The golden ratio is a deceptively intriguing bit of mathematics. The ratio makes numerous appearances throughout history in architecture, nature, art, music, and most importantly mathematics.
It is easiest to understand the ratio by imagining a line. Divide the line into two segments with long segment a and shorter segment b. The segments are said to be in the golden ratio if the ratio of the whole line to the longer part is the same as the ratio of the longer part to the shorter part, or simply if (a+b)/a = a/b. This ratio comes out to be an irrational number (approx. 1.61803...), known as the golden ratio, and is symbolized by the Greek letter φ (phi).
If the lengths of the sides of a rectangle are in the golden ratio, the rectangle is called a golden rectangle. This rectangle exhibits some interesting properties. It is possible to divide a golden rectangle into a square and another (smaller) golden rectangle (pictured below). This process can be repeated indefinitely.
A logarithmic spiral can be drawn and this spiral will closely approximate the Golden Spiral, which is a spiral that gets wider by a factor of φ with ever quarter turn.