So you have a single line of length X, and you want to extent the line to height Y such that you produce a golden rectangle.  Simple as pi pie!

This problem can be solved with some simple algebra, and it is useful if you wish to draw a golden rectangle given a line of any length.  For instance, you may want to incorporate golden rectangles into some artwork you are working on, or you may wish to crop photographs such that they are framed within a golden rectangle.

So, given a line of any length, you can break the line into two parts:

It is easy to see that:

From this, you can calculate that the length (A) of the sides of the square part of the golden rectangle is:

A = (A + B)/1.618

So just extend the line into a rectangle with base=(A+B) and height=(A+B)/1.618

Using this type of reasoning, if you have a square with a side of length A, and wished to extend the length to A+B such that the A+B is the length of the base of a golden rectangle, you can determine the length of B very easily:

Bookmark It

Hide Sites

Drawing A Golden Rectangle

Drawing a golden rectangle is pretty easy using the technique I will show you here.  I used PowerPoint to draw the rectangle here, but you can use this technique to draw a golden rectangle with just about any drawing program.  If you have a compass and a protractor, then I suggest that you try to draw your rectangle on paper.

First, draw a square.  You do not need to know the dimensions of the square, just be sure that all the sides are of equal length:

Next, find the midpoint at the base of the square, and draw a line from the midpoint to the upper-right corner of the square:

Now draw a circle with the midpoint of the base as the center.  Expand the circle such that the line from the base midpoint to the upper-right corner is a radial line from the center of the circle to the circle’s edge:

Now rotate the radial line downward, keeping the edge of the line on the circle:

Continue with the rotation until the radial line is parallel to the base of the square:

Remove the circle.  What is left is the square with an extension at the base:

Draw a rectangle around the base + extension and square height:

Finally, remove the extension and square.  What is left is a perfect golden rectangle:

Very easy and neat.  Congratulations, now you can create great works of art by incorporating golden rectangles into you masterpieces!

Bookmark It

Hide Sites

The Golden Rectangle

The golden rectangle is a mathematical concept that goes back to antiquity. As a definition, the golden rectangle can be described as a rectangle which has a height to base proportion of 1: 1.6180339 (approximately). This ratio also as a special name – the golden ratio. It is also know by several other names, including the divine proportion, the golden mean, and the golden number. In mathematics, it is denoted by the Greek letter phi (φ).

So what is so special about the golden rectangle? Why does it have such a divine proportions? Why am I planning on writing several posts on this seemingly mundane rectangle?

First, let me say that this special rectangle is found in many places. It is found in art, architecture, and nature as well. Look at the Mona Lisa, and you will see that the subjects face is bounded by a golden rectangle. The Parthenon, built in ancient Greece, has several golden rectangles. In nature, the logarithmic growth of nautilus shells is at a rate of phi (φ).

In this section, I will be adding more posts on this subject. I’ll show you how to construct this rectangle, and how to derive phi several ways. This really is a fascinating mathematical concept, and I hope you come back for the post to follow.

Bookmark It

Hide Sites