I love this. A proof without words is, as the name suggests, an image that conveys something without requiring any explanatory text. This one, which Archimedes came up with, shows that 1/4 + 1/16 + 1/64 + 1/256... = 1/3. Seeing the entire square as equalling 1, the green (I can call that green, yeah? It's more of a chartreuse I suppose, but I digress...) squares color in 1/4 of the whole square, then 1/4 of 1/4 ((1/4)^2 = 1/16), then 1/4 of 1/4 of 1/4 ((1/4)^3 = 1/64) etc. and you can see that each green square is one of three of equal size. So in total the square is divided into three equal sets of squares, one of which is colored green, thus the green squares take up 1/3 of the whole square. While perhaps not as rigorous as a traditional proof, it is so very elegant and beautiful and it makes you wonder, can you prove other geometric series this way?
No comments:
Post a Comment