Constructing a Golden Rectangle - Method Two

Now, let's try a different method that will relate the rectangle to the Fibonacci series we looked at. We'll start with a square. The size does not matter, as long as all sides are congruent. We'll use a small square to conserve space, because we are going to build our golden rectangle around this square. Again, please note that the golden area is what your rectangle will eventually look like.

Let's call this square a unit square, and say that it has a side of length 1. Now, let's build another, congruent square right next to the first one:

Now we have a rectangle with with width 1 and length 2 units. Let's build a square on top of this rectangle, so that the new square will have a side of 2 units:

Notice that we have a new rectangle with width 2 and length 3. Let's continue the process, building another square on the right of our rectangle. This square will have a side of 3:

Now we have a rectangle of width 3 and length 5. Again, let's build upon this rectangle and construct a square underneath, with a side of 5:

The new rectangle has a width of 5 and a length of 8. Let's continue to the left with a square with side 8:

Have you noticed the pattern yet? The new rectangle has a width of 8 and a length of 13. Let's continue with one final square on top, with a side of 13:

Our final rectangle has a width of 13 and a length of 21. Notice that we have constructed our golden rectangle using square that had successive side lengths from the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...)! No wonder our rectangle is golden! Each successive rectangle that we constructed had a width and length that were consecutive terms in the Fibonacci sequence. So if we divide the length by the width, we will arrive at the Golden Ratio! Of course, our rectangle is not "perfectly" golden. We could keep the process going until the sides approximated the ratio better, but for our purposes a length of 21 and a width of 13 are sufficient. Let's take one last look at our rectangle:

This rectangle should seem very well proportioned to you, i.e. it should be pleasing to the eye. If it isn't, maybe you need your eyes checked! Go ahead and decorate or color in your rectangle to make it even more attractive if you wish.