Phi Formula Geometric Construction

Phi is most often calculated using by taking the square root of 5 plus 1 and divided the sum by 2:

√5 + 1
2

This mathematical expression can be expressed geometrically as shown below:

Three circle construction

Put three circles with a diameter of 1 (AB and DE) side by side and construct a triangle that connects the bottoms of the outside circles (AC) and the top and bottom of the outside circles (BC).  The dimensions are as follows:

AB = 1

AC = 2

BC = √5

DE = 1

The line BC thus expresses the following embedded phi relationships:

BE = DC = (√5-1)/2+1  = (√5+1)/2 = 1.618 ... = Phi

BD = EC = (√5-1)/2 = 0.618... = phi

This simple and elegant way of expressing the most standard mathematical expression of Phi was discovered and contributed by Bengt Erik Erlandsen on 1/11/2006.

Construction by compass and ruler

Scott Beach developed a way to represent this calculation of phi in a geometric construction:

As Scott shares on his web site:

Triangle ABC is a right triangle, where the measure of angle BAC is 90 degrees. The length of side AB is 1 and the length of side AC is 2. The Pythagorean theorem can be used to determine that the length of side BC is the square root of 5. Side BC can be extended by 1 unit of length to establish point D. Line segment DC can then be bisected (divided by 2) to establish point E.

The length of line segment EC is equal to Phi (1.618 ...).

The length of line segment DB is 1 and the length of line segment BE is the reciprocal of Phi (0.6180...).

Taken together, DB and BE constitute a graphic representation of the Golden Ratio.