Home > Challenge, Geometry > Proofs Without Words

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.


  1. 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) …

  2. November 30, 2011 at 2:00 pm

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

  1. December 30, 2011 at 9:01 am

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