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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3 min

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As Dr. Xie explained, “Shor invented an algorithm to solve prime factors of an integer that can supposedly run on a quantum computer. This algorithm, if run on a large-scale mature quantum computer, can easily solve all these existing cryptosystems' mathematical formulation, which is a problem." The realization of this potential threat has spurred an increased focus on the development of post-quantum cryptography over the past decade. The goal is clear: "We want to have some sort of cryptosystem that is resistant to quantum computer attacks," says Dr. Xie. In 2016, the National Institute of Standards and Technology (NIST) began the process of standardizing post-quantum cryptography. In July 2022, NIST selected four algorithms to continue on to the standardization process, where they are currently being tested for safety and security against quantum computers. The standardization process for these new algorithms is intensive, and two of the candidates that were announced for testing have already been broken during the process. Scientists are in a race against time to increase the diversity of their algorithms and come up with alternate options for standardization. The urgency of this shift to post-quantum cryptography is underscored by recent government action. The White House released a national security memo in 2022 stating that the U.S. government must transition to quantum-resistant algorithms by 2035. This directive emphasizes the critical nature of post-quantum cryptography in maintaining not just personal but national security. Villanova’s Security and Cryptography Lab Once a new algorithm is selected by NIST, it will need to be embedded into various platforms that need to be secured—this is where Dr. Xie’s Security and Cryptography Lab comes in. This lab is actively conducting research into how the newly selected algorithm can be implemented in the most effective and resourceful way. The lab team is working on developing techniques for this new algorithm so that it can be embedded into many different types of platforms, including credit cards and fingerprint technology. However, there are significant challenges in this process. As Dr. Xie explains, "Different platforms have different constraints. A chip-based credit card, for example, has limited space for embedding new encryption systems. If the implementation technique is too large, it simply won’t work.” Another arising issue from this research is security. During the application of this new algorithm, there's a risk of information or security leakage, so Dr. Xie is always on the lookout for developing security issues that could cause problems down the road. The Future of Post-Quantum Cryptography The implications of PQC are widespread and extend far beyond academic research. As Dr. Xie points out, "All existing cryptosystems, as long as they have some sort of function—for example, signing in or entering a password for login—all of these systems are vulnerable to quantum attacks." This vulnerability affects everything from banking systems to small-scale security measures like fingerprint door locks. The scope of this transition is massive, requiring updates to encrypted systems across all sectors of technology. His goal is to ensure that these new cryptographic systems are flexible enough to be applied to everything from small devices like credit cards and drones to large-scale infrastructure like data centers and military equipment. Although researchers are hard at work now, the future of post-quantum cryptography is not without uncertainties. Dr. Xie raises an important question: "When quantum computers become available, will the algorithms we develop today be broken?" While the newly developed algorithms will theoretically be secure, vulnerabilities can emerge when implementing any kind of new security system. These potential vulnerabilities highlight the importance of conducting this research now so that the new algorithms can go through intensive testing prior to being implemented. Despite these challenges, Dr. Xie emphasizes the importance of being prepared for this new reality. "Society as a whole needs to be prepared with this kind of knowledge,” he says. “A new era is coming. With our current security systems, we need to have revolutionized change. On the other hand, we should not be panicked. We just need continued support to do more related research in this field.” More extensive research is required to ensure that our privacy is protected as we enter a new era of quantum computing, but labs like the Security and Cryptography Lab at Villanova are a step in the right direction. Although the “years to quantum” clock is ticking down, researchers like Dr. Xie are well on their way to ensuring that our digital infrastructure remains secure in the face of evolving technological threats.

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