We are looking for a way to calculate the value of , the Golden Ratio. All we know about is that it is the ratio of the length and height of a Golden Rectangle.

What do we know about a Golden Rectangle? Well, we know that it is a rectangle which, when squared, leaves behind another rectangle of the same proportions. That is, the ratios of the lengths and widths of the rectangles is the same.

Based on this knowledge, we can set up a proportion, like so:

Before going any further, let's eliminate one of the variables by assuming that the short side of the larger rectangle (the value x) is equal to 1. As we are only looking for the ratio of the sides, this assumption will not alter the problem in any way. Our proportion now becomes:

By cross-multiplying, we obtain:

Some simple algebraic moving about produces the quadratic equation

The next step is to solve for the value y by applying the quadratic formula. When we do, we obtain the following values:

We can discard the second value because it is negative, and lengths of polygons can not be negative. Now we are left with:

All this so far has been done to find the value of , which is defined as y/x. We have found the value of y, and we said before we started that x was equal to 1. Therefore, the value of y/x is the same as the value of y (because y/1 = y), so:

Remember at least the first few digits of this decimal approximation of , as it will become important in future lessons. Note that this is only an approximation, because is actually an irrational number, meaning that it can not be expressed as the ratio of two whole numbers.

You can try this method again using different values of x, and you will always come out with the same value for .

The Golden Rectangle is proposed to be the most aesthetically pleasing of all possible rectangles. For this reason, it and the Golden Ratio have been used extensively in art and architecture for thousands of years. The most prominent and well known uses of the Golden Rectangle in art were created by the great Italian artist, inventor, and mathematician, Leonardo da Vinci.

Golden Rectangles over image. (scan by Mark Harden)

"The Mona Lisa," undisputably Leonardo's most famous painting, is full of Golden Rectangles. If you draw a rectangle whose base extends from the woman's right wrist to her left elbow and extend the rectangle vertically until it reaches the very top of her head, you will have a Golden Rectangle. Then, if you draw squares inside this Golden Rectangle,you will discover that the edges of these new squares come to all the important focal points of the woman: her chin, her eye, her nose, and the upturned corner of her mysterious mouth. It is believed that Leonardo, as a mathematician, purposefully made this painting line up with Golden Rectangles in this fashion in order to further the incorporation of mathematics into art. In the spirit of mathematics in art, it is also worth mentioning that the overall shape of the woman is a triangle with her arms as the base and her head as the tip. This is meant to draw attention to the face of the woman in the portrait.

Golden Rectangles over image. (scan by Mark Harden)

Leonardo's famous study of the proportions of man, "The Vetruvian Man" (The Man in Action), is also full of Golden Rectangles. Unlike the Mona Lisa, where all the lines of the Golden Rectangle are assumed by the mathematician, in "The Vetruvian Man", many of the lines of the rectangles are actually drawn into the image, at least in part. There are three distinct sets of Golden Rectangles in this painting: one set for the head area, one for the torso, and one for the legs.

To find the first set of rectangles, the one for the head, draw a rectangle whose base goes along the man's neck from shoulder to shoulder (stop at the shoulder lines provided by Leonardo). The top of the rectangle should meet the top of the man's head. This creates the first Golden Rectangle. Once you have that, inscribe a square in the left side of the rectangle, creating a smaller Golden Rectangle on the right side of the man's head. Then do the same with the right side of the original rectangle, creating a long, thin rectangle that runs vertically through the center of the man's head. Note that the smaller Golden Rectangles intersect with the focal points of the head: the eyes.

The second set of rectangles is found in a similar way. This time, all the lines are provided by Leonardo in the painting. Draw a rectangle which runs from elbow to elbow and from neck to waist. This creates a Golden Rectangle. Then, in a similar fashion to the first set, inscribe a square in each side of the rectangle, creating two more Golden Rectangles. Note this time that these new smaller Golden Rectangles intersect with the innermost portion of the man's torso.

For the third set, draw a rectangle whose lower two vertices are at the places where the man's outermost toes touch the outlying circle. The rectangle should extend vertically to the man's waist. This creates yet another Golden Rectangle. Now inscribe squares in the sides of the rectangle as you did before. This time it seems that the two smaller rectangles come to where the man's legs would be if they weren't turned out in that fashion.