The Golden Ratio in Math

            A very popular characteristic feature of nature is its symmetry. It is said that “Nature loves symmetry”. Symmetry is found in different forms, sizes, shapes etc of nature. The golden ratio is an intriguing aspect of symmetry in nature. It exists in the form of a number.  The golden ratio manifests itself in nature in many diverse forms. The subtle existence of golden ratio in things which seem very much unrelated is very interesting. I find this aspect of golden ratio very amazing and the topic very interesting.

            Golden ratio is very important because it is present in many forms. It is present in plants and flowers. It is also present in crystalline structure of solid in the way they are arranged. It is believed to be the basis for aesthetics of many objects. The pyramids are believed to be constructed taking golden number into consideration. The relation between Fibonacci numbers and golden ratio is exploited to make algorithms of computer programs. These things show that it was very important for the past. The golden ratio has a huge unlocked potential which has to be found out and tapped. I feel this potential makes it important for the future.

            The 21^st century is trying to make the grand unification theory. We find a lot of unrelated things which we are trying to unite into a single law during the process. We find golden ratio to be present in things which seem quite unrelated as said before. I think the golden ratio may considerably influence the grand unification theory. The amount of influence depends on the hidden potential of the golden ratio. This is just a possibility of the effect of golden ratio.

            The golden ratio was discovered long back by different people and was used by them. The people who discovered it were not mathematicians alone. The discoverers included sculptors, painters etc. It is used by them in their own professions. The first mathematician who worked on this golden ratio as recorded by history happens to be Euclid while he worked on geometry. Later it was discovered many other mathematicians independently after Euclid.

            The peculiar characteristics of the golden number were the reasons for the discovery of the number in different ways by different people. The number appears every where like in plants, crystalline structure of solids, mathematics. So it is natural to expect to biologists, physicists, mathematicians discover them independently in their own way. It is called with different names as it is discovered by different people who named it differently.

            The golden ratio is also called as golden mean, golden section, or divine proportion and can be defined in a number of ways owing to its peculiar characteristics. The golden ratio is denoted by the Greek symbol “φ”. Let’s see a few ways of defining it.

Golden ratio in geometry (golden section):

            The ratio of length of the whole line segment to the longer segment is equal to the ratio of the length of longer segment to that of the shorter segment.

Consider a straight line segment AB divided at a point

C such that AB: AC = AC: CB. This ratio is equal to the golden ratio “φ”.

 AB: AC = AC: CB

• φ: 1 = (φ _{+ 1} ):φ
• φ²-φ-1=0
• φ = ½( 1 + (5√

= 1.6180…

  The number is not an exact number. It cannot be expressed as a ratio of rational numbers.

  The other root is -0.6180… called -φ

Golden Rectangle:

   A rectangle with ratio of sides x: 1 is present. If it is split into a square and a rectangle such that the resulting rectangle still retains the ratio of sides as x: 1. The rectangle is called as Golden rectangle.

            The value x is equal to φ which is golden ratio.

The successive points dividing a golden rectangle into squares lie on a logarithmic spiral called whirling square.

Golden triangle:

            An isosceles triangle whose legs are in golden ratio to the base is called a Golden triangle

Five pointed star:

            The five pointed star has golden ratio embedded in it as seen in the figure that follows.

            Euclid constructed the pentagon using the golden ratio. (Actually a pentagon is similar to the five pointed star. When we join the five points on a star we get a regular pentagon).

            The following is an equilateral triangle circumscribed by circle. The line segment joining the midpoints of the legs of a triangle is extended to the circle. The ratio of the length of segment with in the triangle to the length of segment between the triangle and circle is equal to the golden ratio.

            AEC is the equilateral triangle. P and Q are the mid points of sides of AE and AC. The segment PQ is extended to meet the circle in R.

PQ: QR = φ

Golden ratio and Right angled triangle:

            Consider a right angled triangle whose sides are in a geometric progression. Let the sides be a, ar, ar² . The ratio is r.

The sides have to satisfy the Pythagorean Theorem. Therefore

                        (ar²)² = (ar)² +a²

=>                   r⁴ = r² + 1

=>                   r² = ½(1+ √5).

                        r²  cannot be –φ because r²  has to be positive.

The sides are a , a√φ, aφ.

            Consider the following figure

            The ratio of the sides of the rectangle ABCD is 1:r. Points E, D, C are such that the triangles AEC, EBF, FCD are of equal area. E and F divide the sides AB and CB in the ratio of 1: φ.

Area of triangle AEC = ½*r(1 + φ)*1 = ½*rφ²

Area of triangle EBF = ½*φ*rφ          = ½*rφ²

Area of triangle FCD = ½*r*(1 + φ)   = ½*rφ²

    We use the fact that “1 + φ = φ^2” in deriving the areas.

                As observed the areas of the triangles are all equal.

Consider the following rectangle

            The lengths of longer side and the shorter side are in the ratio of √5 : 1. Place a square centrally whose side length is 1 unit. This leaves two rectangles on each side.

            Let’s look at the rectangles. The length of shorter side of the rectangle is

½(√5 – 1). The length of longer side of smaller rectangle is 1.

Length of shorter side = ½(√5 – 1) = 0.618…

Length of longer side = 1

            The ratio of longer side to the shorter side = 1: 0.618 = 1.618: 1 = φ:1

            The rectangles formed are Golden Rectangles as seen from the ratio of sides of them

            Let there be regular decagon whose side is “s”.  The radius of circle circumscribing the decagon be “R”. The ratio of circum radius and the side of the decagon is “φ”.

            R/s = R/2Rsin(Π/10) = ½ csc(Π/10) = φ

  Where Π/10 is the angle subtended by a side of the decagon at the centre.

Value of “φ” in trigonometric terms:

            Φ = ½ csc(Π/10) = 2cos(Π/5) = ½ sec(2Π/5)

Golden ratio as a sum of infinite series:

Golden ratio as a nested radical:

Golden ratio as a continued fraction:

  This can also be written as convergence of the recurrence relation

           As n tends to infinity x_n  tends to “φ”. The first term is x_{1 =}  1.

Fibonacci numbers and golden ratio:

         The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21…

The sequence is given by the following relation

          F_n+2 = F_n+1 + F_n

 Where F_n is n^th  number in the Fibonacci sequence.

The Fibonacci sequence and golden number are very much related to each other.

   The golden ratio is given by the limit of the ratio of F_n+1 / F_n as “n” tends to infinity.

 Let x_n = F_n+1 / F_n 

    where φ is the golden ratio.

The powers of φ also satisfy the following equation related to Fibonacci sequence

    φⁿ = F_nφ + F_n-1

The golden ratio as sum of infinite series involving Fibonacci sequence

The golden ratio satisfies the following recurrence relation

                   Φⁿ = φ^n-1 + φ^n-2

Φ also satisfies the following linear recurrence relation

                   Φ(n) = φ(n-1) + φ(n-2)

Nature of the golden ratio as a number:

Is φ a rational number or an irrational number?

         φ is an irrational number.

Proof:

         We prove this fact by the method of contradiction.

Let a, b be two integers with no common factors except 1 such that a/b = φ

We know that φ satisfies the following equation by definition

•          φ²-φ-1=0

here φ = a/b. Therefore the equation turns out to

            (a/b)²-(a/b)-1 = 0

Since b is not zero. We multiply the whole equation with b²    

 The equations turns to be

        =>      a²-ab-b² = 0

        =>      a²-ab = b²

        =>       a(a-b) = b² 

     The left hand side has a factor “a”. The right hand does not have “a” as a factor unless a = 1 by the way we defined a and b.

Rearranging the terms we get the following equation

    =>      a² = ab + b²

    =>      a² = b(a+b)

            By using similar arguments we can say that “b” can be a factor of left hand side only if “b”=1.

            If a=1 and b=1 it does satisfy the basic equation of golden ratio

            φ²-φ-1=0.

               If φ=1 the above equation is not satisfied. Therefore our assumption that φ is a rational number is false. Hence φ is an irrational number.

   Golden ratio is described as one of the two great treasures of geometry the other being Pythagoras theorem.

Sources:

         http://mathworld.wolfram.com/GoldenRatio.html

         www.ransen.com/Articles/Golden/Rectangle.htm

         http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html#PhiRightAng

         http://www.cyberspaceorbit.com/ironhart/root.html