Mathematics

Most students pore over arcane formula such as sin x, arctan y, and manipulate them with much skill, but few understand them. To quote a friend, to be good at mathematics does not mean being smart, quick, or accurate with numbers. To be truly good at mathematics, means that one can invent his own method of calculating and requires an understanding far deeper than what lies on the surface.

In geometry, it is impossible to ignore the significance of Trigonometry, the study of the relationship between angles and sides in a triangle and their functions, and π, the study of the circle.

A circle then, is actually composed of triangles. Sine and cosine are but ratios, ratios of two lies that form a triangle with respect to the origin. There are only two lines that are significant, either the line opposite and the hypotenuse, or the line adjacent and the hypotenuse. The movement and wavelike alternation of sine and cosine, as they move around the unit circle, come to form the perfect circle we know as presently. Note that, when the ratios of the two lines are at zero (0:1) or infinite (1:0), form the four points on the circle on the top, bottom, left and right, because one line has infinitely no length, and the other has the maximum length, or vice-versa.

The other method of constructing a circle and obtaining the value π, or 3.1416 involves taking the limits of inscribed and circumscribed shapes, which is actually integration. For demonstration, we imagine a shape inside a circle, with the circle being the maximum possible area.

The triangle is the most basic of shapes. Next is the square, pentagon, hexagon, and till we get to a polygon with 96 sides, and then towards infinity.

Archimedes was the first, in ancient times, to approximate π rigorously, using bounds between 3.1429 < π < 3.1408. An improved version arose later, by Liu Hui of Wei Kingdom, who used a 3072-gon to obtain the close estimate of π we know as 3.1416.

So the circle is bigger than the inscribed square both in area and circumference, as well as any shape that comes between. Also, the circle has something to do with the square, because the area of any circle, is π multiplied by the square of the radius of the circle.

Because of π’s transcendental nature, the digits not only never repeat and decimal expansion remains infinite, but also no known formulas exist to obtain the exact value of π. This has puzzeled countless mathemeticians and laymen over the centuries, and the nature of the circle remains all the more mysterious. The truth is, we still don’t fully understand the circle.

The chilling and dismaying reality of π leads some to speculate that other bases than 10 should be tried to test if the repeating of the digits are truly random, such as binary, sexagesimal (base 12), and hexadecimal (base 16). Unfortunately, too little data exists at the moment for study. This has lead certain leading scientists to conjecture that, in the Bailey-Borwein-Plouffe formula, that the normality and standard deviation of π in base 2, can be reduced to an explanation of chaos theory.

After thousands of years of study, we come to the conclusion that the circle is related to every other shape, the most complete of them. The circle is the ultimate expression of fulfillment.

The ancient Chinese were the first formulate a philosophy that showed a mature understanding of the circle and apply it. Taoism as well as similar beliefs that arose under the same time period, all place the circle as being the ultimate. Creation is summed as thus: from nothing came one (visually depicted as a circle), from one became two (light and dark arise in the Yin-Yang), and from two, a myriad of things. A holiday was even created to worship the moon, known as the Moon Festival, which celebrates the completeness of a full moon and the coming of Autumn.

The three dimensional version of the circle, or the sphere, is also derived from integrating the circle using circular disks. From planets to billiard balls, from stars to the universe as a whole, all are based upon this basic shape.

The martial arts of Ba Gua and Tai Ji were developed based on the circle, and represent the ultimate flexibility and invulnerable defense against linear approaches of fighting. Everywhere we go in ancient Chinese culture, we find that completeness and roundness are virtues. Although many a straightforward, linear, accurate, predictable, square aspects of life should be retained, the abilities and natural properties of the circle, those which we can apply in our everyday life, should be understood and never be forgotten.

The all-inclusiveness of the circle is what we should all strive to be. Easy it is to act in a linear fashion, by basing our life upon squares and straightness. Difficult, it is to replicate the of creation of all matter, and the curves of the circle. Dissection and analysis is easier than integration and whole understanding. Acting according to fixed rules and laws is easier than adopting a rounded and more flexible approach. In our studies as well as business, we prefer to split disciplines and tasks, but hardly do we know that in the end, in order for everything to work correctly, learning as well as making profit requires the student, or the corporation to work as a whole.

Too often do we offer individual products and packages, but can we do as the Japanese have done, to integrate an entire theme, feel and look, associated with a certain product, whether it be clothing or electronics, to provide customers a complete package and simplicity in their life? Noteworthy companies include Citibank, Sony Corporation, and The Sharper Image, and many top department stores, such as Nordstrom and Macy’s. But even then, the majority of businesses have a long way to go to improve our traditional method of making transactions.