Mathematical properties and applications of pi

¶ … pi is interwoven with the history of humanity. Remarkably, "By 2000 BC, men had grasped the significance of the constant that is today denoted by pi, and…had found a rough approximation of its value," (Beckmann 9). Math historians assume that the study of pi began as an analysis of magnitude: that circles remained circles no matter how big or small. Beckmann suggests that early humans contemplated "the peculiarly regular shape of the circle," which was visible everywhere in nature in "its infinite symmetry," (9). Pi remains a mystery in spite of thousands of years of scholarship and investigation. The number is both irrational (it cannot be represented as a ratio of two integers) and transcendental (it is never the solution of a polynomial equation that involves rational numbers). Pi is remarkable in its scope. Professor Yasumasa Kanada of the University of Tokyo writes computer programs that are designed to calculate pi, and has continually broken his own world records. Kanada has computed pi to well over one trillion decimal places and remains "intent on achieving new world records" (Arndt, Haenel, Lischka, and Lischka 1). Because of Kanada's work, pi is now the mathematical constant "which has been calculated to the greatest number of decimal places," (Arndt et al. 1). In addition to performing the calculations for pure pleasure, Kanada and other mathematicians study pi in search of patterns. Understanding pi would be a significant epiphany, a major evolution in human history.

So far pi has yet to reveal itself fully and the number remains a major mathematical mystery. Pi can be understood easily on its most basic level: that of Euclidian geometry. The fundamental realization that the wider a circle is "across," the longer it is "around" is what led to the discovery of pi in the first place (Beckmann 11). That discovery seems to have occurred in multiple cultures, as pi was studied among the ancient Mesopotamians, Egyptians, and Chinese. The ancient Greeks delved deeply into the study of pi, especially pi's relationship to geometry. Pi was revealed as a constant ratio not just of circumference to diameter but also of radius to area. The existence of both constants was well-known, but the fact that both constants were in fact one and the same number represented a major breakthrough. Arndt et al. note that the ancient Greeks first drew the connection between both ratios as they related to the circle. In 414 BCE, Aristophanes presented the problem known as "squaring the circle," which has become the quintessential problem of pi.

Pi has numerous applications, and not just in the world of geometry. Number theorists hope to discover meaning in the endless stream of digits represented by pi, and pi could in fact be meaningful to the study of theoretical physics. Arndt et al. point out that calculating pi sometimes depends on time as well as space. Pi is also meaningful for probability theories, such as the Wallis product (Arndt et al. 9). Moreover, pi may be related to its transcendental number cousin, e. The value of pi in the engineering fields can be readily appreciated, although the extensive calculations like those computed by Kanada's computers serves "little purpose" for the immediate practical needs of engineers. Likewise, Beckmann claims, "there is no practical or scientific value in knowing more than the seventeen decimal places," (101).

It is highly unlikely, though, that continual research into pi serves little purpose at all. Pi is likely to help mathematicians and scientists unravel some of the most mystifying aspects of the universe. Because of its inherently mystical nature, pi assumes a sort of philosophical role in the study of mathematics. The constant could indeed become relevant to the advancement of astrophysics or to other fields such as artificial intelligence. The potential practical purposes of pi offer some of the reasons why it has captivated mathematicians throughout history.

One of the most remarkable things about pi is that the number remains "one of the oldest subjects of research by mankind," period (Arndt et al. 6). Pi is, therefore, on the level of philosophical discourse because many other mathematical problems elucidated by the ancients have since been solved. Arndt et al. claim that pi is "possibly the one topic within mathematics that has survived the longest," (6). Initial pi explorations may have been prehistoric. Ancient Egyptians and Mesopotamians later developed systems of writing and mathematics that enabled rigorous investigations into crucial problems. In 1650 BCE, ancient Egyptian scribe Ahmes recorded what are likely the first formulas for pi. The formulas are written on what is referred to as the Egyptian Rhind Papyrus (Eymard, Lafon & Wilson).

The Ahmes formulas relate the circle to the square, foreshadowing further investigations into pi by the Greeks. The Egyptians were therefore the first to record attempts to "square the circle," or relate the area of a square to that of a circle in search of a constant variable that could illuminate orders of magnitude. Squaring the circle served a practical function for the ancient Egyptians: accurate architectural engineering and surveying. The great pyramids of Giza are constructed in accordance with the pi ratio, even though the Egyptians never calculated pi to any specific amount. Instead, the Rhind Papyrus essentially reads, "Cut off 1/9 of a diameter and construct a square upon the remainder: this has the same area as the circle," (cited by Blatner 2). Ahmes' calculations were off by less than one percent (Blatner 2). The Babylonians and ancient Hebrews were far less accurate in their representations of the ratio (Blatner 2). Like the ancient Chinese, the Babylonians and Hebrews calculated only a rough approximation of pi, represented by the whole number 3.

However, the ancient Greeks catalyzed the study of mathematics in general, and pi in particular. Aristophanes was the first to articulate the squaring of the circle problem in 414 BCE. Pi was also of concern to Chinese mathematicians. Hippias, Dinostratus, Archimades, and a slew of other Greek mathematicians explored the role and relevance of pi and attempted calculations. In the 5th century CE, Tsu Chhung-Chih came up with the "best pi approximation available for 800 years," (Arndt et al. 5).