# Exploring Pi: From Ancient Civilizations to Modern Records

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## Chapter 1: The Timeless Nature of Pi

The constant pi has been recognized since antiquity and continues to be relevant in various scientific disciplines. Its allure captivates not just scientists, but also authors, filmmakers, and creators. What makes pi so exceptional? Below, I will highlight some fascinating facts about this infinite mathematical constant.

If mathematics is regarded as the queen of sciences, then pi is undoubtedly the crown jewel. It finds utility in geometry, mathematical analysis, and number theory, among other areas. Pi has played a crucial role in the formulation of numerous physical theories and appears across a variety of disciplines.

Who was the originator of pi? The notion that pi was invented is a misconception; it simply exists within the realm of mathematics. It is comparable to elemental substances, gravitational force, or the diffusion of gases — these concepts were not invented but rather uncovered. So, who first recognized pi?

The answer to that question may remain elusive. However, what is certain is that ancient civilizations were aware of and utilized this constant. Evidence of pi's early practical application can be traced back to ancient Egypt. Remarkably, Archimedes' constant (accurate to four decimal places) is reflected in the dimensions of the Great Pyramid of Giza. By examining the ratio of the combined lengths of the base to the height of this monumental structure, one arrives at the value 3.1416.

Was this merely a coincidence? Perhaps, but the presence of pi-related calculations is undeniable in the Rhind Mathematical Papyrus. Compiled prior to 1650 BCE by the scribe Ahmes, likely a replica of an even older text from the era of Pharaoh Amenemhat III, this document encompasses over 80 mathematical challenges. Among them, it describes the area of a circle being equivalent to that of a square, where the square's side was 8/9 of the circle's diameter. This configuration yields a pi value of approximately 3.1604.

The Babylonians also demonstrated practical uses for pi in ancient Mesopotamia. One stone tablet, dating back to 1900–1680 BCE, suggests the circumference of a circle with a diameter of 1 was approximated to 3.125. In the Indian text Shatapatha Brahmana, dated between the 8th and 6th centuries BCE, the astronomer Yajnavalkya provided a pi value of 339/108, translating to 3.139.

Additionally, the Bible references pi in the "Second Book of Chronicles," which describes a circular basin with a diameter of ten cubits, a height of five cubits, and a circumference of thirty cubits. It seems the creator of this basin believed the ratio of circumference to diameter was roughly 3.

For most ancient cultures, pi was considered a practical tool in agriculture and construction. This perspective shifted only in the 3rd century BCE when Archimedes began to explore the constant's properties. Likely the first to analyze pi mathematically, the Greek scholar calculated its value as 22/7, limiting precision to two decimal places. He achieved this by determining the lengths of the sides of two shapes (regular polyhedra), one inscribed within a circle and the other circumscribed around it. This groundwork allowed him to compute the arithmetic mean of their perimeters, revealing an approximate length of the circle.

### Section 1.1: Understanding the Value of Pi

In elementary education, we learn that pi is roughly 3.14. This approximation signifies that it is not an exact figure. As an irrational and transcendental number, pi is infinite.

The idea of its infinite nature was first proposed in the 12th century by the Jewish philosopher and physician Moses Maimonides. Madhava, an Indian mathematician, made the first use of infinite series in calculus around 1400. In 1596, Ludolph van Ceulen provided an approximation of pi accurate to 20 decimal places, and by the end of his life, he extended this to 35 decimal places.

The first indication of pi's irrationality emerged in the 12th century, but it wasn't until 1767 that the Swiss mathematician Johann Lambert provided proof. The transcendence of pi was established by German mathematician Ferdinand Lindemann in 1882, solving the squaring the circle problem.

The last approximation calculated using "analog" methods was conducted by Ferguson in 1946, achieving precision up to 620 decimal places. Subsequent calculations leveraged computer power. Notably, John von Neumann used the ENIAC computer in 1949 to calculate pi to 2037 decimal places, a process that took around 70 hours.

## Chapter 2: Pi in the Modern Era

The first video titled "How did Archimedes ACTUALLY calculate pi? Pi day 2021" delves into the historical methods used by Archimedes to derive the value of pi and its significance.

The second video, "An Introduction to Archimedes' Discovery of Pi," provides an overview of Archimedes' contributions to mathematics and his role in understanding pi.

The most recent record for the longest known expansion of pi, listed in the Guinness World Records, stands at 100 trillion digits, calculated by Google engineers in 2022 over a span of 157 days. This achievement surpassed the previous record set in 2021 by researchers from the University of Applied Sciences of the Grisons, who computed 62.8 trillion digits with a high-performance computer at the DAViS computing center.

Before that, Google engineers had held the record in 2019 with an expansion of 31.4 trillion digits. In the realm of mathematics, the pursuit of pi's expansion has evolved into a specialized discipline.

### Section 2.1: The Symbol of Pi

The iconic symbol π originates from the Greek word περίμετρον (perimetron), meaning "perimeter." It was introduced by Welsh mathematician William Jones in 1706 and first appeared in his monograph "Synopsis Palmariorum Matheseos." Initially, the symbol did not gain immediate acceptance, but it began to gain recognition in the scientific community after Leonhard Euler featured it in his 1737 work "Analysis."

### Section 2.2: Celebrating Pi Day

Notably, pi has its own celebration, known as Pi Day, observed annually on March 14. This tradition began in American academic circles and has since spread globally. The date was intentionally chosen, as pi is approximately 3.14, and in the American date format, March 14 is written as 3.14.

Interestingly, March 14 also marks the birth of Albert Einstein and the death of Stephen Hawking, adding further significance to the day.

### Section 2.3: Pi in Culture and Art

The influence of pi extends beyond mathematics and science, inspiring works in culture and art. One prominent example is Darren Aronofsky's film "Pi," which tells the story of a mathematician on the verge of uncovering a code that holds immense power. In literature, Carl Sagan's novel "Contact" presents a creative portrayal of extraterrestrial communication, incorporating the concept of pi.