## 11/11/11 — Equilateral Triangle Day!

While others celebrate the number 11 on this special day, I prefer to honor the Equilateral Triangle.

Last year, on 10/10/10, I celebrated the symmetry of the equilateral triangle.  This year, I offer a favorite Proof Without Words.  Well, a proof with some words.  In any event, we will use equilateral triangles to prove that the following infinite series

$\frac{1}{4}$ +  $\frac{1}{16}$  +  $\frac{1}{64}$  + $\frac{1}{256}$  +  . . .

is $\frac{1}{3}$.

Consider the following diagram.

Notice that the largest blue equilateral triangle is $\frac{1}{4}$ the area of the entire equilateral triangle.  The next largest blue triangle is $\frac{1}{4}$ of $\frac{1}{4}$, or $\frac{1}{16}$ of the entire triangle.  The next largest blue triangle is $\frac{1}{64}$ of the original triangle, and so on.

So, the sum of the blue triangles is

$\frac{1}{4}$ +  $\frac{1}{16}$  +  $\frac{1}{64}$  + $\frac{1}{256}$  +  . . .

Let’s call this S.

Now, here’s the magic:  the sum of the red triangles is also S!  This is true because for every blue triangle, there is a congruent red triangle right next to it.  Similarly, the sum of the yellow triangle is also S.

When you put all the blue, red, and yellow triangles together, you get the original triangle, whose area is 1.  Thus, 3S = 1, and so

$S = \frac{1}{3}$

Therefore, we have

$\frac{1}{4}$ +  $\frac{1}{16}$  +  $\frac{1}{64}$  + $\frac{1}{256}$  +  . . .  = $\frac{1}{3}$

Happy Equilateral Triangle Day!

1. Whered says:

• MrHonner says:

I recognize Armistice Day, but the mathematics that naturally comes to mind (total losses due to war) is too much of a downer to write about.

2. Thanks… I love this one… only wish we were in school today– I would use it with my students!! Oh well, guess I will need to wait until next year 12/12/12

• MrHonner says:

I love this one, too–a student showed this to me last year.

And make sure to use it on 12/12/12! Otherwise, you’ll be waiting a long time for the next opportunity.

3. chrisharrow says:

I think the image is gorgeous as a proof without words.

4. [...] Glaz incorporating the Fundamental Theorem of Arithmetic. And of course the calender sparked a lot of posts, our personal favorite being the one of Freakonometrics [...]