Proofs Without Words
Here are two Proofs Without Words I whipped up in Geogebra. I’ve been thinking about infinite geometric series lately, and these are two lovely, well-known, visualizations of two amazing infinite sums:
In a square of side length 1 (and therefore, area 1), cut the square in half; then cut one half in half (that’s a quarter); now cut one of the quarters in half (that’s an eighth); and so on and so on and so on (this puts the infinite in infinite sum). Eventually you’ll fill up the whole square! So this is a demonstration of the following amazing, and somewhat counterintuitive, fact that
Similarly, this diagram
is a visual representation of the following sum:
As any good, lazy mathematician would say: the details are left to the reader.
Click here to see more in Geometry.
The sum below is equivalent to the square one:
3/4 + 3/16 + 3/64 + … =
(1/2 + 1/4) + (1/8 + 1/16) + (1/32 + 1/64) …
Cool! I wonder if that equivalence could be visualized in an elegant way.