## 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.