Why Is Math So Fascinating?

[morphism]

The axiom of choice and everything equivalent to it, and everything it can do.

 

[dontdisturbmycircles]

Thanks for posting your work chroot, I didn't have time to think about why it works, but I suppose it's fairly simple. I thought it was kind of neat myself, although I do admit, it is simple (as it was supposed to be I guess.)

I look at your list of math that amazed/intrigued you, and I can't wait to figure some of the stuff out. I know that I will be studying taylor series very shortly and have already had small doses of information as to why e is so special (basically just that it's slope at x=0 is 1).

 

[chroot]

(basically just that it's slope at x=0 is 1).

e is a contant. Its slope is everywhere zero.

- Warren

 

[dontdisturbmycircles]

Woops sorry, yes, e^x. The slope of y=e^x at (0,1) is 1. (derivative).

 

[0rthodontist]

Getting my first rough inkling of the incredible size of the field of math.

 

[dontdisturbmycircles]

Sorry to repost, but I meant the slope of the tangent line.

My bad.

 

[Quaoar]

http://en.wikipedia.org/wiki/Graham%27s_number" , which is so big that it has its own special notation.

 

[Office_Shredder]

Holy frick. That's a big number.

I'm going to prove tomorrow that g₆₆ > 1. I will utilize g₆₅ in the process, thus breaking the record.

 

[gravenewworld]

The Church-Turing Thesis.

The topology trick with the strip of paper where you twist it 180, 360, and 540 degrees then cut it down the middle to get different results. that was totally mind blowing.

 

[moose]

The topology trick with the strip of paper where you twist it 180, 360, and 540 degrees then cut it down the middle to get different results. that was totally mind blowing.

Im kinda interested to know what you're talking about...

 

[z-component]

The topology trick with the strip of paper where you twist it 180, 360, and 540 degrees then cut it down the middle to get different results. that was totally mind blowing.

Like the Möbius strip?

 

[gravenewworld]

Like the Möbius strip?

Yes

Im kinda interested to know what you're talking about...

Get a piece of regular 8x11 paper. Cut 1" wide strips (so the length of the strip is 11"). Take a strip and turn one end 180 degrees. Tape it to the other end. Get a pair of scissors and cut it down the all the way down the middle until you end where you started cutting(i.e.cut it down the "spine"). What did you get? Repeat the same procedure but take another strip and put a 360 degree twist and a 540 degree twist in it. What did you get? This little grade school trick still completely shatters my mind.

 

[mathwonk]

as a kid, i waS READING A DONALD DUCK COMIC BOOK AND HE haD THE IDEA TO GET RICH from a double your money back offer. a store offered double money back if a hair restorer failed to work. he figured he could not grow hair, being a duck, and remarked that if he doubled a dollar 20 times he'd be a millionaire. i checked and he was right. i was amazed.

of course he lost. he took the stuff back for his first or second refund and the guy ridiculed him and rubbed the hair grow tonic all over his body and kicked him out. a day later he popped out in hair all over and had to pay double to get the hair remover.

 

[DaveC426913]

as a kid, i waS READING A DONALD DUCK COMIC BOOK AND HE haD THE IDEA TO GET RICH from a double your money back offer. a store offered double money back if a hair restorer failed to work. he figured he could not grow hair, being a duck, and remarked that if he doubled a dollar 20 times he'd be a millionaire. i checked and he was right. i was amazed.

of course he lost. he took the stuff back for his first or second refund and the guy ridiculed him and rubbed the hair grow tonic all over his body and kicked him out. a day later he popped out in hair all over and had to pay double to get the hair remover.

That's so ... mind blowing.

 

[Quaoar]

Another thing that blew my mind: Using residues to solve integrals that I couldn't solve before without Mathematica. That was pretty neat.

 

[Gelsamel Epsilon]

The version that 'Add a foot to the circumference' thing that I heard was something like there is a circle with circumference of Some really really big X if you added 1 meter to that circumference then could you fit a person through it or something like that.

 

[FrogPad]

http://en.wikipedia.org/wiki/Graham%27s_number" , which is so big that it has its own special notation.

haha :) that was really interesting

 

[mathwonk]

that is a great list chroot! those are all wonderful examples of math gems.I remember another life changing moment, when in college i noticed in a footnote to courants calculus book, about page 27, the proof of the formula for the sum of the rth powers of the first n positive integers, by induction. i was blown away, as this footnote had more content than the entire book of any course I had taken in high school. i knew I wasn't in kansas any more, to paraphrase dorothy.

another experience was sitting in the univ library at vanderbilt and eading the proof there are infinitely many primes.

another was the day we proved the number of rationals is the same as the number of integers, but less than the number of reals, by cantors diagonal arguments. this still seems mind boggling.

a sad sequel was learning recently that cantor was so far ahead of his time he was ostracized for these fantstic insights to the point where he became depressed and died unhappy. later hilbert is said to have exclaimed "we will never be driven out from this paradise cantor has built for us!"

i myself obtained admittance to an advanced honors calc class taught by john tate, at harvard entirely based on my knowledge of cantors proof of the uncountability if the reals. up to that point in my interview he was politely ushering me out the door as a hopeless dunce.

 

[Alkatran]

sum of 2^n from 0 to i-1 = (2^i) - 1
I realized it when I was reading a short story in middle school about a man playing some game with the devil, and the devil casually remarked that if he drank the next glass he would get 100$ more than all he had already gotten.

It really bugged me, because I remember we had to do that sum once, and we did it term by term! Before long I had figured out how to do it for any base and any starting point, but I still can't believe we had to do that sum!

 

[dontdisturbmycircles]

Just wanted to add that I learned why e is so important today. :P Since when differentiating an exponential function of the form a^x you get ax*(the value of f'(x) at x=0). So since the slope of the tangent line drawn at x=0 for the graph e^x is 1, it's derivative is the simplest of all possible, itself. :) (Chroot said that discovering why e is important was neat, and I agree)

 

[Gza]

I think most of the "mind blowing" within learning about math has truly come about after learning quantum mechanics. You really gain an appreciation for how deep and beautiful math can be. I feel that learning quantum has been the only way for me to really grasp linear algebra, as well as many other bits and pieces of mathematics.

 

[turbo]

Graphical representations of the intersection between real and imaginary numbers are not only mind-blowing, but can be beautiful, or even creepy. Here is a plate of alien fried eggs. I've got dozens and dozens of images - all different, and many of which took hours to construct.

http://img299.imageshack.us/img299/2563/eggsye2.jpg

 

[Evo]

Graphical representations of the intersection between real and imaginary numbers are not only mind-blowing, but can be beautiful, or even creepy. Here is a plate of alien fried eggs. I've got dozens and dozens of images - all different, and many of which took hours to construct.

http://img299.imageshack.us/img299/2563/eggsye2.jpg

Beautiful, yet creepy. What exactly is it?

 

[turbo]

Beautiful, yet creepy. What exactly is it?

It is an image that I created with Fractal Magic. You get full control over layers, colors, transparency, etc. There is a HUGE learning curve, but it can be worth it. Here is another one that should appeal to the sisterhood. I chose the colors because they looked like octopi/squid colors, and there are some artifacts (seahorses) that look more like traditional Mandelbrot images.

http://img299.imageshack.us/img299/2779/squidglowch7.jpg

 

[DaveC426913]

As someone who played with Mandelbrot and Julia set generators for a while, I'd say some of those shapes don't look like bona fide fractal constructs - notably the left and right edges of the image.

Is Fractal Magic taking some "liberties" with its designs?

 

[Evo]

It is an image that I created with Fractal Magic. You get full control over layers, colors, transparency, etc. There is a HUGE learning curve, but it can be worth it. Here is another one that should appeal to the sisterhood. I chose the colors because they looked like octopi/squid colors, and there are some artifacts (seahorses) that look more like traditional Mandelbrot images.

http://img299.imageshack.us/img299/2779/squidglowch7.jpg

Wow, post more!

 

[turbo]

As someone who played with Mandelbrot and Julia set generators for a while, I'd say some of those shapes don't look like bona fide fractal constructs - notably the left and right edges of the image.

Is Fractal Magic taking some "liberties" with its designs?

That is the "magic". You get to apply symmetries, inversions, etc to make your images. Most of the horsepower comes from choosing which functions you want to apply in each layer, where to set the transparency limits and how to implement those, and what color schemes and gradients to apply. You can of course recover Mandelbrot and Julia sets in their original iterations and make beautiful images playing with only palettes and gradation slopes, etc, but that is not really satisfying. As I noted in another post recently, my "take" on creativity has a decidedly technical bent.

 

[GeoMike]

Getting my first rough inkling of the incredible size of the field of math.

I agree. When I was younger "Calculus" seemed like the upper limit (no pun intended) of math -- if you knew Calculus you were a math genius. I was amazed (and overwhelmed) when I began seeing how much more existed beyond basic calculus.

-GeoMike-

 

[turbo]

Don't mess with the bug that rules the universe.
http://img100.imageshack.us/img100/128/ant1ui4.jpg

If you're going to burn in hell (see Steve Vai's "The Audience is Listening"), you might as well have an idea what eternity in fire looks like.
http://img100.imageshack.us/img100/6914/hellfirenonsymrt7.jpg

This one was inspired by my spiderplant.
http://img100.imageshack.us/img100/1174/spiderplantkm2.jpg

I should have titled this one "Gene Simmons"
http://img299.imageshack.us/img299/665/spiral4yk3.jpg

 

[Evo]

Turbo, those are really incredible!