Machine Intelligence of Pi From Geometrical Figures With Variable Parameters Using SCILab

Introduction

The value of pi is of great importance in the calculation of various mathematical formulas, derivations and modeling since ancient times. It is a constant generally denoted by the greek letter π. It is, in general defined as the ratio of the circumference to diameter of a circle. There are many parallel definitions to support the existing and most prevalent definition. It is an irrational number with the value being calculated approximately as 3.14159. The value of pi was first estimated by Greek philosopher Archimedes, for which the value is also referred sometimes as Archimedes’ constant. Since pi is an irrational number, it cannot be represented by any common fraction. However the value 22/7 is commonly used to give an approximated value of it. It has certain features to characterize the value such as it is a recurring decimal number where the numbers do not repeat within a fixed interval. This implies that the in the computational value of pi, the numbers occurring after the decimal point are always random in nature, validating the statistical randomness and supposition of the numerical value. It is not the root of any polynomial whose coefficients are rational making the value a transcendental number. This trancedity of pi had been a great challenge for mathematicians in earlier times to manually calculate the value using inscribed polygon within a circle which was circumscribed inside another polygon, where straight edges and compasses had been used to get a mere approximation of the value. Today the value of pi plays a significant role in calculation areas and volumes of various 2-dimensional and 3 dimensional shapes with greater accuracy. In analytical mathematics, the value is defined using the infinite multiple set of integers belonging to the unbounded spectrum of of real number where the value is represented simply as a period or Eigen value without any reference of 2D and 3D structures.

With the advent of technology over various years, numerous new approaches had been developed by mathematicians for the calculations. With the increased computational power and higher efficiency of the implementation systems, the representation of the value had been extended to many numbers. Various efficient algorithms have been developed and implemented using the workability of super computers. The various approaches to calculate the value of pi theoretically include using of polygons, infinite series and calculus for the approximation.

The calculation of pi used the concept of infinite series where a number of terms of a particular sequence or pattern had been used . An infinite series is an infinite sequence of the sum or product of the terms involved. It did not involve any formula but required a large number of terms for the computation purpose. There are many infinite series which have been used. They are

The Viette’s Series is one of the first infinite series of the form represented by equation (1).

The Wallis Series is of the form represented by equation (2)

Next came the Leibnitz Series which was represented by equation (3) as

One of the most popular series is the Nilkantha Series of the form represented by the equation (4) as

Apart from the above mentioned series, there are many more infinite series used for the calculation of the value. One of the main problems of using infinite series is that a large number of terms are required (generally more than 300) to attain higher accuracy. By using the computational power of supercomputers these calculations have been made much easier.