The Golden Ratio M.V.N. Murthy, The Institute of Mathematical Sciences, Chennai The golden ratio, also called golden mean, is a number which is denoted by the Greek letter \phi and is given by It is a very special number not only in mathematics but has played a role in arts and architecture over a long period of time. The symbol \phi (prounced phi) was chosen in honour of the great Greek sculptor and artist Phidias (480-430 BC) who is said to have employed this ratio in his designs. Phidias is considered the greatest sculptor in ancient Greece who designed many monuments in ancient Greece in the 5th century BC. One of them is the Parthenon in Athens, shown on the cover. The ratio of the long side to the height is supposed to be in the golden ratio. Parthenon is one of the monuments found on a hill in Athens as part of a complex of buildings called Acropolis. The floor plan of the Parthenon and its elevation are supposed to contain dimensions in the golden ratio. This number was supposedly also known to the Egyptians. In the great Pyramid of Giza in Egypt the ratio of the slant height to half the length of its (square) base equals the golden ratio to a very good approximation. Over centuries many monuments, including the famous Mosque of Kairouan in Tunisia, were built with many dimensions in the golden ratio. Golden ratio in Mathematics Golden ratio has very interesting mathematical properties. Most students are familiar with pi which appears every where in mathematics. It is an irrational number which cannot be expressed as the ratio of two integers. The golden ratio is yet another irrational number which pops up everywhere, not just in mathematics, just like pi. In algebra The easiest way to realise the golden ratio is to consider the solutions of the quadratic equation x^2-x-1=0. It has two solutions given by x=(1+\sqrt5)/2 and x=(1-\sqrt5)/2. The second solution is negative; the first solution is just the golden ratio. In geometry In geometry it can be realised in terms of the ratios of segments of a line. Consider a line which is divided into two segments of length a and b; b is smaller than a. Suppose a and b are such that the two ratios, (a/b) and (a+b)/a are the same. Call (a/b)=x. Then which is the same as the quadratic equation one of whose solutions is the golden ratio. A nice way of seeing it is to pair a rectangle of sides a and b with a square of side a (see figure). Then the two sides are in the golden ratio. In arithmetic The golden ratio can be expressed in many interesting forms. For example it can be expressed as a series of terms like This form is known as the continued fraction and we can obtain approximations to the golden ratio by summing successive terms up to the accuracy we need. For example summing the first few terms successively yields 1/1,2/1,3/2,5/3,8/5,13/8,21/13,34/21...,1.6180339887. The precise value emerges only after summing up to infinite terms. If we notice carefully the sequence of approximations here, you can guess that it is the ratio of the successive numbers given by the sequence of numbers 1,1,2,3,5,8,13,21,34,55,89,... It is easy to see that each number in this sequence is the sum of previous two numbers. This famous sequence is known as the Fibonacci sequence. So the successive approximations to the golden ratio are simply the ratio of successive numbers in the Fibonacci sequence. The approximation becomes better and better as we go up in this sequence until we reach the golden ratio. There is yet another interesting form of the golden ratio---it can be written as square root of 1 iterated many times, also called infinite surd: Squaring both sides, we have y^2=1+y, which leads to the same quadratic equation, one of whose solutions is the golden ratio. Interestingly you can check that there is one more quadratic equation which leads to infinite surd form, namely y^2=2+y, whose solution can be written as Not only in mathematics and artitecture, the golden ratio finds a place in art as well. The famous painter Leonardo da Vinci provided illustrations, containing many figures in golden ratio, for the famous book on mathematics "On the Devine Proportion" written by Luca Pacioli around 1497 in Milan. More recently the famous painter Salvador Dali used the golden ratio in his masterpiece, "The Sacrament of the Last Supper". The two sides are explicitly set in the golden ratio. (Source Wikipedia). It is normally accepted that objects made according to this ratio have a pleasing, aesthetic, appeal which may explain the frequent appearance of this ratio in everyday designs like, shape of post cards, photographs, book shelves, even in the design of cars. Till recently television screens were also made in this proportion, until replaced by the wide screens of today. Many people think that even in nature, in the shape of shells or flowers, or in the arrangement of leaves on stems, the golden ratio makes its presence felt. It is truly a thing of fascination for all people.