Part II: the Pentagon

The regular pentagon possesses more unusual and complex mathematics than do figures like the square and its derivatives, or the equilateral triangle and the hexagons. For ancient geometers the pentagon became the source of much consternation and sacred symbolism.

One reason was that the pentagon was the source of a harmonic ratio Section known as the Golden Section or Golden Ratio or simply the Greek letter q . q is labeled harmonic because it yields many mathematically and aesthetically fascinating relationships. It abounds in both ancient architecture and in natural patterns.

Like P (the ratio of a circle's diameter to its perimeter) q is an irrational number, which can only be expressed as an infinite, non-repeating decimal.

Constructing the Golden Ratio
 

Procedure: Draw line CB perpendicular to AB. Line CB should be half the length of AB. This length can be determined by bisecting AB. Strike an arc from C of radius CB to intersect AC at d. From A strike an arc of radius AD to intersect AB at E. The ratio EB/AE is the Golden Ratio.

Triangle ABC may be any right triangle whose perpendicular legs have a 2:1 proportion. The ratio EB/AE derived from the triangle constructed above equals the ratio of the diagonal of a regular pentagon to its side, and can be used to construct that pentagon.

The pentagon from the Golden Ratio
 
 

Procedure 1: Draw a line equal to BE from the previous construction. This is the first side of the pentagon. From B and E strike intersecting arcs of radius of length AE also taken from the previous construction. The intersection A is a third vertex of the pentagon. From points B and A and from points E and A strike pairs of intersecting arcs of radius BE. These intersections will establish the last two vertices of the pentagon.

 

Procedure 2: Draw a circle of radius AE. Construct a radius perpendicular to the diameter and mark the distance BE on the radius. From an endpoint, A, of the diameter srike an arc through B to intersect the circle. The distancde from A to the intersection denotes one side of an inscribed pentagon. Srike similar arcs to complete the pentagon.