Golden Section

Pronunciation: /ˈgoʊl dən ˈsɛk ʃən/ Explain

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Manipulative 1 - Golden Section Created with GeoGebra.

The golden section is the ratio that divides a line segment into two parts such that the ratio of the length of the whole to the length of the larger segment is the same as the ratio of the length of the larger segment to the smaller segment: phi = a/b = (a+b)/a.. The golden section is also called the golden ratio, golden mean and divine proportion. The value of the golden section is represented by the Greek letter φ (phi). The value of the golden section is

phi=(1/2)(1 + sqrt(5)) which is approximately 1.61803398.

A golden rectangle is a rectangle where the ratio of the length of the sides is φ. A golden triangle is an isosceles triangle where the ratio of the length of the legs to the length of the base is φ. The golden triangle was used by Euclid to approximate the value of φ.

Derivation of the golden section

The value of the golden section is derived using the quadratic formula. Table 1 shows the derivation.

StepFormulaDescription
1phi = a/b = (a+b)/a.Start with the definition of the golden section.
2a/b = (a+b)/a implies a^2=b(a+b).Cross multiply.
3a^2=ab + b^2Apply the distributive property of multiplication over addition and subtraction.
40 = -a^2 + ab + b^2Use the additive property of equality to add -a2 to both sides of the equation.
5-x^2 + bx + b^2 = 0Use the substitution property of equality to substitute x for a.
6x=(-b +- square root(b^2-4(-1)b^2))/(2(-1))Apply the quadratic formula.
7x=(-b +- square root(b^2+4b^2))/(-2)Simplify the multiplication
8x=(-b +- square root(5b^2))/(-2)Simplify the addition inside the parenthesis
9x=(-b +- b*square root(5))/(-2)Simplify the square root.
10x=b(-1+-squareroot(5))/(-2)Use the distributive property of multiplication over addition and subtraction to pull b out of the fraction.
11x=b(1-+squareroot(5))/2Multiply the numerator and denominator of the fraction by -1.
12x=b(1+squareroot(5))/2Since a negative number does not make sense here, change the ± to +.
13a=b(1+squareroot(5))/2Use the substitution property of equality to substitute a back in for x.
14a/b=(1+squareroot(5))/2Use the multiplication property of equality to multiply both sides of the equation by 1/b.
15phi=(1+squareroot(5))/2Since definition of the golden section states φ=a/b, use the substitution property of equality to substitute φ for a/b.
Table 1 - Derivation of the golden section.

Representations of φ

Many different ways to represent φ have been found. One way to represent φ is the repeating fraction

phi = 1+(1/(1+1/(1+...)))

Educator Resources

Cite this article as:

McAdams, David E. Golden Section. 7/11/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/g/goldensection.html.

Image Credits

Revision History

7/10/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
2/8/2010: Added "References". (McAdams, David E.)
11/26/2008: Changed equations to images. (McAdams, David E.)
11/18/2008: Changed manipulative to GeoGebra. (McAdams, David E.)
8/4/2008: Changed equations from images to Hot_Eqn. (McAdams, David E.)
6/6/2008: Initial version. (McAdams, David E.)

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