Approaching one-third using only halves

Origami paper-folding has been a hobby of mine ever since I folded my first paper airplane. Coincidentally, origami is a rich source of all kinds of math problems in areas as diverse as geometry, algebra, calculus, and topology. One problem results from trying to fold a piece of paper into thirds. It's easy to fold a piece of paper in half, all you need to do is line up the opposite sides of the paper and crease. From there, it is easy to fold a paper into fourths, eighths, sixteenths, and so on, just by lining up different creases and folding. But what about folding a piece of paper into thirds? If you don't have a ruler handy, folding paper into thirds must be done by eye, which can lead to error. If only there was a way to reach 1/3 by only folding paper in half repeatedly. Can you think of a way? My friend Luke Nimtz did. See if you can figure it out. Answer is below.

We can approach 1/3 by starting at 1/2, removing 1/4, adding 1/8, removing 1/16, adding 1/32 and so on, adding or removing exactly half of the previous distance. Here's a diagram including platypuses.

The blue rectangle is our sheet of paper. Following the arrows we get this pattern:
We start from 1/2.
1/2 - 1/4 = 1/4
1/2 - 1/4 + 1/8 = 3/8
1/2 - 1/4 + 1/8 - 1/16 = 5/16
1/2 - 1/4 + 1/8 - 1/16 +1/32 = 11/32
1/2 - 1/4 + 1/8 - 1/16 +1/32 - 1/64 = 21/64
I made a black line at each of these fractions in the figure and you'll notice that they start piling up in one spot, getting closer and closer to the red dotted line which happens to be located at 1/3. (For proof, each platypus image is 1/3 the size of the rectangle.) If we continued this pattern forever we would eventually reach the red line exactly. We say that the series converges to 1/3.

So here is the proposition:
1/2 - 1/4 +1/8 - 1/16 + 1/32 - 1/64 + 1/128 - 1/256 + ... (and so on forever) = 1/3

This is an infinite sum. If we group the sum like this:
(1/2 - 1/4) + (1/8 - 16) + (1/32 - 1/64) + (1/128 - 1/256) +....
And then subtract inside the parentheses, we get:
1/4 + 1/16 + 1/64 + 1/256 + ....

This is the same series as before, it's just written in a different way. Now that we have gotten rid of the subtractions, we are taking one-fourth, adding one-fourth of one-fourth, adding one-fourth of one-fourth of one-fourth, and so on. Each term we are adding is a power of the first term. This is an example of a geometric series. Thankfully, sums of infinite geometric series are easy to calculate, as long as they start with 1. Ours doesn't, but we can fix that by adding 1, calculating the sum, then subtracting 1 from our answer.

So we'll calculate the sum of this series, with the 1 added at the front.
1 + 1/4 + 1/16 + 1/64 + 1/256 +...

t is well known that the sum of a geometric series = 1 / ( 1 - r), where r is the "ratio". (Our ratio is 1/4.)
1 / (1 - 1/4) = 1 / (3/4) = 4/3. Subtracting the 1 that we added, we end up with 1/3. Ta da.

So we have a proof that our series approaches 1/3. For a more intuitive explanation, take a look at this diagram from Wikipedia's geometric series page:

Think of the diagram this way. First we take a square and divide into four equal squares. We then take one of those four squares and divide it further into four squares, going on forever until we get the diagram on the left. Each time we divide a square, we get three squares and one "leftover" that gets divided further. Eventually, the "leftovers" are so small that they don't matter anymore, and we are left with three groups of squares of equal size that fill up the original, showing that our series is indeed 1/3.







The result is elegant and pretty easy to see with squares, but we can use other shapes and other fractions. What if, instead of using 1/4 as our ratio, we used 1/7? You can figure this out using the geometric series formula above, or by thinking about dividing up shapes. Here's a diagram showing a rectangle divided into sevenths repeatedly.

Each time we divide a rectangle into sevenths, we save six pieces and split up the last one further. The seventh rectangle is split into sevenths, so each of those rectangles is one-seventh of one-seventh of the original. Just the red colored rectangles in this diagram are 1/7 + (1/7)(1/7) + (1/7)(1/7)(1/7) + (1/7)(1/7)(1/7)(1/7) of the whole image. There is a set of these rectangles for each of the six colors, so we can see that 1/7 + (1/7)(1/7) + (1/7)(1/7)(1/7) + (1/7)(1/7)(1/7)(1/7) + ..... = 1/6.

This works the same for any ratio. When we divide our area into x parts, we save (x-1) and divide the remaining one further. You should be able to predict by now what would happen if we used a ratio of 1/9. What about 1/43? What about 1/278?