Featured Post from Villanova University

The Life of Pi

Mar 13, 2024

5 min

The digits of pi are infinite. It’s an irrational number—one whose decimal never ends and never repeats. Over time, with brilliant minds and brilliant technology, humans have been able to calculate its digits further and further, now forming a 100-trillion decimal tail.


In many ways, the infinitely long decimal expansion of pi is a metaphor for its limitless applicable value. With new ways of thinking, measuring and computing, the significance of pi has permeated throughout the study and use of mathematics and countless other disciplines. Pi is a mathematical constant by definition and also because of its ubiquity.


“To offer an Augustinian-themed analogy, pi is to mathematics as Gregor Mendel’s peas are to biology,” said Katie Haymaker, PhD, associate professor of mathematics and statistics at Villanova University. “At first glance, Mendel’s experiments convey some basic understanding of the function of genetics. However, the way genes are expressed is incredibly complicated, and scientists continue to study genetics and gene therapies today. Similarly, mathematicians today study objects that are historically tied to the development of ideas inspired by explorations of pi. Pi is one gateway to a garden of mathematical possibilities.”


Dr. Haymaker currently teaches a course called “Mathematics for Human Flourishing,” inspired by the Francis Su book of the same title. Her class discusses math in everyday life and how engaging in mathematical exploration can develop virtues like studying ethics or other humanities can and also expand quality of life.


“I don’t have class this Pi Day, but usually I would share a little something about pi if I did,” she said.


So, what exactly is pi? The easy answer is that it’s the ratio of the circumference (the length all the way round) to the diameter (the length across) for any circle.


“If you measure across a circle with a piece of rope, how many pieces of the same rope would you need to measure around?” Dr. Haymaker explained.


The never-ending answer starts with 3.14, hence the common celebration of Pi Day on March 14.


This value has been studied for millennia: The ancient Egyptian Rhind Papyrus, dating to around 1650 BCE, offers a very close estimation for pi as 3.16049. Even the Bible, in 1 Kings 7:23, a circular vessel is described, and the ratio of circumference to diameter calculated to exactly three.


“There are interesting explorations by biblical scholars about why this number is not exactly the value that we now know as pi, including that the brim around the vessel accounts for the discrepancy,” Dr. Haymaker said.


Over time, novel methods for approximating pi were discovered, advancing humankind’s understanding of its value, and leading to various other paths of mathematical study. The famous Greek mathematician and inventor Archimedes, for instance, discovered a way to approximate pi’s value by use of a regular polygon (a closed geometric figure made of equal straight lines and angles).


In those times, measuring a circle was not well defined, according to Dr. Haymaker. By placing the regular polygon inside the circle, the straight lines can be measured, those straight lines can be split to form a regular polygon with more measurable sides, and so forth. The more sides, the closer the measurement is to the true circumference of the circle.


“That whole idea of approximating to the actual value is the main idea of calculus,” she said. “This notion by Archimedes predated that entire field.”


Later, formulas were developed that surpassed Archimedes’ technique. A better approximation of pi was discovered around the year 450 by Chinese mathematician Tsu Ch’ung-chih, arriving at the easy-to-remember fraction 355/113.


“This is one rational approximation to pi, and it’s also a fun Pi Day party trick because it’s the best approximation you can get with a fraction like this whose denominator is less than 10,000,” Dr. Haymaker said.


These are just two examples of the many contributions made to advance the understanding of pi. Even the use of the Greek symbol to describe the ratio, which was popularized by 18th-century mathematician Leonhard Euler, was instrumental. Prior to that, there had been no agreed-upon symbol and the concept was often described using only words.


In the thousands of years since the first recorded approximation of pi, the methods of its approximation—and pi itself—have been applied to multiple other fields of mathematical study such as trigonometry and calculus. For Dr. Haymaker, pi was important in the development of her own understanding of mathematics as well.


“I have learned to expect the unexpected when it comes to pi,” she said. “It shows up in all sorts of places and it inspires us to dig deeper into why.”


Today, pi can be calculated to 100-trillion decimal places (though only a dozen or so are needed for even the most accurate applications). On Pi Day 2023, 21-year-old Rajveer Meena memorized the first 70,000 digits, breaking a Guiness World Record in a mind-boggling 10-hour recitation.


“I think that pi is fascinating to people because its digits behave in a random way and they go on forever,” Dr. Haymaker said. “If there is a string of numbers that is special to you—take 1842 for example—it exists somewhere in the digits of pi. In exploring pi, it feels like you are exploring the infinite, which naturally inspires mystical feelings about the number.”


By the way, the string of numbers 1842 (the year Villanova was founded) appears at position 1738 counting from the first number after the decimal, according to the Pi-Search Page.


So, whether it’s on March 14, July 22 (because the fraction 22/7 is a rational approximation of pi) or whenever you might celebrate, look down at that beautiful circular dessert you will cut into and think about “why pi(e)?”


“Some people may scoff at Pi Day as being separate from ‘real’ mathematics,” Dr. Haymaker said. “But there is a joyfulness in celebrating this day that represents a deep connection that people have to mathematical discovery. After all, a person wrote the Rhind Papyrus, and it is people who program the computers that searched for the 100 trillionth digit of pi. This quest to understand is a deeply human endeavor.


“To quote Francis Su, ‘the pursuit of math can, if grounded in human desires, build aspects of character and habits of mind that will allow you to live a more fully human life and experience the best of what life has to offer.’ So, if eating pie and other round foods on March 14 inspires someone to learn, ask questions, pursue answers and see themselves as an explorer of mathematics, then it is indeed a day to celebrate.”


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Since this lawsuit was filed, it has been a popular topic of public discourse, much of which has centered on the product—a crustless peanut butter and jelly sandwich—itself. Is that what this is truly about? Professor Moorad: Much of the commentary has been focused on the argument of whether Smucker's is permitted to have a monopoly of peanut butter and jelly sandwiches, or if Trader Joe's can actually infringe upon the Uncrustables product without necessarily using the actual trademarked name. While both discussions are legitimate conversations folks could have while munching on the delicious snack products, they are not necessarily the relevant legal claims at the crux of this lawsuit. Q: Before we get into what those relevant legal claims are, Smucker’s has filed dozens of trademarks in its 128-year history. What sorts of intellectual property do these trademarks generally protect? 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GX13+1 is a binary system consisting of a neutron star orbiting a K5 III companion star—a cooler giant star nearing the end of its life. Neutron stars are small, incredibly dense cores of supergiant stars that have undergone supernovae explosions. They are so dense, Dr. Neilsen says, that one teaspoon of its material would weigh about the same as Mount Everest. Because of this, they yield an incredibly strong gravitational field. When these highly compact neutron stars orbit companion stars, they can pull in, or accrete, material from that companion. That inflowing material forms a visible rotating disk of gas and dust called an accretion disk, which is extremely hot and shines brightly in X-rays. It’s so bright that sometimes it can actually drive matter away from the neutron star. “Imagine putting a giant lightbulb in a lake,” Dr. Neilsen said. “If it’s bright enough, it will start to boil that lake and then you would get steam, which flows away like a wind. It’s the same concept; the light can heat up and exert pressure on the accretion disk, launching a wind.” The original purpose of the study was to use XRISM to observe an accretion disk wind, with GX13+1 targeted specifically because its disk is persistently bright, it reliably produces winds, and it has been well studied using Chandra— NASA’s flagship X-ray observatory—and other telescopes for comparison. XRISM can measure the X-ray energies from these systems a factor of 10 more precisely than Chandra, allowing researchers to both demonstrate the capabilities of the new instrument and study the motion of outflowing gas around the neutron star. This can provide new insights into accretion processes. “It's like comparing a blurry image to a much sharper one,” Dr. Neilsen said. “The atomic physics hasn't changed, but you can see it much more clearly.” The researchers uncovered an exciting surprise when the higher-resolution spectrum showed much deeper absorption lines than expected. They determined that the wind was nearly opaque to X-rays and slow at “only” 1.4 million miles per hour—surprisingly leisurely for such a bright source. Based on the data, the team was able to infer that GX13+1 must have been even brighter than usual and undergoing a super-Eddington phase. So much material was ejected that it made GX13+1 appear fainter to the instrument. “There's a theoretical maximum luminosity that you can get out of an accreting object, called the Eddington limit. At that point, the radiation pressure from the light of the infalling gas is so large that it can actually hold the matter away,” Dr. Neilsen said, equating it to standing at the bottom of a waterfall and shining light so brightly that the waterfall stops. “What we saw was that GX13+1 had to have been near, or maybe even above, the Eddington limit.” The team compared their XRISM data from this super-Eddington phase to a set of previous observations without the resolution to measure the absorption lines directly. 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