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

Pope Leo XIV Paid Public Homage to Saint Augustine in Algeria, but Has Invoked His Teachings Since Election, Says Augustine Scholar Paul Camacho featured image

6 min

Pope Leo XIV Paid Public Homage to Saint Augustine in Algeria, but Has Invoked His Teachings Since Election, Says Augustine Scholar Paul Camacho

During his historic trip to Africa in April 2026, Pope Leo XIV toured the archaeological ruins of Annaba, Algeria, capping what he referred to as a “special” journey to the country. Annaba, where the ancient city of Hippo Regius once stood, is not far from where Saint Augustine—patron of the Augustinian order to which the Pope belongs—was ordained, made bishop and crafted his most influential theological works. Walking in his footsteps, Pope Leo—who referred to himself as a “Son of Augustine” upon his election to the papacy—toured the ruins with other members of the Order and his traveling party. It was a personal and symbolic moment, but for those paying close attention to his early pontificate, it represented only the most recent of many tributes he has paid to Augustine and his teachings. “In reading through Pope Leo’s addresses and his homilies, his messages and his video recordings, there is barely one that goes by without him referencing Augustine in some way,” said Paul A. Camacho, PhD, associate director of The Augustinian Institute at Villanova University, and scholar of Augustine’s teachings. When the Pontiff accepted the Saint Augustine Medal from the Augustinian Province of St. Thomas of Villanova in August 2025, he made it clear why. “As Augustinians, we strive every day to live up to the spiritual example of our spiritual father, Saint Augustine,” he said to the Province. “To be recognized as an Augustinian, it’s an honor held dearly. So much of who I am, I owe to the spirit and teachings of Saint Augustine and I am grateful to all of you for all the many ways your lives exhibit a deep commitment to the values of Veritas, Unitas and Caritas.” Veritas Examples abound related to all three values and how Pope Leo has invoked Augustine in speaking about them. Dr. Camacho cites a handful he found particularly noteworthy. For example, Veritas first appeared in an especially prominent way during an early address for the June Solemnity of the Holy Trinity and Jubilee of Sport. Then, Pope Leo said, “This combination of Trinity and Sport is somewhat unusual, yet the juxtaposition is not inappropriate. Every good worthwhile human activity is in some way a reflection of God’s infinite beauty, and sport is certainly one of these.” “For Saint Augustine, the Trinity and wisdom are intimately connected,” Dr. Camacho said, also mentioning that Augustine authored an entire book on the subject titled “On the Trinity.” “Divine wisdom is revealed in the Most Holy Trinity and wisdom always leads us to Veritas—the truth," he said. “What Pope Leo goes on to do is say that the life of God, the Trinity, is a life of play…It’s a dance, and dance is a sport, and our own way of participating in that kind of activity is a participation in life of God.” When Pope Leo later spoke to the International Foundation of Catholic Universities—an occasion for which the motto was “Choreographers of Knowledge”—he referenced that dance again. “Pope Leo invites us to think of our relationship to Veritas not as something to master,” Dr. Camacho said. “But rather as a light that draws us out of ourselves towards it, in a dance of relationship with it and one another.” Unitas During a September Mass for the General Chapter of the Order of Saint Augustine, Pope Leo prayed that those in attendance be given “the gift to listen, the gift to be humble and the gift to promote unity within the Augustinian Order, and throughout the Order, throughout the Church and the world.” Earlier in the summer, he touched on unity during the Jubilee of Families, Children, Grandparents and the Elderly, commenting on the Gospel of John, in which “Christ prays that we may all be one.” “This being one is the greatest good we can desire, for this universal union brings about among his creatures the eternal communion of love that is God himself: the Father who gives life, the Son who received it and the Spirit who shares it… the Lord does not want us in this unity to be a nameless and faceless crowd,” Pope Leo said. “What Pope Leo expresses here is that unity is not conformity, but rather community—the kind of community in which each one of us becomes more of who it is that we are the more we love and work for what is good in common,” explained Dr. Camacho. “This is a deeply Augustinian insight.” In one general audience with members of various religious groups, Pope Leo relayed “Unity has always been a constant concern of mine, as witnessed by the motto I chose for my episcopal ministry.” He was referencing, Dr. Camacho says, the Latin phrase In illo uno unum, displayed on his coat of arms. “That phrase—‘in the one [Christ] we are one’—comes from Saint Augustine’s commentary on the Psalms. Pope Leo comes back again to this idea that when we love and follow the truth that is Christ, we simultaneously become more uniquely individual, and our unity becomes genuine community, and not mass conformity.” Caritas In his Opening Homily for the Beginning of the Pontificate, Pope Leo followed his salutations with these words: “I greet you all with a heart full of gratitude at the beginning of the ministry that has been entrusted to me. Saint Augustine wrote: ‘Lord, you have made us for yourself, and our heart is restless until it rests in you.” According to Dr. Camacho, Augustine taught that the solution to the restless heart is Caritas, or rightly ordered love that finds its rest in God. Augustine’s thinking about love is developed in a social dimension when he distinguishes between two different kinds of political communities, which he calls the “Earthly City” (in which pride and self-love dominate), and the “City of God” (in which love of God and neighbor form the community). Augustine’s enduring and influential work “De Civitas Dei,” or “City of God,” articulates this distinction between false and true loves. At the 16th Annual Meeting of the International Catholic Legislators Network, Pope Leo suggested that we look to Augustine, citing his own witness of social upheavals in his time, and that we consider Augustine’s reminder that we must attend to these “two orientations of the human heart.” “Ordo amoris—order of love,” explained Dr. Camacho. “This is Augustine’s notion that following Christ is not a matter of private devotion, but rather demands a dilation of our hearts: our loves must move out to embrace the poor, the marginalized, the downtrodden, the forgotten. In Catholic Social Thought, this is the principle of solidarity: while the things that we love begin with what is most intimate to us, we are called to expand our love outward. One of the things Pope Leo is trying to say is that Augustine’s thought remains a living inspiration for how we might organize our societies.” Speaking to the International Interparliamentary Union in late June, Pope Leo intimated that Saint Augustine speaks of “your responsibility to promote and protect, independent of any special interest, the good of the community, the common good, particularly by defending the vulnerable and the marginalized.” “Augustine’s idea was that what makes a community is what its members love in common,” Dr. Camacho said. “In other words, Caritas—love—is fundamental to how we think about how we relate to one another. Not just on a one-to-one, personal level or in terms of a private relationship to my own good, but rather how we relate to our communities, and what we build together for the common good.” “Bringing Augustine Back to the World” One of the great gifts of the Augustinian Order, says Dr. Camacho, is that “because of its charism of community, it draws individuals to become religious from all walks of life, and not just those who have extensively read or studied the work of Augustine.” It is especially noteworthy, he says, just how entrenched Pope Leo’s words and actions are in his long-time study of Augustine. “It is quite clear that Pope Leo loves Augustine, not just as an exemplar, as a patron saint of the Order, or as a doctor of the Church—though he is all of these. Leo loves Augustine as a companion: a man whose life reminds us that discipleship is about fidelity and not about perfection; a thinker who is rich in philosophical and theological insight; and a priest and bishop who cared deeply for his community. “In his pontificate, it is already clear that Pope Leo, this great “Son of Augustine,” is bringing Augustine back to the Church and to the world as a vital source of wisdom and inspiration.”

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

4 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 to St. John Henry 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 St. 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.

View all posts
Powered by