Great Theorems of Mathematics

[AvalonXQ]

My guess: when you think of the circle's perimeter as a line from [0-1], that rotation is not a continuous function.

I'm pretty sure there's no bijection that maps [0-1] to a circle.

 

[ben m]

I'm pretty sure there's no bijection that maps [0-1] to a circle.

Can't you cut the circle in one place and unroll it? You basically get x = theta/2 pi and theta = 2 pi x. Oh wait---we need to include 0 and 1, and it looks like this only includes one or the other, is that the problem? Can it be patched up somehow?

 

[ctamblyn]

Brouwer's theorem doesn't apply to circles. In it's general form, it applies to any convex, compact subset of a Euclidean space - any closed ball, for example. But a circle fails, as it isn't convex.

 

[readme.txt]

0.999... = 1 is pretty simple and easy to understand proof :

1/3 = 0.333...
<=>
3 x 1/3 = 3 x 0.333...
<=>
1 = 0.999...

 

[ddt]

How does it apply to a circle?
f(x) = x + pi [rotate the whole circle 180 degrees]
... seems to map one circle to the other with all x != f(x). What am I missing?

Brouwer's theorem doesn't apply to circles. In it's general form, it applies to any convex, compact subset of a Euclidean space - any closed ball, for example. But a circle fails, as it isn't convex.

Okay, I phrased that wrong. I meant the (closed) disk, not the circle.

 

[ctamblyn]

Okay, I phrased that wrong. I meant the (closed) disk, not the circle.

Sorry, I guessed you did. I was just being an evil pedant .

 

[ctamblyn]

Can't you cut the circle in one place and unroll it? You basically get x = theta/2 pi and theta = 2 pi x. Oh wait---we need to include 0 and 1, and it looks like this only includes one or the other, is that the problem? Can it be patched up somehow?

Ok, so you want a bijection from the closed unit interval to the unit circle. This is easy if you first find a bijection between [0, 1] and [0, 1). If you abandon continuity, this can be done with a function f defined like so:
For any x in [0, 1] such that x = 1/n for some positive integer n, define f(x) = 1/(n + 1).
For all other x in [0, 1], define f(x) = x.

This gives the required bijection from [0, 1] to [0, 1). It's easy then to get to the unit circle.

 

[AvalonXQ]

If you abandon continuity,

My mistake. What I meant to say was that there's no continuous bijection from [0,1] to the unit circle. Hence, it doesn't satisfy the theorem.
But it turns out that what was meant was "disk", not "circle". And I agree it works for a solid disk.

 

[I Ratant]

Looking at the definition of bijection....
http://en.wikipedia.org/wiki/Bijection
.
Huh?
So what?

 

[readme.txt]

Looking at the definition of bijection....
http://en.wikipedia.org/wiki/Bijection
.
Huh?
So what?

For any 2 sets of finite or infinite number of elements, if you find a single bijective function, it means the 2 sets have the same amount of elements. Some infinite are "bigger" than others.

 

[I Ratant]

And this amazingly uninteresting feature of number would affect me how?
Or anything at all?
I read a lot of book on math and its history, and it's MEGO when they get these angels/pinhead theories.

 

[ctamblyn]

While we're on the subject of bijections, here's another theorem:
http://en.wikipedia.org/wiki/Cantor–Bernstein–Schroeder_theorem
It was this one which guaranteed the existence of a bijection from [0, 1] to [0, 1), albeit not a nice continuous one.

 

[ctamblyn]

And this amazingly uninteresting feature of number would affect me how?
Or anything at all?
I read a lot of book on math and its history, and it's MEGO when they get these angels/pinhead theories.

I doubt that infinite cardinals have much bearing on any practical issue. For some people though, that is precisely the attraction .

 

[readme.txt]

And this amazingly uninteresting feature of number would affect me how?
Or anything at all?
I read a lot of book on math and its history, and it's MEGO when they get these angels/pinhead theories.

Set theory is amazingly useful in algebra.

Example :
Let's say you have a set, which contains elements respecting 2 operations (we'll call it a "ring"). These elements are used to calculate the angular position of "objects" (quaternions). Quaternions are very useful in video game programming, GPS and animation. This ring has an infinite number of elements and the quaternion ring is very hard to work with because it is very unintuitive and its operations are hard to calculate. Well, if you can find another ring which totally contains different elements and 2 different operations and if you can find a morphism (a bijective function that also has other characteristics) between your quaternion ring and your new ring (let's call the second ring, the ring of rotation matrices), well it means the 2 rings are similar (if not "the same ring", but with different elements and operations). This means you can start with an initial quaternion, use your morphism (bijective function) to "convert" into rotation matrices, work with the ring of rotation matrices, use the inverse function and go back to quaternions, without calculating ANY quaternion. Rotation matrices are way easier to use, its operations are more intuitive and its elements are easy to use as well.

Now, knowing that and having proved it, you can avoid using quaternions in your video game if you think it's easier to use rotation matrices.

NOTE : This was only an example. Most video game programmers prefer using quaternions over rotation matrices.

And by the way, quaternions are also "ANGELS/PINHEAD THEORY", since those contain hypercomplex units (or "hyper-imaginary" numbers, if you want).

 

[ctamblyn]

Just thought I'd add this:
"The abstract mathematics of today is the advanced physics of tomorrow."
Can't remember where I heard it, and it's not likely to always be true, but it has often turned out to be the case. Riemannian geometry, group theory and complex analysis spring to mind, and there are probably other good examples. Outside physics, even number theory has found practical applications in cryptography. I still can't imagine in what field infinite cardinals would play a practical role, but who knows? And anyway, some people simply enjoy a good puzzle .

 

[readme.txt]

Just thought I'd add this:
"The abstract mathematics of today is the advanced physics of tomorrow."
Can't remember where I heard it, and it's not likely to always be true, but it has often turned out to be the case. Riemannian geometry, group theory and complex analysis spring to mind, and there are probably other good examples. Outside physics, even number theory has found practical applications in cryptography. I still can't imagine in what field infinite cardinals would play a practical role, but who knows? And anyway, some people simply enjoy a good puzzle .

you are absolutely right. The mathematical theory behind the GPS technology was discovered hundred of years before the invention of GPS and people thought those theories would never apply to anything in the world.