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.”


Powered by

You might also like...

Check out some other posts from Villanova University

Solar Déjà Vu: Eclipses, Like The One in August, Follow Predictable Cycles, Explains Villanova Astronomer featured image

3 min

Solar Déjà Vu: Eclipses, Like The One in August, Follow Predictable Cycles, Explains Villanova Astronomer

As the moon passes in front of the sun on August 12, 2026, viewers along the eclipse’s path of totality in Iceland, Greenland and Northern Spain will be treated to more than two minutes of daytime darkness. For many, it will be their first experience in totality. For the eclipse itself, it’s just another slightly different chapter in a millennium-long rinse and repeat. That’s because every eclipse belongs to what is called a saros series—a “family” of eclipses that follow a predictable schedule. A saros series begins and ends with partial eclipses near opposite poles of the Earth, and in between may feature annular, hybrid or total eclipses—like the kind occurring this August—across the world. “Then the series will end,” said Frank Maloney, PhD, associate professor of Astrophysics and Planetary Science at Villanova University. “New saros series are always coming into being, and old ones fading from existence. While they are here, they usually have between 70-73 eclipses over a span of 1,200-1,300 years.” Within the series is what is known as a saros cycle. It’s a period of roughly 18 years, 11 days and eight hours, when a natural synchronization of the moon’s three lunar phases arises. That synchronization can predict both lunar and solar eclipses, which follow very similar geometrics to the ones that precede and follow in their series. But why, then, aren’t all eclipses in a series over the same viewing area? The answer lies more with the movement of the Earth than with the eclipse. Within each series, the biggest cause of change from eclipse to eclipse comes from that last eight hours. It’s only the length of a workday, but each cycle, the Earth has rotated an additional 120 degrees from the previous. While the eclipse itself remains very similar to its predecessor and successor in the series, the viewing area does not. “Every fourth eclipse, or roughly every 54 years, you are back to where you began,” said Dr. Maloney. “That fourth eclipse in a saros is referred to as an exeligmos. But even that eclipse won’t follow the exact same path for a few reasons, including slight variations in the moon’s orbital node and distance from Earth.” Same region, but different track. The August 2026 eclipse will pass over Greenland, Iceland and Northern Spain, but its exeligmos from July 10, 1972 passed over parts of Russia, Alaska and Northeast Canada. Ed Guinan, PhD, ’64 CLAS, professor of Astrophysics and Planetary Science, witnessed that one with a Villanova contingent on the Gulf of St. Lawrence in Nova Scotia. Fifty-four years from now, this eclipse will be only a partial one, as the saros series begins to wane. The upcoming August event, in fact, is the penultimate of 10 total solar eclipses in Saros 126. The 10th will be in August 2044—the next one visible in the contiguous United States, seen in totality in parts of Montana, North Dakota and South Dakota. Saros 126 began with a partial eclipse on March 10, 1179, and will end with another partial on May 3, 2459. Humans understood these patterns long before this one began. In the first millennium BCE, the Chaldeans discovered the saros cycle by “carefully keeping record of observations” of eclipses over time, explained Dr. Maloney. “Our lives are linked with the sun. To the ancients, having the sun disappear was believed to be a really bad thing, like a demon devouring it. That goes for lunar eclipses too, which caused the moon to turn red and really bother people. When they could begin to predict them, total eclipses ceased to be the powerfully bad omens that ancients believed they were. It was still a mystery as to why it happened, but at least it was predictable, which reduced people's anxieties.” Now, the anxiety stemming from eclipse viewing is saved for the fierce competition for hotels along the path of totality and threats of cloudy weather. “But don’t worry,” Dr. Maloney said. "If you miss one, you might be able to see it ‘nearby’ in another 54 years.”

Ahead of America250, Villanova Historian Reveals How Independence Hall Almost Didn't Survive featured image

4 min

Ahead of America250, Villanova Historian Reveals How Independence Hall Almost Didn't Survive

Philadelphia’s Independence Hall has long occupied an outsized place in the American imagination. The space where the Continental Army was established, the Declaration of Independence adopted and the United States Constitution ratified, the site was once described by President Abraham Lincoln as the source “where were collected together the wisdom, the patriotism, the devotion to principle, from which sprang the institutions under which we live.” In July, these hallowed grounds will yet again take center stage, as the country observes its semiquincentennial, or America250, celebration. In due course, House lawmakers will gather at the landmark for a special commemorative event, mayors from across the U.S. will march to the gates in a show of civic pride and solidarity, and thousands of visitors will flock to the site daily in appreciation for its significance to the cause of “Life, Liberty and the pursuit of Happiness.” However, while Independence Hall’s role in the national saga will go widely remarked and recognized, the building itself has a story that remains largely unknown. According to Whitney Martinko, PhD, associate professor of History and director of the Albert Lepage Center for History in the Public Interest at Villanova University, the “cradle of American democracy” almost never survived the country’s infancy. “Early on, the challenge was about two things,” says Dr. Martinko, who specializes in public history, historic preservation and the early U.S. “One was about ownership of what was called the ‘Old State House,’ because it was the former statehouse in the colony of Pennsylvania. And the second was about the development of the city around it.” As Dr. Martinko explains, in the early 19th century, Independence Hall—then the Old State House—was under the control of the Commonwealth of Pennsylvania, which had shifted its governmental seat from Philadelphia to Harrisburg by 1812. To fund the construction of a new capitol building in the wake of the move, Pennsylvania legislators seriously contemplated selling the site to private enterprise, with the surrounding area undergoing a development boom. “Today’s Independence Mall was built up entirely,” says Dr. Martinko. “In the 18th century, it was full of buildings, shops and houses, and by the 19th century, it had become a huge furniture district and a heart of commerce in many ways.” As plans were drawn up to deliver the hall to the highest bidder, local resistance quickly emerged. Opposed to the landmark’s loss, citizens of Philadelphia and municipal leaders rushed to the defense of the building and its lawn, arguing that their preservation entailed a necessary public good. “Everyone looked to this site as the heart of the new nation. It’s a historic site. It’s an important building. People thought of it as one of the great pieces of Georgian architecture at the time,” says Dr. Martinko. “It was also seen as a civic space, as people gathered there on Election Day. And its lawn was highly valued, with green, open space considered important even then, for air circulation. So, it was really seen as a political space, a civic space and a green space that was important for the well-being of Philadelphians and the health of Philadelphia.” Deliberations over the fate of Independence Hall would continue for a period of five years, up until 1818. After a spirited public campaign, a settlement was finally reached when the City of Philadelphia purchased the plot from the Commonwealth of Pennsylvania for $70,000 (roughly $1.85 million in today’s currency). In essence, the deal would forevermore secure Independence Hall’s place within the pantheon of great American shrines, parks and monuments. However, in a terrific irony, it would also eventually lead to the loss of a different piece of history: Between 1950 and 1967, the 19th-century development projects that once threatened Independence Hall became a casualty of the city’s efforts to make the “birthplace of America” an urban focal point, with the creation of Independence Mall. “Those buildings were all torn down in the mid-20th century, when Ed Bacon and the City Planning Commission decided to make Independence Hall a major attraction,” says Dr. Martinko. “There were debates surrounding this issue as well. The Jayne Building was one of the 19th-century buildings that was demolished and that is most well-known. So, there’s this sense of preserving 18th-century history through the demolition of 19th-century architecture.” As the nation approaches its 250th anniversary, the near loss of Independence Hall and the removal of its 19th-century neighbors stand as striking examples of the ways in which what we value, and how much we value it, evolves over time. What’s more, the historic threats to Philadelphia’s most famous site serve as a poignant reminder of the delicate nature of public memory and preservation—and the fact that the places we treasure today may not always be with us tomorrow. “Even though it seems absurd to us now, we’re still seeing debates over the line between redevelopment and connection with the past,” concludes Dr. Martinko. “It’s not that no one saw the value of Independence Hall, or that they didn’t see it as historic. It was just this debate that a lot of very reasonable people continue to have today: Is this what really needs to be preserved? And how should it be preserved?”

100 Years After the "Launch" of Aerospace Industry, Villanova Faculty Continue to Innovate the Sector featured image

4 min

100 Years After the "Launch" of Aerospace Industry, Villanova Faculty Continue to Innovate the Sector

In 1926, Robert H. Goddard launched the world’s first liquid-fueled rocket in Auburn, Mass. Goddard’s 10-foot-tall rocket was airborne for just 2.5 seconds, reaching speeds of 60 miles per hour before landing 184 feet away from the launch site. A century later, the aerospace industry is booming, with new technology and missions making headlines every day—some with incredible success, and others encountering challenges that send scientists back to the drawing board. In February 2026, NASA delayed the launch of Artemis II—its next mission to the moon—citing issues with helium flow in the rocket’s systems. By April, the mission was on track again, and Artemis II completed the first crewed flight to the moon in more than half a century. Crew members for the upcoming Artemis III mission were also recently announced, as well as a timeline and overview for Artemis IV, the first planned crewed mission to the lunar South Pole in 2028. Amid these successes and setbacks, researchers continue to innovate the field and develop new technologies designed to help expand our knowledge of the vast universe. That innovation comes from diverse and unique places, including Villanova University. Research in Flight Student interest in aerospace led to the creation of a Master of Science in Aerospace Engineering (MSAE) program at Villanova University, which began in the Fall 2025 semester. Sergey Nersesov, PhD, associate professor of Mechanical Engineering, was instrumental in the inception of the MSAE program and helped develop courses in space flight mechanics, applied aerodynamics, aerospace structures analysis and advanced flight dynamics and control, among others. The idea for the graduate program was inspired by the growing popularity of the College of Engineering’s Minor in Aerospace Engineering. The minor attracts students from across the university, drawing from other majors and colleges at Villanova. For example, Dr. Nersesov recently collaborated on a research project focused on spacecraft and satellite control systems with Aedan Disanto ‘26 CLAS, an astrophysics and planetary sciences major and aerospace engineering minor. “If you look up at the sky, sometimes you see satellites chasing each other,” said Dr. Nersesov. “Dynamics and control researchers develop algorithms to ensure proper spacing between the satellites so they can function correctly.” The spacing between satellites is crucial to avoid collision, which is also a potential issue when a spacecraft approaches a space station to dock. In this situation the velocities, rotation and orientation of both vehicles are carefully controlled so that docking mechanisms align correctly, which requires up to 12 variables to be coordinated simultaneously. Dr. Nersesov and Disanto analyzed the algorithms needed to guarantee perfect satellite function and built upon them, discovering more efficient ways to operate vehicles in space. This summer, Dr. Nersesov and his students will also begin designing a prototype for a new kind of drone. Typical drones use ample amounts of energy to become airborne and capture photos or video content because they rely entirely on thrust to hold themselves up. To improve effectiveness, Dr. Nersesov and his students aim to create a drone in the style of an airplane, with vertical takeoff and landing (VTOL) capability. The drone will take off vertically, like a helicopter, but then transition to flying horizontally like an airplane, allowing lift from the wings to reduce the energy needed to stay airborne. As a result, it could stay in the air up to ten times longer than a hovering drone. While the project focuses on a single aircraft design, it represents the type of forward-thinking research driving the aerospace field today. Aerospace Engineering with Biology Elsewhere, Qianhong Wu, PhD, chair of Mechanical Engineering in the College of Engineering, is exploring a concept called super-lubrication, inspired by the way red blood cells move through the human body. Blood cells travel through capillaries narrower than their own diameter without damaging themselves or the vessel walls. A soft, porous layer called the endothelial glycocalyx within the vessels allows cells to glide through, reducing friction. In studying this biological process, an idea emerged that could potentially be translated to the aerospace field. Dr. Wu’s team is currently applying their deep understanding of biomechanical processes to applications that might reduce aerodynamic friction on aircraft surfaces by more than 90 percent. This lower friction may also improve fuel efficiency and extend flight endurance for drones or other aircraft. “Our work is an example of how thinking outside your traditional field can lead to innovation,” said Dr. Wu. “Sometimes the solution comes from a completely different subject, like biology.” A Century of Momentum One hundred years after Goddard’s brief but groundbreaking flight, aerospace innovation has expanded far beyond its earliest experiments. Today, progress in the field depends not only on major missions and milestones, but also on the steady work of researchers refining systems and exploring new ideas. At Villanova, that work is taking shape across disciplines—from spacecraft control systems to biologically inspired materials. Together, these efforts reflect how the field continues to evolve through collaboration and creativity.

View all posts
Powered by