# Proofs Without Words

Here are two of my favorite *Proofs Without Words*. I’ve been thinking about infinite geometric series a lot 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*.

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## 6 Comments

## Rick · November 30, 2011 at 12:46 am

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) …

## MrHonner · November 30, 2011 at 2:00 pm

Cool! I wonder if that equivalence could be visualized in an elegant way.

## Taylor · December 27, 2012 at 3:42 pm

“I wonder if that equivalence could be visualized in an elegant way.” Ever the teacher, Mr. Honner.

## MrHonner · December 31, 2012 at 11:51 am

I learned long ago that continually asking questions is a great way to deflect attention!

## Quinn Culver · May 5, 2013 at 12:39 pm

Hi,

I used one of these pictures here: http://math.stackexchange.com/questions/382295/prove-by-mathematical-induction-that-1-1-4-1-4n-4-3/382302#382302. I found it via a google search. Please let me know if that’s not okay.

Quinn

## MrHonner · May 5, 2013 at 9:09 pm

It’s fine, and I appreciate you letting me know, Quinn.