Proofs Without Words

Published by patrick honner on

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:

infinite series -- square

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 squareSo this is a demonstration of the following amazing, and somewhat counterintuitive, fact that

infinite series sum 1

Similarly, this diagram

infinite series -- triangle

is a visual representation of the following sum:

infinite series sum 2

As any good, lazy mathematician would say, the details are left to the reader.

Related Posts

patrick honner

Math teacher in Brooklyn, New York


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


I used one of these pictures here: I found it via a google search. Please let me know if that’s not okay.


    MrHonner · May 5, 2013 at 9:09 pm

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

Leave a Reply

Your email address will not be published. Required fields are marked *


Get every new post delivered to your Inbox

Join other followers: