delight in the beauty of definite integral. There is something a little mystical in math, it is true, but only because in those truest moments of wonder do I sit there and smile and say, "Hah! It works, but I did not see it coming." Only then can I feel small enough to wonder at the abstract Truth of math, guarded by God himself in The Book. As my dear idol Goedel taught me, it may seem to obvious, trivial even, but yet, it shall always elude us. Our system has taken on a life of it's own!

Natalieeeeee: Do not forget that other mystical number so popular lately, φ. (1+5^(1/2))/2 ~ 1.618... The Fibonacci's, whence it can be derived, are a truly fascinating lot.

I think the complex numbers are the most beautiful part of mathematics. I love how they provide such an elegant way of describing the solutions to differential equations. They also crop up in so many other areas including fractals and dynamical systems.

Also gotta love e^(i*pi) = -1.

It’s funny though. I remember hating the square root of negative one when I first saw it in my high school math text. Only later did I realize that complex numbers have nothing to do with the ridiculous notion that negative real numbers have square roots. Complex numbers are ordered pairs that behave a certain way under the operation of multiplication. They are not square roots of negative REAL numbers. Square roots of a negative REAL numbers don’t exist, square roots of complex numbers with negative real components do exist.

What is prettier: definite integrals or the people who can do them?

I found that definite integrals were pretty but so were the girls I met in my calculus II course