The Golden Ratio a Place in History
Written for: Dr. J. Wilson
Ma 669, Winter Quarter
Written by: Sandra M. McAdams
This essay began with the simple notion that I would explore the "golden ratio" and find some constructions and applications for its use. However as I began tracing down its origins I found that according to mathematics historians much of the history of classical mathematics could be written around the idea of the golden section. According to Proclus (410-485 A.D.) whose works inform us of the history of Greek geometry, Eudoxus' (370B.C.) works added to the number of theorems that Plato originated concerning the section. Proclus' work is the first reference for the name of such a division of a line segment. Proclus also included in his commentary that the solution to the quadratic equation generated by this geometric approach was "ancient and the discovery of the Muse of the Pythagoreans" which was 100 to 150 years before Eudoxus and Euclid. Suffice it to say this is a concept that has reached over the ages to intrique people.
With Euclid and many ancient mathematicians numbers were represented by line segments. In his discussions he would reduce quadratic equations down to the geometric equivalency and solve them by applying theorems on areas. To construct the Golden Rectangle (which uses the golden ratio or golden section) he would construct a rectangle of unknown height X so that its base lies on a given line segment AB. Additionally there was the constraint that the area of the rectangle either exceeds a specified value by X squared or it falls short of that value by X squared. The ratio of the length to the width distinguishes a Golden Rectangle from other rectangles. The following construction can be found in Proposition 5 of Book II of The Elements.
Constructions in today's geometry texts are similar (see below). Assigning a unit of 1 to the length of the rectangle allows us to find an easy way to write a proportion of the lengths in relation to one another.
Given a square of side length X, the Golden Rectangle is a rectangle in which the length and width are given in the following proportions: 1/X = X/Y. Also because Y = 1-X, the proportion may be written as 1/X = X/(1-X). The ratio of the length to the width 1/X is the golden ratio.
To construct a golden rectangle construct a unit square ABCD. Construct the midpoint of side AD using M as center and radius MC, draw an arc which intersects AD at point E. Construct EF perpendicular to AE and complete the Golden Rectangle ABCD.
The Golden Rectangle was considered by the early Greeks to be one of the most aesthetically pleasing geometric forms. It was used in the Parthenon constructed in Athens in the 5th century B.C. and has been used for centuries by architects. The ratio is also found through out the ages in art, music and nature.
What is the golden ratio? It has been described verbally by Eves, " A point is said to divide a line segment into a golden section when the longer of the two segments formed is the mean proportional between the shorter segment and the whole. The ratio of the shorter segment to the longer is called the golden ratio. As we have one algebraic description of this relationship is 1/x = x/ 1 -x. State the product of the means and extremes for this proportion and it becomes 1-x = x^2. Of course the quadratic equation that develops from this is x^2 +x -1 = 0. What are the solutions to this quadratic? This is the point at which the ancient Greeks did not like the outcome. It is an irrational number which they chose to illustrate as a length of a segment. One real issue was that they had no way to represent irrational numbers other than line segments. To find the golden ratio they simply drew it. Today we would use the quadratic formula to solve this and find . This is also a point which I found in some texts the writer agreed with the solution 1/X = .618034..., however in other texts the ratio was referred to as 1.618034... It is an interesting characteristic of this number that these happen to be reciprocals for one another. In most high school texts that I perused I found the ratio to be .618034... Of course strict application of the solutions would yield .618034... and -1.618034... I did not see a discussion of why the ratio could not be negative, however in application to areas etc. a negative value would not make sense.
Another interesting relationship can be found by the following exploration:
This system of rectangles was constructed using a specific pattern.
1. Give the dimensions of the next three rectangles in the sequence.
2. Find the value of the ratio of the width to the length of each. What are the values of these ratios approaching?
1. Solution & Discussion: What you should find is the that the next three have the dimensions 8,13; 13,21; and 21,34. This then produces a sequence 1,2; 2,3; 3,5; 5,8; 8,13; 13,21; 21,34;...Note if we think of these as (x,y) pairs to find the next rectangle dimension you simply use the previous y and then the sum x + y. This gives the value as (y, x+y). It is interested to see these graphed as ordered pairs on a graphing calculator. If you calculate the linear regression for the first nine terms it produces the equation y = 1.61519X + .05973. Note the slope of this line is exceptionally close to the reciprocal to the golden ratio given in many books ( 1.618034). Another interesting side note is that the Fibonacci Sequence is related to this set of numbers. A recursive formula for the Fibonacci Sequence is let a1= 1 and a2 =1, then a(n) = a(n-1) + a(n-2) for n > 2.
2. Solution & Discussion: The values for the ratios of the given sides are .5, .667,.6, .625, .619, .6176. As we increase the number of entries in the sequence we will find that the ratio approaches our golden ratio of .618.
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