Skip to content

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:

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.


  1. Rick says:

    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) …

  2. MrHonner says:

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

  3. Quinn Culver says:


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


Leave a Reply