Representations of pi (π) help scientists use values that are close to the real thing without having to store a million digits. Creating a new pi required using a series, which is a structured set of terms that either converge to a single expression or diverge. In a new study, physicists use principles of quantum mechanics to build a new model of the abstract concept of π. Or, more accurately, they built a new model that happens to include a cool new representation of π. But what does this mean, and why do we need different representations of π?
Because quantum mechanics looks at tiny particles, one at a time, even simple questions can have complex answers that require massive amounts of computing power. High-tech video games and movies like Avatar can take days or more to render, and that’s still not up to par with reality. In a new paper published in the peer-reviewed journal Physical Review Letters, physicists Arnab Priya Saha and Aninda Sinha describe their new version of the quantum model that reduces complexity but maintains accuracy, Popular Mechanics reports.
This is called optimization. Think of how early internet video buffered in chunks of similar colors, or how classical animators drew static bodies with separate moving parts on top, or how people cut corners of square walkways until they get the shortest path down a dirt path. We are surrounded by optimization and optimizing behavior.
The Number Pi. LiveJournal
Saha and Sinha combined two existing ideas from mathematics and science: the Feynman diagram for particle scattering and the Euler beta function for scattering in string theory. The result is a series—something represented in mathematics by the Greek letter Σ surrounded by parameters.
Series can eventually generalize into general equations or expressions, but they don’t have to. And while some series diverge—that is, the terms keep alternating with each other—others converge to one approximate, specific result. That’s where pi comes in. The digits of pi extend to infinity, and pi itself is an irrational number, meaning it can’t really be represented as a whole fraction (the one we often learn in school, 22/7, isn’t very precise by 2024 standards).
But this number can be represented quite quickly and well by a series. This is because the series can continue to build up values down to the smallest digits. If a mathematician composes the terms of the series, he can use the resulting abstraction to do mathematical calculations that are impossible with the approximation of π, and are truncated to 10 digits by a standard desk calculator. The sophisticated approximation enables the work with nanoparticles that originally inspired these scientists.
“In the early 1970s,” Sinha said in a statement from the Indian Institute of Science, “scientists briefly explored this line of research but quickly abandoned it because it was too complex.”
But mathematical analysis like this has come a long way since the 1970s. Today, Sinha and Saha are able to analyze an existing model and remake it with altered terms. They can plot the sequence and see that it converges to π in far fewer terms than expected, making it easier for scientists to run the series and use it for further work.
All of this requires decades of fundamental work in the field, and a great deal of work showing that certain mathematical moves work where others don’t. It’s a commentary on the continuous and collaborative nature of mathematics, even when the result is a working model that can help scientists. Our ability to make meaningful approximations has grown along with our ability to solve complex problems directly.
“Doing such work, although it may not have immediate application in everyday life, provides the pure pleasure of doing theory for the sake of doing it,” Sinha said in a statement.
The number represented by pi (π) is used in calculations when dealing with anything round (or nearly round), such as circles, spheres, cylinders, cones, and ellipses. Its value is essential for calculating many important quantities about these shapes, such as understanding the relationship between a circle’s radius, its circumference, and its area (circumference = 2πr; area = πr2).
Pi is also used in calculations to determine the area of an ellipse and to find the radius, surface area, and volume of a sphere. There are many round and nearly round objects in our world; finding the exact value of pi helps us build, manufacture, and work with them more accurately, The Conversation reports.
Historically, people had only very rough estimates of π (like 3, or 3.12, or 3.16), and while they knew they were just estimates, they had no idea how far off they were.
The search for the exact value of π led not only to increased accuracy, but also to the development of new concepts and methods, such as limits and iterative algorithms, which then became fundamental to new areas of mathematics.
Between 3,000 and 4,000 years ago, people used trial-and-error approximations of π, without doing any math or taking into account possible errors. The earliest written approximations of π are 3.125 in Babylon (1900–1600 BC) and 3.1605 in ancient Egypt (1650 BC). Both approximations start at 3.1—close enough to the actual value, but still relatively far off.
The number Pi
The first rigorous approach to finding the true value of π was based on geometric approximations. Around 250 BCE, the Greek mathematician Archimedes drew polygons around both the outside and inside of circles. Measuring their perimeters gave upper and lower bounds on the range containing π. He started with hexagons; by using polygons with increasingly larger numbers of sides, he eventually calculated three exact digits of pi: 3.14. Around 150 CE, the Greco-Roman scientist Ptolemy used this method to calculate the value of 3.1416.
Independently, around 265 CE, the Chinese mathematician Liu Hui created another simple iterative algorithm based on a polygon. He proposed a very fast and efficient approximation method that yielded four digits of accuracy. Later, around 480 CE, Zu Chongzhi adopted Liu Hui’s method and achieved seven digits of accuracy. This record stood for another 800 years.
In 1630, Austrian astronomer Christoph Grienberger arrived at a 38-digit number, which is the most accurate approximation achieved by hand using polygonal algorithms.
The development of infinite series methods in the 16th and 17th centuries greatly improved people’s ability to approximate pi more efficiently. An infinite series is the sum (or, much less commonly, the product) of terms of an infinite sequence, such as ½, ¼, 1/8, 1/16, … 1/(2n). The first written description of an infinite series that could be used to calculate π was set out in Sanskrit verse by the Indian astronomer Nilakantha Somayaji around 1500 CE, with a proof provided around 1530 CE.
The Number Pi. Komsomolskaya Pravda
In 1665, English mathematician and physicist Isaac Newton used an infinite series to calculate pi to 15 digits, using a calculus he co-discovered with German mathematician Gottfried Wilhelm Leibniz. The record continued to be broken after that. He reached 71 digits in 1699, 100 digits in 1706, and 620 digits in 1956—the best approximation achieved without the aid of a calculator or computer.
In parallel with these calculations, mathematicians were also investigating other properties of π. Swiss mathematician Johann Heinrich Lambert (1728-1777) was the first to prove that π is irrational—it has an infinite number of digits that never appear in a repeating sequence. In 1882, German mathematician Ferdinand von Lindemann proved that π cannot be expressed as a rational algebraic equation (for example, π²=10 or 9π4-240π2+1492=0).
Bursts of computing even more digits of π followed the adoption of iterative algorithms, which repeatedly create an updated value using a calculation performed on a previous value. A simple example of an iterative algorithm allows you to approximate the square root of 2 as follows, using the formula (x+2/x)/2:
(2+2/2)/2 = 1.5
(1.5+2/1.5)/2 = 1.4167
(1.4167+2/1.4167)/2 = 1.4142, which is already a very close approximation.
Progress toward more digits of pi occurred with the use of the Machin-type algorithm (a generalization of the English mathematician John Machin’s formula developed in 1706) and the Gauss-Legendre algorithm (late 18th century) in electronic computers (invented in the mid-20th century). In 1946, ENIAC, the first general-purpose electronic computer, calculated pi to 2,037 digits in 70 hours. The most recent calculation showed pi to more than 13 trillion digits in 208 days!
It is widely accepted that for most numerical calculations involving pi, a dozen digits provide sufficient accuracy. According to mathematicians Jörg Arndt and Christoph Haenel, 39 digits are sufficient for most cosmological calculations, since this is the precision needed to calculate the circumference of the observable universe to within the diameter of a single atom. Therefore, more digits of pi have no practical use in calculations; rather, the current pursuit of more digits of pi is to test supercomputers and numerical analysis algorithms.
There are also fun and simple methods for estimating the value of π. One of the most famous is the Monte Carlo method.
The method is quite simple. To try it at home, draw a circle on a piece of paper with a square around it. Imagine that the sides of the square are 2 long, so its area is 4; the diameter of the circle is therefore 2, and its area is π. The ratio between their areas is π/4, or about 0.7854. Now take a pen, close your eyes, and place dots on the square at random. If you do this enough times, and your efforts are truly random, eventually the percentage of your dots landing inside the circle will be close to 78.54% – or 0.7854. You have now joined the ranks of mathematicians who have been calculating π for centuries.
Pi Day was created in 1988 in San Francisco when Larry Shaw, the legendary tech curator at the city’s Exploratorium, saw a connection between March 14 (3.14) and pi (3.14159…). Add to that the fact that March 14 is Albert Einstein’s birthday, and you have a ready-made celebration.
The number pi seems simple: it’s the ratio of a circle’s circumference to its diameter, but it’s anything but. The number has an almost mystical quality that has sent mathematicians down rabbit holes trying to find more digits of pi, as well as ways it intersects with the rest of mathematics. Here are five incredible facts about the magical number in honor of Pi Day.
There are several claims about the very first discovery of pi. Historians point to the Babylonian and Egyptian cultures as likely discoverers of the concept around 2000 BC. Archimedes further refined it around 250 BC in his treatise On the Measurement of a Circumference.
In 250 AD, Chinese mathematician Liu Hui determined in a work known as the Nine Chapters on the Mathematical Art that the ratio of a circle’s circumference to its diameter must be greater than three. Using a 96-sided polygon, Liu Hui determined that the ratio must be greater than three. In fact, he calculated the first five digits: 3.1416.
In the West, Archimedes became known as the discoverer of the constant. For centuries, 22/7 (a fractional approximation of π) was known primarily by its full Greek name, περιφέρεια. This translates as “periphery,” which makes sense given how the constant relates to a circle.
The Number Pi. Belarus Today
But in the early 1700s, Welsh mathematician William Jones decided to simplify the whole endeavor. In 1706, Jones published Synopsis Palmariorum Matheseos, a beginner’s text on calculus and infinite series. Since the constant is infinite, Jones began shortening it to simply π. Jones eventually rose through the math community, befriending legendary names Isaac Newton and Edmund Halley. But the name didn’t stick.
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It wasn’t long before pi began its rise to mathematical fame. The idea of an infinite number was appealing to many, especially as scientific and technological discoveries flourished during the Industrial Revolution. For some, like William Shanks, pi became an obsession. Shanks was born in 1815 in rural England. Little is known about his life, but he became the landlord of a private boarding school in a small village called Houghton, which was primarily known at the time for coal mining. However, this didn’t interest Shanks much. Instead, he devoted his free time to calculating and determining more and more digits of pi. He was not a mathematician, but that didn’t stop him from spending his mornings writing down calculations and his afternoons checking them.
Over time, he made impressive progress. In 1853, he published a book entitled Contributions to Mathematics, Involving Chiefly the Straightening of the Circle , which gave 607 decimal places for π, the first 500 of which were independently verified.
In 1873, Shanks reached the pinnacle of his powers with pi. He calculated to 707 decimal places, a record that stood until the advent of the electronic computer. But there was another humiliation: in 1944, a mathematician named D. F. Ferguson independently reviewed Shanks’s work. There was an error. Ferguson discovered that Shanks had misplaced two terms, throwing off his 528th number.
The sheer computing power of computers changed mathematics forever. Two mathematicians offered an early demonstration of this power in 1962 using the early IBM 7090 computer. First released in 1959, the 7090 was an all-transistor system. This was a completely new concept in 1959, when most computers still used vacuum tubes, and the 7090 could perform calculations six times faster than those vacuum tubes. It could be rented for as little as $63,500 a month (about $700,000 in today’s money).
The 7090’s customers were primarily institutions like the Department of Defense and NASA. In 1961, mathematicians Daniel Shanks and John Wrench used it to reach digits of pi that William Shanks couldn’t imagine: 100,000. According to an academic paper showcasing their work, the 7090 took eight hours and 43 minutes to complete the calculation. That may seem like a long time, but when you compare it to Shanks’s lifelong work on the topic, it’s easy to see how the computer would revolutionize mathematics. Versions of the 7090 would later be used on the Gemini and Mercury space missions.
IBM 7090. A Day in Tech History
The obsession with pi continues into the modern era. Because it is an irrational number, it is not the end of pi, and the pursuit can go on indefinitely. Across the world and centuries, Shanks found a common soul in Yasumasa Kanada, a professor in the computer science department at the University of Tokyo. In 2002, Kanada set a new record: pi to 24 trillion decimal places.
It took Canada’s team five years to develop the program used to produce their result. And while their record has since been broken, Canada’s efforts show why the fascination with pi endures. At a certain point, the number loses any practical or even academic use. But the challenge to achieve more with it shows a human determination that has spanned centuries. Today, the record stands at a whopping 62.8 trillion decimal places.
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