The Golden Ratio Part 3 – Constructing the Golden Ratio

The Golden Ratio has been embedded in art throughout history. In antiquity it can be found in the great temple to Athena Parthenos (the Parthenon). In the Middle Ages it can be found in the design of grand cathedrals such as Notre Dame and in the Renaissance prominent artists used it most of their works. This post will show how to construct the Golden Ratio using only a straightedge ruler and a compass, the tools used by classical Greek Geometers.

1. First Draw a square
2. Mark a new point E on the base of the square so that E is equal distance from C and D
3. Draw a line from E to B and now draw a circle with the radius equal to the distance from point E to point B centered at E
4. Extend the base of the square to intersect the sides of the circle, call one of these intersections name the left one F and the right one G.

The ratio of the line CG and the line AC is actually the golden ratio. If we were to assume that each side of the square is equal to one, then the distance from E to B is actually about $\sqrt{5}/2$ because of the Pythagorean theorem. This means that the E to G is the same distance, and since C to E is $1/2$ we get a total of $(1+\sqrt{5}) / 2$ which is the golden ratio.

This construction can be continued to form the golden rectangle and from the golden rectangle we can also construct the golden spiral.