Pi: The History and Fascination of an Extraordinary Number

The search for π is deeply rooted in the human spirit. The relationship between a circumference and its diameter, symbolically represented by the Greek letter π, often intervenes in mathematics, physics, statistics, engineering, architecture, biology, astronomy and even in the arts.

The π is hidden in the rhythms of acoustic waves like those of the sea, and is omnipresent both in nature and in geometry. The English mathematician Augustus De Morgan once wrote, about the π, “this mysterious 3,14159...which enters from every door and window and is located under every roof”.

The children of the school, normally, do this simple experimentation: a cord wrapped around the periphery of a circle is just over three times longer than its diameter. By measuring more precisely the pupils discover that the value of the piece of cord that exceeds the triple of the diameter is more than an eighth of the diameter but less than a quarter.

The oldest existing documentation of this report was left by an Egyptian scribe named Ahmes around 1650 BCE, in what is known today as the Rhind Papyrus. Ahmes wrote: “ Remove 1/9 to a diameter and build a square on the remaining part; this square has the same area of the circle”. As we know that the circle area is equal to πr, if this area is the square of 8/9 of the diameter, Ahmes' text implies that the circumference ratio to the diameter is equal to 16 – 9 = 3,16049... Ahmes' implied value differed less than 1 percent from the true value of about 3,141592, manifesting considerable precision for that time. However, this result had no diffusion. A thousand years later the Babylonians and the ancient Jews continued to use value 3, which was much less accurate. The formulas contained in the Papiro Rhind are also the first documented case of an attempt to “frame the circle”, that is to build a square with the same area of the circle.

Historians of mathematics often attribute the value of π= (256/81). In fact, there is no direct evidence that the Egyptians considered a constant number, and much less that they tried to calculate it. They were certainly interested in finding the relationship between the circle and the square, probably in order to measure with precision land and buildings.

The Bible gives us very clear information about the value achieved by ancient Jews. In the Old Testament, The Kings,7:23, we read about the altar built in the temple of Solomon: “Then he made the molten sea: ten cubits from one side to another, that is, completely round; his height was five cubits and a cord of thirty cubits surrounded him around the other.” This step (which is almost identical to π Chronicles,4:2) indicates that the ratio of circumference to diameter is 3; it was probably written around the 6th century BCE (although it describes the temple built in the 10th century BCE).

After in Egypt the scribe Ahmes had recorded his formulations, for a thousand years no one devoted many reflections to the relationship between circles and squares. All that time Egyptians and Babylonians felt that the elementary understanding of the relationship was sufficient for the purpose of surveying and building. The study of the measurement of the circle was resumed with renewed commitment in the fourth century BCE by the Greeks. The first Greek thinker to try to find a definitive relationship between a circle and a square was Anassagora of Clazomene (500-428 BCE). The square of the circle was one of the oldest mathematical problems.

Shortly after Heraclea’s Antiphon and Brison, Socrates’ contemporaries (469-399 BCE), attempted to find the area of a circle using a brilliant new idea: the principle of exhaustion.

If you take a hexagon and double its sides turning it into a dodecagon, and then you double them again, and again, sooner or later you will have a polygon with a number of sides so large that you have turned into a circle. First Antifonte formed the area of a circle, calculating the area of the successive polygons - from the number of increasingly larger sides - in it inscribed. Then Brison made a second revolutionary step, calculating the areas of two polygons, one inscribed in the circle and the other circumscribed to it. He hypothesized that the circle area should be between the areas of the two polygons: this was probably the first time a result was determined using lower and higher limits. A couple of centuries later, the challenge was resumed by the Syracusan Archimede ( 287- 212 BCE) one of the greatest thinkers of history, extraordinary mathematician, physical and inventor.

When he turned his attention to the circle, Archimedes used in his calculations methods of exhaustion of Antiphon and Brison. He concentrated on the perimeters of the two polygons instead of their areas, thus finding a approximation to the circumference of the circle. He doubled four times the sides of two hexagons, obtaining two polygons of 96 sides, of which he calculated the perimeters. Subsequently he published his discoveries in the book “Misura del rim”.(v. appendix 6): “The circumference of each circle is triple in diameter, plus a smaller part of a seventh in diameter and greater than ten seventy-nineths” (prop.3). Archimedes knew he could describe only the upper and lower limits of the relationship, but if you make an average of the two values you get 3,1419, with an error of less than three decimilles of the real value.

In the history of mathematics there is disagreement on the problem if to calculate the lower limit of the relationship as equal to 211875/67441 (i.e. about 3,14163) was Apollonius of Perga (big mathematician, thirty years younger than Archimedes) or the same syracusan, based on the “Misura of the circle”. But whoever did the calculations, this is the last recorded value before the famous astronomer Tolomeo (87-165) used, more than two centuries later, the less accurate value of 3 17/120 (about 3,14167).

At the height of their empire (27 BCE- 476 CE), the Romans often used for π the value of 3 1/8 (though knowing that 3 1/7 was more accurate), because for their legions it was easier to use 1/8 (which is a half a half a half). In fact, a Roman treaty of land surveying contains even the following instructions for the square of the circle: “You divide the circumference of a circle into four parts and take one as a side of a square; this square will have the area equal to the circle”. This implies that π= 4. Knowing these things, we are surprised that the Romans could build their great monuments.

It is known that China was home to one of the oldest scientific and mathematical civilizations. But although in the 12th century BCE the Chinese mathematics had reached good levels, the Chinese continued to use in their calculations the value of π=3. The authentic advances of China in the measurement of the circle would have been only partially century years later. Ch’ang Hong, minister and astrologer of Emperor An-ti in the first half of the π century CE, before he died, in 139, wrote that the square of the circumference of a circle stands at the square of the perimeter of the square circumscribed as 5 stands at 8. Using a unitary circle (a circle with a diameter of 1), we have that π /16= 5/8, so that by performing the calculation we find that the implied value of π is equal to V10 (i.e. about 3,162). Although it is far from accurate, the V10 value has become the most popular approximation for π throughout Asia for many years. Wang Fau (229-267) adopted for π the value of 3.156. Liu Hui, in 263, using the exhaustion method with a polygon of 3072 sides, found for π the value of 3,1416.

The astronomer of the V century Tsu Ch’ung-chih, using polygons inscribed at least 24.576 sides (with every probability he departed from a hexagon and doubled the number of sides eleven times: 6×2 ), dedusse ch eπ approximately 355/113 (about 3,1415929). This value differs only 8 millionths of 1 percent from the accepted value of 3,141592653589...No one would have found a more accurate value for over a thousand years.

Around 530 CE, the great Indian mathematician Aryabatha found an equation to calculate the perimeter of a polygon of 384 sides; a ratio between circumference and diameter of V9,8684 (= 3,141414). He wrote Aryabatha that if at it is equal to the side of a regular polygon of n sides inscribed in a circle of unitary diameter, and b is the side of a regular polygon inscribed with 2n sides, then b=V[1/2-1/2V(1-a )]. This is the equation used to find its well-known value of π.

The largest Indian mathematician of the Vπ century, Brahmagupta calculated the perimeters of the inscribed polygons of 12, 24, 48 and 96 sides, obtaining, respectively, the values of V9,65, V9,81, V9,86, V9,87. Then, armed with this information, he made a leap of faith assuming that, at the approach of the polygons to the circle, the perimeters, and therefore the π, they would be approximated to V10. It was, of course, totally wrong. It seems strange that it did not realize that its square roots were converging towards a significantly smaller number of 10 (in fact the square of π is only slightly greater than 9,8696). The square root of 10 was however the value he adopted, and it was the value that spread from India to Europe, and that it was used in Middle Ages by mathematicians from all over the world, perhaps also thanks to the fact that it is so easy to transmit and remember.

In the 9th century, philosophy and science were prospering in Islamic cultures, especially in current Iraq, where he lived and taught one of the greatest mathematicians, Abu ‘Abd-Allah ibn Musa al-Khwarizmi. In his works he used the values of 3 1/7, V10 and 62,832/20.000, attributing the first to the Greeks and the other two to Indian mathematicians. Most importantly, in his writings he used Indian figures, later also known as Arabic, including zero and decimal comma.

In 1085 Alfonso VI of Castile torn the city of Toledo to the Arabs and, with it, a great Library. The sovereign promoted the Latin translation of scientific works from Arabic, Greek and Hebrew. Even the crusaders of the XI-XIII century brought books and teachings to their homeland. At the beginning of the 12th century, Adelardo of Bath translated into Latin the Elements of Euclide, the Almagest of Ptolemy, the works of al-Khwarizmi and introduced in the West the Arabic numerals and its notation.

In 1202 Leonardo Pisano (Fibonacci) wrote the Liber Abaci, which contributed to the spread of Arabic numerals in Europe and in 1220, in Practica geometrye, Fibonacci used the approximate value of π of 1440/ (458 1/3) or 864/275 (about 3,1418). The philosopher Alberto of Saxony (1316-1390) wrote in the “De quadratura circuli” that the ratio of the circumference to the diameter was exactly 3 1/7. In the middle of the fifteenth century Cardinal Niccolò Cusano claimed to have squared the circle exactly, finding that the ratio of the circumference to the diameter was 3,1423. His method would later be proved false by Regiomontano (Johannes Muller, 1436-1476). In 1579 Viète used the experimental method of Archimedes of polygons inscribed and circumscribed to establish that π was greater than 3,1415926535 and less than 3,1415926537. To obtain this result he doubled the sides of two hexagons sixteen times, finding the perimeter of the polygons, inscribed and circumscribed, of 393.216 sides each. But although its value, exact to the tenth decimal figure, was the most exact measurement obtained until then, the greater conquest of Viète was to express π using an infinity product. This was, perhaps, the first time an infinity product was used to describe something; it was also one of the first steps in the subsequent evolution of the philosophy towards advanced trigonometric identity and towards infinitesimal calculation. Three Dutch mathematicians from the late sixteenth century also used the polygon method to calculate π. In 1585 Adriaan Anthonisz found that 377/120>π>333/106. In decimal notation, this means 3,14167>π>3,14151). Eight years later Adriaan van Roomen determined π to the fifteenth decimal, using inscribed polygon of more than one hundred million sides! Finally, Ludolph van Ceulen spent several years calculating π to the twentieth decimal figure using the same method as Archimedes, with the difference that its polygons had more than 32 billion sides each (60×2 ). When he died in 1610, van Ceulen had calculated 35 decimal digits, with the same methods used by mathematicians for thousands of years.

The exhaustion method was too uncomfortable to be used by many others in an attempt to proceed further. In 1621 the Dutch mathematician Willebrord Snell found a method of calculating based more on intelligence than resistance. While his predecessors had every time doubled the number of sides of a polygon, Snell found a better approximation using the same number of sides. Simply inscribing and circumscribing a hexagon to a circle, he could determine that π is between 3,14022 and 3,14160. Using a 96-sided polygon, Snell managed to determine the value of π up to the sixth decimal number and with a little more work managed to verify the 35 decimal figures of van Ceulen. Christian Huygens simply inscribed a triangle managed incredibly to equal Archimedes' approximation for the value of π; with a hexagon managed to determine nine exact decimal digits, using the limits 3,1415926533 and 3,1415926538.

The English mathematician John Wallis, a contemporary of Huygens, again faced the problem of finding the area of a circle. Wallis' equation, like Viéte's, is an infinity product, but differs from the fact that it involves only rational operations without any need for square roots. In the seventeenth century many other great mathematicians lived: Pascal, Keplero, Cavalieri, Fermat, to name only a few. Each of them provided an important piece to the solution of the puzzle and approached the very important innovation of the infinitesimal calculation.

James Gregory found an extremely elegant solution to the calculation of rainbows, which led to a completely new method of calculating π: the series of rainbows. Three years after Gregory found this new solution, the German Leibniz discovered independently the rainbow series. Leibniz was one of the fathers of infinitesimal calculation. The other father was Newton (1642-1727).

In order to determine the circumference ratio to the diameter, elementary calculations were no longer sufficient. The infinitesimal calculation and rainbow series allowed mathematicians to perform much faster calculations than polygon measurement; in fact the calculation of only four terms of one of the Newton series gives 3,1416. Soon the real problem became that of efficiency: finding an equation that converged on π with maximum speed. At the end of the seventeenth century, having these new instruments, the search for the decimal figures of π made a sharp leap forward. In 1699 Sharp found 72 decimal digits; in 1706 Machin 100 decimals; in 1719 Lagny calculated 127 digits (but only 112 were correct). Seventy-five years later, Vega calculates 140 digits.

Then, in the middle of the eighteenth century, he briefly turned his attention to the calculation of π one among the greatest and most mathematical prolific of all time, Leonhard Euler (better known as Eulero).

Eulero found many rainbow formulas and series to calculate π, used a method to calculate 20 decimal digits in one hour. After the brilliant steps ahead of Eulero, the Nineteenth century seems very poor if we merely consider the progress made in the methods for calculating π.

In fact, only at the beginning of the 20th century another mathematician would find a new set of equations to be applied to the problem. However, data hunters continued to find a growing number of figures: Callet 152 (1837), Rutherford 208 (1841), Clausen 248 (1847), Rutherford 440 (1853), Shanks 607 (1853), Shanks 707 (1873).

In 1945 D.F. Ferguson calculated 530 digits of π with a formula with rainbows. This result was the result of a full-price admission year of work with paper and pen, at the average pace of just over one figure per day. In 1947 Ferguson, with the help of one of the first table calculators, had found 808 digits of π. In 1948 Smith and Wrench found the thousandth decimal digit of π. In 1949 G. Reitwiesner, J. von Neumann and N.C. Metropolis used the Eniac computer, with 19,000 valves and hundreds of thousands of resistors and capacitors, to calculate 2037 digits of π. This calculation took only seventy hours with an average of one digit every two minutes. With the advent of electronic computers, in 1954, it was possible to calculate 3089 digits in only thirteen minutes (about 4 digits per second). In 1958 the first 704 digits in just 40 seconds., The first 10,000 digits in an hour and forty minutes. In 1961 with an Ibm 7090 100,265 digits were found with an average time of 3 digits per second.

In 1973 J. Guilloud and M. Bouyer found the millionth figure. In 1982 the value of π was found until 8,388,608 ° (= 2 ) decimal in just under seven hours. The combination of increasingly powerful computers and Gauss-Brent- Salamin algorithm have launched π calculations towards stratospheric heights. As we write, Kanada and Takahashi have calculated and verified more than 51 billion decimal figures of π, establishing a new world record.

The fact that there is no use in any concrete application that is not to test a new computer. A better understanding of the nature of π can instead be important for understanding physics, geometry and mathematics.

Attempts to understand the nature of π have committed many mathematicians. One of the most important developments was the demonstration that π was an irrational number, demonstration given in 1767 by J.H. Lambert (1728-1777). We remember that irrationals are those real numbers that cannot be written as a quotient of two whole numbers, that is, they are not fractional numbers. It is quite simple to show that numbers such as V2 or V3 are irrational, but Lambert had to wait in the eighteenth century to have the demonstration of the membership of π to that category. Its discovery assumes special importance if one thinks that rational numbers ( fractions) have a decimal development that can be finished or periodic; that is, decimal digits or end at a certain point, or are only followed by zeros, or show a continuous repetition of a certain block of numbers. Now, if it were rational, it should show one of these two behaviors, so sooner or later, the decimal development should be definitively determined. Proving that Lambert was irrational instead guaranteed that his decimals would never end. As if it was not enough, in 1882 F. Lindemann demonstrated that π was not only irrational, but also transcendent. Synthetically we can say that the set of all real numbers can think it divided into two exhaustive subsets and that exclude each other: algebraic numbers and transcendent numbers. We can identify algebraic numbers with the familiar amounts we encounter in arithmetic or elementary algebra. The whole numbers, for example, are all algebraic, like all fractions (rational numbers), their square, cubic roots and so on. A number is said to be transcendent if it is not algebraic: if it is not a solution, that is, of no polynomial equation with whole coefficients. The division between algebraic and transcendent numbers is a strong dichotomy, such as sex between human beings: either you are male or you are female, there are no middle ways. A fundamental point is that no transcendent number can be built with line and compact. Lindemann's merit was to prove that π is a transcendent number. In other words, π is not algebraic and therefore is not even edible. Lindemann's discovery showed that the square of the circle, a problem that had occupied mathematicians from the time of Hippocrates until modern times, was a lost cause. The line and the compass alone are insufficient to turn the circles into squares.

The story of π also allows us to talk about one of the most important mathematicians of this century, S. Ramanujan ( 1887-1920). In Ramanujan's theories there is an anticipation of the method that is the basis of the most recent calculations of π, although in order to apply it concretely it was necessary to wait for the development of efficient algorithms, modern supercalculators and new ways to multiply numbers. After almost eighty years, scientists and mathematicians are still engaged in studying the fascinating equations of this genius, applying them to everyday problems and using them to generate other algorithms, designed to be efficiently applied by computers. These are iterative equations: they allow to reintroduce in the formula the results of the calculation to have an even better approximation. The results are amazing because every time you turn the algorithm you can double or quadruple the number of relevant figures. Ramanujan, like most mathematicians, could not resist the temptation to explore π, and his great insights allowed considerable progress in the study of the number.

The figures of π follow infinity in a river that appears quite random. Beyond the taste of establishing a certain type of record, it might seem that the attempt to calculate millions of decimal places of the number is entirely ocious. Thirty-nine digits of π are sufficient to calculate the circumference of a circle that encloses the high-price admission universe known, with an error not exceeding the radius of a hydrogen atom. It is difficult to imagine physical situations requiring a greater number of figures. So why are the mathematical and computer experts not satisfied with the first 50 decimal figures of π? You can give different answers. One is that the calculation of π has become a kind of parameter for processing: it serves as a measure of refinement and reliability of the computers that carry out it. In addition, the search for increasingly precise values of π leads the mathematicians to discover unexpected and interesting implications of the theory of numbers. Another reason, more sincere, is simply the existence of π: “because there is”. In fact, π is a fixed theme of culture mathematics for more than two and a half millennia. Moreover, there is always the possibility that these calculations serve to shed light on some of the mysteries surrounding π, a universal constant still not well known despite its relatively elementary nature.

At the end of the 20th century we must not forget that this long mathematical journey to the discovery of π began from a short treaty written 2225 years ago by the unsurpassed Archimedes of Syracuse: “The measurement of the circle”.

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