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For Merkw, Urchun, nemo and other Mathmos....

What are some fascinating mathematical observations, theorems, conjectures and other things you've come across?

I've not gone too in-depth in too many sub-fileds but I find a lot of math facts fascinating.

Here is a short list:
1) The fact that there are many mathematical statements that are implicitly "undecidable."
2) The fact that group theory is so incredibly versatile at capturing symmetries in the universe.
3) The intimate connection between Gaussians and sinusoids.
4) The central limit theorem, linear regression and other reasons the Gaussian is so "natural"

I figure if one of us lists something another finds fascinating, we can take turns explaining (attempting to) them to each other, and or exploring the idea in depth when we find it fascinating but don't really understand it yet (most on my list are in the second category).

What do you say?

2. Heh mathematics is so vast I wonder how many topics will be brought up that most of the rest of us are not familiar with. For example I don't think I've ever encountered Gaussians (or if I have, then I have forgotten). Here are some things off the top of my head that I find interesting:

*Formally I believe that a set is defined to have cardinality Aleph1 if its cardinality is equivalent to the cardinality of the power set of Aleph0. I have seen it mentioned that the cardinality of the real numbers is Aleph1, but I don't think this has been proven. (Or if it has been proven I would like to know.)
*I find terms like real, irrational, and imaginary to be amusing. For example the real numbers have some features that seem unrealistic. Both the rationals and irrationals are dense within the real numbers, but the rationals have the same cardinality as the natural numbers, while the irrationals have the same cardinality as the real numbers.
*Fractals - I just think they are neat. They represent well ordered chaos to me.
*I've seen a hypercube which is an abstraction of a forth dimensional object, but I'd like to see more elaborate forth dimensional representations.
*I've found Flatlands to be an interesting book. It is a simple fictional book about how a three dimensional person would interact in a two dimensional world. It is implied that one can extrapolate how a forth dimensional object would be observed by three dimensional people.

3. Originally Posted by The_Liquid_Laser
Heh mathematics is so vast I wonder how many topics will be brought up that most of the rest of us are not familiar with. For example I don't think I've ever encountered Gaussians (or if I have, then I have forgotten).
I'm sure you've come across them.

Gaussians are often called "normal distributions" (though I usually reserve this for the mean=0, std. dev.=1 Gausssian of 1 dimension) or "the bell-curve."

Gauss may is one of the greatest mathematicians to ever live (Euclid and Euler may be the only ones more influential). So I like calling the curves Gaussians.

Originally Posted by The_Liquid_Laser

Here are some things off the top of my head that I find interesting:

*Formally I believe that a set is defined to have cardinality Aleph1 if its cardinality is equivalent to the cardinality of the power set of Aleph0. I have seen it mentioned that the cardinality of the real numbers is Aleph1, but I don't think this has been proven. (Or if it has been proven I would like to know.)
This wikipedia article covers the Continuum hypothesis fairly well

Originally Posted by The_Liquid_Laser

*I find terms like real, irrational, and imaginary to be amusing. For example the real numbers have some features that seem unrealistic. Both the rationals and irrationals are dense within the real numbers, but the rationals have the same cardinality as the natural numbers, while the irrationals have the same cardinality as the real numbers.
I always found these things pretty cool. There was a puzzle about getting out of hell some day by guessing the pair of numbers the devil was thinking about. It's rather simple, but it proved that this set (a pair of integers) is countable (and therefore had the same cardinality as the numbers used to count) . Of course, it is a short hop to rationals being countable.

The matching of real-numbers to irrationals is a bit less obvious, but the cut Dedekind cut formulation of rational numbers seems to make it pretty clear.

The "real" numbers have quite a bit more application that the integers or rationals, since most sciences assume a continuum (at least at mid to large scales).

Originally Posted by The_Liquid_Laser

*Fractals - I just think they are neat. They represent well ordered chaos to me.
*I've seen a hypercube which is an abstraction of a forth dimensional object, but I'd like to see more elaborate forth dimensional representations.
*I've found Flatlands to be an interesting book. It is a simple fictional book about how a three dimensional person would interact in a two dimensional world. It is implied that one can extrapolate how a forth dimensional object would be observed by three dimensional people.
Keith Devlin should be name "math evangelist of the decade," imo. Your list seems like a list of the books he's written.

4. I really like LeGrange's theorem and everything it implies, especially regarding groups of prime order. I think proofs by induction and proofs by contradiction are attractive.

Whenever I use the floor function, I feel dirty.

5. I freeking love analysis and set theory. When I was a wee-lad taking algebra for the first time, I was completely appalled that a function was capable of taking a segment of the real line and mapping it to a curve that was longer than the original segment. Take, for instance, f(x) = 4/3*x. When x is between 0 and 3, the arc-length of the function is 5. How does it take something of length 3, and just by mapping points linearly, turn it into something of length 5 without ripping or taring anything?! Impossible!

Needless to say, when I finally got into an analysis course I was in heaven. And then Cantor's Diagonalization proof blew my mind.

The Banach-Tarski paradox is vaguely related, although even more absurd when you first hear about it.

Another oddity that freaks people out at parties is Gabriel's Horn, an object with finite volume but infinite surface area. Imagine: you can fill it full of paint, but never have enough paint to cover the surface! What's nice about this is it only takes knowledge of basic calculus to figure out, and if you're clever you can find the higher-dimensional variants as well.

A cool fact I proved accidentally while taking a number theory class is that the radius of the circle inscribed by any Pathagorean triangle is always an integer.

Actually, number theory might be my favorite subject of all within mathematics. I took my first class in it after about a year of grueling, highly-technical mathematics and physics courses. I described the sensation of being introduced to the subject for the first time to my friend (who's an astronomy major) to be akin to studying astrophysics for an entire year, deep within a library, buried in a dizzying pile of books, and then walking outside for the first time and seeing the stars again. That sort of simple beauty is greatly enhanced once you've been exposed to how enormously complicated life is.

My other love within math is dynamics and probability, which is probably what I'd go into if I went to graduate school. There's immensely interesting overlaps in the theory of zeta functions, spectral theory, ergodic theory, and dynamical systems, which all sort of tie together my interests in number theory, dynamics, and probability; although I admittedly am mostly ignorant of what they are, exactly.

I haven't taken abstract algebra or complex analysis yet, but my advisor says I'll love them. I've read a great deal about both topics, but I've been too busy lately to self-educate myself much. How have you guys liked those courses?

6. Originally Posted by Urchin
I really like LeGrange's theorem and everything it implies
That's on my short list of favorite theorems, if not at the top.

Originally Posted by nemo
I haven't taken abstract algebra or complex analysis yet, but my advisor says I'll love them. I've read a great deal about both topics, but I've been too busy lately to self-educate myself much. How have you guys liked those courses?
Complex analysis is like real analysis but confined to R2 and making specific use of the definitions of e and i. Should be fun for you. I never took a real analysis class (advanced calc. and complex analysis were the closest I got) because I was Discrete Math major. I bought a couple of Real Analysis and Topology books later, but I haven't gotten around to reading them yet.

After spending that much time in analysis-land, abstract algebra will be a cake-walk for you. It is a lot of fun too, a lot of applications to number theory (and to seemingly every branch of mathematics).

I got math as an auxiliary degree while focusing on computer engineering. So I missed out on a lot of the analysis classes. Part of me wants to go back to school for a math degree focused on analysis.

I especially want to cover some heavy-duty set theory, topology, real analysis and measure theory.

Another part of me want to go back and study non-linear dynamics, chaos theory, catastrophe theory and the like. After spending time as an EE masters and all the assumptions of linear dynamics, I can't help but wonder how the non-linear world is described.

7. Originally Posted by ygolo
After spending that much time in analysis-land, abstract algebra will be a cake-walk for you. It is a lot of fun too, a lot of applications to number theory (and to seemingly every branch of mathematics).
I'm still in high school, but I worked most of the way through an abstract algebra book last year, and I'm working through real analysis now. I feel like I've got my brain half-dipped into something metaphisical when I understand a really abstract concept. There's nothing else like it.

I also did a tiny bit of number theory last year. I picked up most of what I learned to help with a problem I was trying to crack.

8. Originally Posted by nemo
A cool fact I proved accidentally while taking a number theory class is that the radius of the circle inscribed by any Pathagorean triangle is always an integer.
I missed this on the first read.

Generating pythogorean triples in the usual way by setting m and n to relatively-prime naturals...

a=m^2-n^2, b=2mn, c=m^2+n^2,

and calculating the in-radius in the usual way ...

s=(a+b+c)/2=m^2+mn=m(m+n)
r^2=(s-a)(s-b)(s-c)/s=(n^2+mn)(m^2-mn)(mn-n^2)/[m(m+n)]=
(m^2n^4-m^4n^2-mn^5+m^3n^3)/[m(m+n)]=
[m^2n^2(n^2-m^2)-mn(n^2-m^2)]/[m(m+n)]=
[(m^2n^2-mn)(n^2-m^2)]/[m(m+n)]=
(mn^2-n)(n-m) which is an integer.

My way is inelegant as usual. It's been a while since I was a math major, but I think that could be made rigorous.

I'm sure yours was far more elegant, considering you did it by "accident."

9. Originally Posted by ygolo
I missed this on the first read.

My way is inelegant as usual. It's been a while since I was a math major, but I think that could be made rigorous.

I'm sure yours was far more elegant, considering you did it by "accident."
Oh, I'm terrible. I can "see" a proof almost instantly, but to verbalize it I have to wrestle it out still bloody and kicking and screaming. It can take me weeks to write a proof for something I solved in 2 minutes.

Many of my other friends are much better at it than I am. I figured it was always an ENTP vs. INTx thing.

10. Originally Posted by nemo
Oh, I'm terrible. I can "see" a proof almost instantly, but to verbalize it I have to wrestle it out still bloody and kicking and screaming. It can take me weeks to write a proof for something I solved in 2 minutes.

Many of my other friends are much better at it than I am. I figured it was always an ENTP vs. INTx thing.
I wonder if that is your Ne at work.

I was rather plodding. I would try hard to find counter examples of the theorem I was trying to prove almost immediately. As I kept failing, I tried to notice what was causing that failure and that is usually the crux of the proof I needed. Then I chose a proof strategy that are basic restatements/easy inferences/backward chains form hypotheses and and conclusion. Then I tried to link the "crux" to the two ends of the proof strategy.

If I started having trouble again, I used the source of my trouble to try to construct counter-examples again to what I was trying to prove. If the theorem is true, I would have further trouble constructing counter-examples, and would find another "crux" to the problem and try to link that in (perhaps reconsidering the proof strategy). I could continue like that for hours and hours.

Pretty pedantic huh?

I only immediately "saw" proofs for easy theorems. Of course the more math I know/remember, the easier the inferences/backward chains from the hypothesis/ conclusion was to come up with.

Like in the theorem you mentioned above. The pythogerean generator was a key fact, as was the formula for the in-radius. Those were just things I remembered. I would have been hard-pressed to notice the patterns needed from attempted counter examples to solve that problem through my usual grind.

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