Thursday, December 20, 2012 10:29:11 PM
@MacGuffin: I'm a physicist, in case you`ve missed the thousands of posts where I`ve made that obvious.
Draw a square in a piece of paper. Bend that piece of paper. OMFG a 2D figure projected in 3D curved space! Holy sh*t that must be the work of the devil for you.
Your picture only demonstrates that it is true in 2D Euclidean space. Good luck demonstrating that the areas remain the same when the square lives in curved space, here`s a hint: they don`t. Ergo, your drawing doesn`t say WHY, by any chance, the sum of the areas MUST be unequivocally the same. Believe it or not, questions like that keep us mathematicians up at night. No meds necessary though.
Tuesday, December 18, 2012 3:01:43 PM
Why it's true is still a subject of research, as is my understanding, since it doesn`t have to be true (and in fact it isn`t in all kinds of spaces, it is only true in flat/euclidean space)
Stay off those drugs, dude. Hint: a *square* (which Pythagoras` Theorem makes heavy use of) is a two-dimensional shape. It only exists in Euclidean space. Otherwise it`d be a cube. Or a tesseract. Or an n-dimensional hypercube. The diagram below demonstrates exactly why Pythagoras` Theorem works - namely that exactly the same amount of space is left over when you surround a square of side length equal to the hypotenuse or two squares with side lengths equal to the lengths of the opposite and ajacent sides laid corner to corner with a larger square of the same size in each case. It is not a "matter of research", it`s a mathematical proof that`s been fully understood for thousands of years.