Monday, January 15, 2007

is mathematics art?

while mathematics currently deals with patterns FAR SIMPLER than the insane complexity of human life/death/birth/love/hate... the act of doing mathematics and the results do share many formal properties with art that does deal with the meaty thick of human life. of course we should recognize that there is plenty of shit that's called art that has little to do with the mature meaty drama of human life, but i'll hold to the def of art as work that DOES.

in essence, a mathematician is staring at something and it's gnawing at him, he's COMPELLED to draw out some mystery! (irrationaly? why bother to do math at all?) and he FEELS something going on or FEELS that something is true. his job is to communicate that HUNCH to himself (logically ?), to convince himself that his hunch is true, and then to someone else and convince them. this is called a math proof.

take a 300 level math course and hand in proofs to your proff. he will sometimes turn them back and say: "there, you haven't really explained that step, spell it out more". hand in another proof and he might hand it back saying:"why did you spell out that step so much? it's obvious!" the whole process is very SUBJECTIVE!

a math proof is a poem. a poem is a means of communication of some hard-to-put-your-fingers-on INNER experience you have to someone else so that they too might experience what you did. same with math proof, it's a way to communicate some inelucable inner hunch.

often to understand someone's proof you have to go through all sorts of rituals of imagination, drawing pictures, working out examples, before you feel that same final sense of satisfaction that the original mathematician acheived, that the hunch is true. often you get a good willie off the flash of insight when you GET IT.

have you taken 2nd semester calculus? have you had to do tricky trig integrals and trig substitution integrals? these take many creative leaps of the imagination.

let me then describe the process of a proof i did many years ago:

we had a problem from highschool math team: to prove that on a perfect haxagonal lattice you couldn't make a perfect square out of any 4 vertices.

(try to prove it!)

we didn't figure it out.

i had it in the back of my mind over the years, and every now and then thought about it.

one day 5 years later, my girlfriend asked me why i coulndnt solve it. i asked "hey where did you hear about that problem?" she said, "your friend told me.." feeling my manhood was at stake here, i attempted to do so that evening.

the first stroke of creativity occured to me that the hexagons were messy and somehow i imagined that peices were missing, so i popped centers in each and drew the rays and made it into a tesselation of equilateral triangles. extra points so the problem should be easier, but if i could prove it wasn't possible with extra points, it certainly wasnt possible without 'em.

my first attempt crashed into an algebra error, so i changed approaches. that didn't work. tried another thing... then at 3am staring at the picture of a tesselation of triangles with a sqaure marked out by four vertices it came to me why not make more squares? so then i imposed a tesselation of sqares on the dammed thing. (this was a little free wheeling as i wasn't yet 100% sure i was guaranteed the squares could tesselate out of the vertices from just ONE sqaure) still not sure what to do. throughout this process i had this vague notion that the ultimate issue was that squares have to do with the sqaure root of 2 , while the hexagons/equilateral triangles had to do with the square root of 3. and they weren't compatible!

so now what? at 3am staring at this i suddenly saw the triangle's lines as lines of irrational slope on the square tesselation and what really came to me was how you could wrap the square tesselation up into a torus (this was a pretty wacky picture and very far from the original hexagonal tesselation) and i remembered that the lines of irrational slope go winding around forever on the torus never crossing or coming back to their original point. so those irrational sloped lines each emanating from each corner of ONE of the squares can NEVER meet any other corners of the squares.. something is a foot... so then i saw the diagonall lines from all the triangles. so many of them... all threading through all the corners of the square tesselation going on infinitely but they couldn't intersect all the corners, but there where way more squares' corners going off to infinity. seemed like there where too many corners for lines, a contradiction! because each corner of the square is made from the triangle's lines! of course this feeling was still a hunch! i had to demarcate the appropriate area of squares - n - wide and calculate the number of irrational lines, then calculate the number of sqare vertices in that region. yup they didn't match up.

ergo insoluble.

my friends thought it was a wild looking proof, very goofy.

it was so silly that it didn't sit well with me, there ought to have been a simpler way just having to do with the two squrae roots! it still gnawed at me.

the next morning i found the algebra error in my first attempt which dealt more directly witht he sqare roots and finally i found a third proof having to do with abstract Rings of sqqre root of 2, and you couldn't express square root of 3 in it.

quite a beutifull bit of work.

i still like the first wacky proof though.

you gotta draw the pictures and fill it in! hope it's understandable what i did! well that's the point, probably at this point it aint such a good poem, i probably have to spell some more things out more starkly.

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