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

Villanova Tax Experts Break Down Legislative Changes, Best Practices Before Filing Deadline featured image

4 min

Villanova Tax Experts Break Down Legislative Changes, Best Practices Before Filing Deadline

It's time to collect your W-2s, 1098s and 1099s: On April 15, Americans are required to submit their annual tax returns—recapping earnings, income and life events from the past calendar year. Yet, as filers prepare their records and statements for 2025, they should anticipate some substantial departures from the 2024 season. According to Stephen Olsen, JD, faculty director of the Graduate Tax Program at Villanova University Charles Widger School of Law, and Luke Watson, PhD, associate professor of Accounting in the Villanova School of Business, one of the most significant drivers of change is the One Big Beautiful Bill Act (OBBBA), signed into law on July 4, 2025. A sweeping piece of legislation, OBBBA has instituted a number of new tax rules and regulations, including a deduction of $6,000 for taxpayers over 65 years of age, a deduction for certain overtime payments up to $12,500, a deduction for certain tips up to $25,000 and an increase in the state and local tax (SALT) deduction cap from $10,000 to $40,000. Given these policies, most taxpayers should expect a modest decrease in taxes owed this season, and refunds on average should trend higher than in 2025—something borne out by early data from the U.S. Department of Treasury. Of course, the full extent and range of the windfalls have yet to be discovered. "It is unclear if the average refund will be as high as estimated by the administration or what the actual distribution of those tax benefits will be," says Professor Olsen. "Not every taxpayer will receive a significant benefit, and there are so many factors that could cause someone's tax bill to increase, including increased taxable income, the loss of other deductions, changes in filing status and claimed dependents." "There are also many restrictions and phaseouts that changed with OBBBA," adds Dr. Watson. "For example, many seniors and tipped workers paid little to no income tax even under prior law, so they would not necessarily see much benefit with the new legislation." As Dr. Watson's remarks reflect, the question of who exactly qualifies for and benefits from OBBBA's provisions has preoccupied the general public for quite some time. In particular, the deduction-related measure tied to tips, or the "no tax on tips" policy, has sparked curiosity. "Like many things in the Internal Revenue Code, 'no tax on tips' sounds simple, but the actual law is a bit more nuanced," says Professor Olsen. "Only workers who are in an occupation that 'customarily and regularly' receives tips qualify for this deduction. This includes lots of people in transportation, personal services and the food, beverage and hospitality industries—even some in entertainment. "Certain professions, including those in healthcare, law, accounting, financial services and consulting, will likely be prohibited from taking the deduction… Importantly, qualifying workers will need to be able to prove their tip income to take the deduction." "It is also worth noting that, due to the generally lower-wage nature of tipped jobs and the historical underreporting of tips on returns, many taxpayers earning tips paid little to no tax on tips in prior years," says Dr. Watson. "So, they would not experience much of a change under OBBBA." Navigating new provisions like this one, during a process and time already known for provoking anxiety, can be a daunting prospect for many taxpayers. What's more, administrative adjustments at the Internal Revenue Service (IRS) could potentially exacerbate levels of stress. "IRS staffing remains lower than in past years, which could impact customer service and processing of returns," says Professor Olsen. "In addition, the current administration has elected to eliminate the IRS Direct File tool for income taxes that was available last year. Taxpayers who used the tool last year will need to find other options for the preparation of their income tax returns." For those concerned about filing and worried about abiding by the new rules and regulations, Professor Olsen and Dr. Watson highlight the importance of remaining organized, maintaining a game plan and, if necessary, seeking assistance. "First, taxpayers should be proactive," says Professor Olsen. "Start gathering information as soon as possible and start the process of preparing your returns as early as possible. That will provide you with more time to troubleshoot issues or find other information you may not have initially gathered." "Free resources are also available for taxpayers," says Dr. Watson. "There is a federal volunteer-run program called Volunteer Income Tax Assistance (VITA) that operates throughout communities to prepare tax returns for free, and many of the big tax prep companies do offer a free version of their software, despite trying very hard to sell a paid version. "That said, the best advice is to keep thorough tax and financial records throughout the year. Then, seek help—such as through VITA—when you need it. The better your records, the easier it will be for VITA or others to assist you."

Elevation of John Henry Newman to Doctor of the Church Stands Out Among Pope Leo’s First-Year Actions featured image

3 min

Elevation of John Henry Newman to Doctor of the Church Stands Out Among Pope Leo’s First-Year Actions

Pope Leo XIV’s first year as leader of the Catholic Church was marked by observation, listening and careful communication, and was largely devoid of major doctrinal or political action. As reasonably expected of a new pontiff, he issued his first major document—the apostolic exhortation Dilexi te. He also embarked on his first international trip, traveling to Türkiye and Lebanon, where he celebrated the 1700th anniversary of the Council of Nicaea. But perhaps publicly overlooked was his elevation of a 19th-century saint to Doctor of the Church—a designation reserved for saints found to have made significant contributions to doctrine and spirituality through writings and teachings. After approving it in July 2025, Pope Leo issued that designation of Newman on All-Saints Day, making him only the 38th individual given the title. “The move to make St. John Henry Newman a Doctor of the Church was already underway, but Pope Leo verified it and actually carried it out, bringing Newman into a newfound kind of prominence,” said Michael Moreland, PhD, Professor of Law and Religion at Villanova University Charles Widger School of Law, and a scholar of St. Newman. A Bridge Between Anglican and Catholic Theology Newman, considered an influential individual in the shaping of modern Catholic theology and education, was not always Catholic. Born in England in 1801, Newman became an Anglican priest in his mid-20s. He later was a prominent leader of the Oxford Movement—one intended to recover elements of traditional Catholic heritage in Anglicanism. It led to the birth of Anglo-Catholicism and caused many Anglicans to convert to the Roman Catholic faith, including Newman himself. Newman was received into the Catholic Church in 1845 and was ordained a Catholic priest in 1847. He would go on to become an influential theologian for his contributions to the ideas of faith, conscience and doctrine. “He established what he called the "evolution of the doctrine"—the idea that the deposit of faith is not something immobile, but something that grows in awareness over time,” said Luca Cottini, PhD, professor of Italian Studies at Villanova University. “He [Newman] was also a person of the university; an intellectual,” said Dr. Moreland. “He was known for his voluminous writings on all kinds of issues, including his famous book ‘The Idea of a University.’" In 1878, Pope Leo XIII made then-Father Newman—who was not even a Bishop at the time—a Cardinal. Like the current Pope Leo, the elevation of Newman was one of Pope Leo XIII’s first acts of government. “It’s an astonishing resemblance between our current Pope and his namesake,” Dr. Cottini said. “Both recognized this important theologian and his contributions to Catholicism within the first year of their papacy.” Newman died in 1890 and is remembered as an influential theologian, scholar and an important bridge between Anglicanism, Catholicism and the modern world. He was beatified in 2010 by Pope Benedict XVI and canonized by Pope Francis in 2019 before Pope Leo XIV elevated him to Doctor of the Church. Newman’s Promotion Shrouded in Significance Beyond its relative rarity—only five individuals have been promoted to Doctor of the Church since 1971—Newman’s elevation is significant in many ways, according to Dr. Moreland. “He was an Anglican by birth who converted to Roman Catholicism,” he said. “He was someone from the English-speaking world, not from continental Europe, and he was from the 19th century, which is relatively recent in this context.” Newman is only the second Doctor of the Church from England, and aside from Thérèse of Lisieux, born in 1873, is the only Doctor of the Church born after 1700. Beyond the significance related to Newman himself, Pope Leo’s recent action underscored a critical focus of the Church, and evidenced its personal importance to the pontiff himself. “It highlighted Newman’s role in education,” Dr. Moreland says. “That is something Pope Leo has been formed by in important ways: as a seminary rector, seminary professor and as part of the Order of Saint Augustine, which values education highly.” At the Mass elevating Newman to Doctor of the Church, Pope Leo also named Newman a co-patron saint of Catholic Education, joining 13th-century priest and theologian St. Thomas Aquinas. He then added Newman’s feast day of October 9—the day he converted to Catholicism in 1845—to the General Roman Calendar, thus bringing his memorial to the global Church. “I think Pope Leo elevating Newman to a Doctor of the Church, along with these subsequent actions, signifies the emphasis he is going to place on education during his papacy,” Dr. Moreland said.

A Century and a Half of Connectivity: Professor Mojtaba Vaezi Reflects on the Evolution and Future of Communication Technology featured image

3 min

A Century and a Half of Connectivity: Professor Mojtaba Vaezi Reflects on the Evolution and Future of Communication Technology

On March 10, 1876, Alexander Graham Bell spoke the first words ever transmitted over telephone: “Mr. Watson, come here; I want you.” This simple request to Bell’s assistant, Thomas Watson, marked a significant milestone in direct person-to-person communication. Now, 150 years later, this message has paved the way for advanced cellular technology in the form of satellites, wireless networks and the personal devices we carry everywhere. For Mojtaba Vaezi, PhD, associate professor of electrical and computer engineering at Villanova University and director of the Wireless Networking Laboratory, Bell’s few words spoken over telephone marked the beginning of an ongoing technological revolution. “One hundred fifty years ago when telephone communication first started, there was essentially a wired line and a transmitting voice,” said Dr. Vaezi. “That simple, basic transmission has transformed the field of communication technology in unimaginable ways.” According to Dr. Vaezi, five shifts have defined the past century and a half of communication technology: wired devices to wireless, analog to digital, voice to data, fixed landlines to mobile phones and human-to-human communication giving way to an increasing focus on machines and artificial intelligence. Early wireless networks were built around one device per person. Today's networks must support multiple devices per person, plus the technology behind innovations such as smart homes, driverless cars and even remote surgery. “Applications are much more diverse now, so communication has to follow,” said Dr. Vaezi. “A big portion of communication now, in terms of number of connections to the network, is from machine to machine—not human to human or even human to machine." The growing number of connections can cause a host of issues for users. When multiple users share the same wireless spectrum simultaneously, their signals interfere with one another—a problem that is becoming more acute as the number of connected devices increases exponentially. Dr. Vaezi’s research at Villanova focuses on developing techniques that allow multiple users to transmit messages on the same frequency at the same time and still be understood. Another vibrant research area of Dr. Vaezi’s involves Integrated Sensing and Communication (ISAC). This field of study focuses on integrating wireless communications and radar so they can function within the same spectrum. “Historically, radar and wireless communication work in different bandwidths or spectrums and use separate devices. Although they are related, they happen in different fields,” said Dr. Vaezi. “Almost every communication scheme that has been developed has focused on this: How can we better utilize the spectrum?” ISAC is increasingly important as new innovations like driverless cars become fixtures in everyday life. These vehicles rely on radar to continuously scan for hazards, and when a hazard is detected, a signal must be sent to trigger safety mechanisms. Currently, the radar and communications systems operate on separate bandwidths using separate hardware. Dr. Vaezi's research explores how both functions could be housed in a single device running on one shared spectrum. Areas of study like Dr. Vaezi’s that focus on machine to machine communication are becoming increasingly relevant as communication technology evolves and moves away from simple person to person messaging. As for the next big milestone in communications, Dr. Vaezi is looking ahead to the implementation of 6G by 2030, though he tempers expectations. For most users, the change will feel modest, amounting to slightly faster device speeds. The most massive shift with 6G will be the amount of added coverage in areas that previously did not have network accessibility. “Say you order a package and it’s coming from somewhere abroad,” explained Dr. Vaezi. “6G will add network coverage over oceans, so you’ll be able to track your package in real time using that satellite technology.” The sixth generation of cellular technology will continue to connect our world and optimize current communications to accommodate more users and devices that need network access each day. It is far different from Alexander Graham Bell’s historic phone call 150 years ago. That brief exchange over a single wired line laid the groundwork for a communications ecosystem that now supports billions of devices, complex data networks and emerging technologies yet to be seen. It also serves as a reminder that despite how far communication technology has come, and how complex it has gotten, it all shares a common, simple goal: to transmit information from one point to another.

View all posts
Powered by