The Art of Mathematics

Ted Dintersmith, author of Aftermath: The Life-Changing Math that Schools Won't Teach You, argues that much of what we learn in math can be outsourced to machines. That frees people to understand the mathematical ideas that underlie much of modern life. An example is estimation, such as political polls. Understanding the various metrics that fill the news requires creativity coupled with logic. Learning to ask the right questions is more important than, say, learning to do closed-form integrals.

Josh Cole discussed Zeno's paradox that says motion is not possible since you must go half-way, and then half of the remaining distance, and so on, infinitely. Aristotle and others struggled with this, using ideas like "potential infinity" that hint at ideas that would appear in calculus hundreds of years later. The ancient Greeks were focused on the counting numbers, so they expect things to happen in succession. In particular, the Arrow Paradox asks at what point in time does an arrow move.

Joseph Bennish discusses Euler’s formula, which involves pi, e, the imagery i, 0 and 1, a beautiful formula that unites disparate types of numbers. We can think of e raised to an exponent as compound interest or a function with a remarkable property. We can extend the properties we expect from exponentials to imaginary numbers, which gives us periodicity instead of the usual steadily rising exponential growth.

Aris Winger, Math Professor and Executive Director of the National Association of Mathematicians, has experienced first hand how math can save students' lives by uplifting them. Our education system can move beyond workbooks and help students, all students, think crisper and understand what's happening in the world.

Beyza Aslan, Associate Professor of Math at the University of North Florida, crochets mathematics. This turns abstractions, such as hyperbolic geometry, into something that can be touched, felt, manipulated, and experimented with. Her work as been exhibited at the Joint Mathematics Meetings.

Ben Cornish, host of The Mathematicians Podcast, discusses Pythagorean triples, integers that can be the sides of a right triangle. There are infinitely many primitive triples, as he proves. This concept has been around even before Pythagoras and across cultures. Yet, there are always new questions to ask. Answering one involves, surprisingly, complex numbers. We leave you with an open conjecture.

Joel David Hamkins, author of Proof and the Art of Mathematics, presents the game Buckets of Fish, which seemingly will go on forever. Yet he presents a proof that it will always come to an end. In fact, he proves it using contradiction, mathematical induction, and even transfinite ordinals. Why do mathematicians like to do multiple proofs of a single statement? He also gives a winning strategy for the game and proves it works.

Krystal Taylor, Associate Professor of Mathematics at Ohio State University, discussed the surprising characteristics of fractals, "infinity in a box." They may have fractional dimension, which varies depending on how it's measured. An infinite perimeter may enclose a finite area. Yet they are not just mathematical oddities--they appear in nature and have practical applications.

Alon Amit addresses the various facets of mathematics. Is it an art or a science? Both? Neither? Is it invented or discovered? Why is math that's developed for purely aesthetic reasons so often a useful tool for the real world? He likes that there are not simple, one-way answers. He challenges the listeners to post questions to Quora that surprise and delight him.

Alon Amit, prolific Quora math answerer, discusses how Artificial Intelligence might change the role of the mathematician. AI will make mathematics more efficient but it can't do math in a deep sense at present. It can't perform logical reasoning or even know if it's wrong. However, there are recent advances in proof verifiers. They may eventually be able to check complex proofs like the recent alleged proof of the ABC Conjecture.

Cindy Lawrence is the Director and CEO of the National Museum of Mathematics in New York City. She and a former math professor built it up from a grass-roots museum started by math teachers. The Museum, soon to move into a 30,000 square foot space, appeals to both those who love and hate math. Attendees learn that math is beautiful, fun, and surprising--"That's so cool!"

Veselin Jungic, teaching professor of mathematics at Simon Fraser University, introduces undergraduate math minors to contemporary math research. The focus is Ramsey theory, an area of current research activity that brings together multiple areas of math, deals with big ideas, proves complete chaos is impossible, and is built on human stories. Some students extended or corrected ongoing research. Others used their artistic talents to express the patterns of mathematics through, for example, a graphic novel or a poem.

Joseph Bennish discusses math as a "concept factory." The concept of prime numbers came from a desire to break numbers down to their simplest atoms. This simple concept led to simple questions like the twin prime conjecture that no one has been able to answer. Those questions in turn led to deep research. The concepts of new geometries grew out of failed attempts to prove that Euclid's geometry was the only geometry. Gauss' "most wonderful theorem" of surfaces led to Riemann's higher dimensional manifolds. This, combined with Minkowski's space-time geometry, led to Einstein's relativity, "the most beautiful theory of physics."

Jeanne Lazzarini tells us how a clockmaker used an egg to win the competition to build the dome of the Florence Cathedral. The Cathedral had had a huge gaping hole for a hundred years since no one knew how to build such a large dome. His solution involved the equation for a hanging chain and parallel lines that meet.

Math is in a sense the science of patterns. Alon Amit explores the question of what exactly is a pattern. A common example is the decimal digits of pi. The statement that they have no pattern seems to be either obvious or completely untrue. We explore the spectrum of pattern-ness from simple repetition to total randomness and finally answer the question about pi. We also discuss analogy, which powers mathematical exploration.

Alon Amit joins us on the antipode of Pi Day to counter the myths and mysteries of this most famous irrational number. There's nothing magical about a non-repeating string of digits. The real and profound mystery is the ubiquity of pi. It shows up in places that have nothing to do with circles, such as the sum of the reciprocals of the squares of the integers and the normal bell-shaped curve.

Kate Pearce, a post-doc researcher at UT Austin, talks about her experience teaching math in a women's prison. Her remedial college algebra students came in with negative experience in math, so she devised ways to make the topics new. The elective class called, coincidentally, The Art of Mathematics, explored parallels between math and art, infinity, algorithms, formalism, randomness and more. The students learned to think like mathematicians and gained confidence in their abilities in abstract problem solving.

Alon Amit, prolific Quora math answerer, argues that an honest representation of mathematical ideas is enough to spark interest in math. It's not necessary to exaggerate the role of math; the golden ratio does not drive the stock market, the solution of the Riemann hypothesis will not kill cryptography, and Grothendieck did not advance robotics. History and seeing the thought process and the struggle behind the tight finished proof are ways to make math compelling.

Dave Cole, the author of the Math Kids series of books, talks about introducing kids to math as a fun challenge and puzzle beyond the rote memorization they've come to expect. Kids who like to read are enticed by puzzles and mysteries. Möbius strips, Pascal's triangle, and other concepts that are new to them, make them marvel, "Is this math?" They see patterns and learn to make and even prove conjectures.

Neil Epstein, Associate Professor of Mathematics at George Mason University, introduces us to the fractions used by the ancient Egyptians, well before the Greeks and Romans. The Egyptian fractions all had a unit numerator. They could represent any fraction as a sum of unique unit fractions, a fact that was not proved until centuries later. These sums inspired conjectures, one of which was proved only recently, while others remain unsolved to this day. Recent work extends these concepts beyond fractions of integers. Human heritage goes way back, but is still inspiring modern research.

Jeanne Lazzarini joins us again to introduce us to the mathematician Luca Pacioli, whose views of numbers and shapes influenced Leonardo da Vinci, leading to a period of art and invention. His book, De Divina Proportione, is the only book ever illustrated by da Vinci. The Renaissance was a period when mathematicians studied art and artists studied mathematics. As da Vinci said, "Everything connects."

Alon Amit, probably the most prolific answerer of math questions on Quora, shares his reasons for his deep involvement. He seeks to share the journey, the exploration and stumbles of solving a problem. He's especially drawn to questions that will teach him things, even if he never completes the answer. He also shares his joy of problem solving with kids through Math Circles. One example problem, involving only 4 dots, can be worked on by a young child, yet affords deep exploration.

Lee Kraftchick continues his tour of books about math written for the non-mathematician like himself. We also can't let go of Gödel Escher Bach. Lee cites an opinion piece in the Washington Post, titled, "The Problem with Schools Today is Too Much Math," which gives a very narrow view of what math is. He counters it with a response (see theartofmathematicspodcast.com) and more books that demonstrate that math provides "pleasures which all the arts afford." He also discusses books about math and the real world and compilations of the broad range of mathematics.

Lee Kraftchick discusses some of his favorite books for non-mathematicians to explore the breadth of mathematics. These books range from very old to current. Some discuss beautiful proofs, whether math is invented or discovered, and how to think. Lee and Carol agree on the number one greatest book for mathematicians and non-mathematicians alike. See the full list at theartofmathematicspodcast.com.

Jeanne Lazzarini talks about kaleidoscopes and the mathematics that makes them work. This "beautiful form watcher" uses the laws of reflection to make ever-changing repeated symmetries. The use of more mirrors, rectangles, cylinders or pyramids create even more complex patterns.

Ethan Zhao and Edward Yu are the winners in mathematics of the prestigious Davidson Fellow Scholarships, awarded based on projects completed by students under 18. Ethan's project was on learning models and Edward's was on combinatorics. It was math contests and the MIT Primes program that gave them the background to do original research in high school, an experience most mathematicians don't get until graduate school. They also discussed the accessibility of math. You can come up with interesting problems while staring out the window. You can invent your own tools.

Lawyer Lee Kraftchick discusses the search for truth and basic principles in the legal community and the surprising parallels and similarities with the same search in the math community. Mathematical and legal arguments follow a similar structure. Even the backwards way an argument is created is the same.

Lee Kraftchick, a lawyer with a math degree, discusses some of the surprising parallels between the fields. Math is used directly to make statistical arguments to rule out random chance as a cause. He gives examples from his experience in redistricting and affirmative action. Math is used indirectly in legal reasoning from what is known to justified conclusions. Math reasoning and legal reasoning are remarkably similar. He invites lawyers to set aside the usual "lawyers aren't good at math" stereotype and see the beauty of the subject.

Jeanne Lazzarini looks for math in the real world and finds the Fibonacci sequence and the closely related Golden Ratio. These appear as we examine plants, bees, rabbits, flowers, fruit, and the human body. These natural patterns and pleasing symmetries find their way into the arts. Does nature understand math better than we do?

Brian Katz, from California State University Long Beach, invites us to explore the various layers of ordinary sounds, informed by a little calculus. The limited frequencies that come out of the wave equation are what separates sounds that communicate (voice, music) from noise. These higher notes are in the sound itself and you can hear them (but alas, not on this compressed podcast audio). Brian has provided links to hear these layers of pitches at theartofmathematicspodcast.com

Jeanne Lazzarini, who has visited us before to talk about tessellations, discusses a new mathematical discovery that even earned a mention on Jimmy Kimmel. It's a shape that can be used to fill the plane with no gaps and no overlaps and, most remarkably, no repeating patterns.

Lawrence Udeigwe, associate professor of mathematics at Manhattan College and an MLK Visiting Associate Professor in Brain and Cognitive Sciences at MIT, is both a mathematician and a musician. We discuss his recent opinion piece in the Notices of the American Mathematical Society calling for "A Case for More Engagement" between the two areas, and even get a little "Misty." He's working on music that both jazz and math folks will enjoy. We talk about "hearing" math in jazz and the life of a mathematician among neuroscientists.

Joseph Bennish returns to dig into the math behind the Fourier Analysis we discussed last time. Specifically, it allows us to express any function in terms of sines and cosines. Fourier analysis appears in nature--our eyes and ears do it. It's used to study the distribution of primes, build JPEG files, read the structure of complicated molecules and more.

Joseph Bennish, Professor Emeritus of California State University, Long Beach, joins us for an excursion into physics and some of the mathematics it inspired. Joseph Fourier straddled mathematics and physics. Here we focus on his heat equation, based on partial differential equations. Partial differential equations have broad applications. Fourier developed not only the heat equation but also a way to solve it. This equation was used to answer, among other questions, the issue of the age of the earth. Was the earth too young to make Darwin's theory credible?

Jim Stein, Professor Emeritus of CSULS, returns to complete his (admittedly subjective) list of the ten greatest math theorems of all time, with fascinating insights and anecdotes for each. Last time he did the runners up and numbers 8, 9 and 10. Here he completes numbers 1 through 7, again ranging over geometry, trig, calculus, probability, statistics, primes and more.

Jim Stein, Professor Emeritus of California State University Long Beach, discusses some bets that appear to be 50-50, but can have better odds with a tiny amount of seemingly useless information. Blackwell's Bet involves two envelopes of money. You can open only one. Which one do you choose? We learn about David Blackwell and his mathematical journey amid blatant racism. Another seeming 50-50 bet is guessing which of two unrelated events that you know nothing about is more likely; you can do better than 50-50 by taking just one sample of one of the events. Dr. Stein then discusses how mathematics shows up in some surprising places. Mathematics studied for the pure joy of it often finds surprising uses. He gives some examples from G. H. Hardy as well as his own research.

Joseph Bennish, Prof. Emeritus of CSULB, describes the field of Diophantine approximation, which started in the 19th Century with questions about how well irrational numbers can be approximated by rationals. It took Cantor and Lebesgue to develop new ways to talk about the sizes of infinite sets to give the 20th century new ways to think about it. This led up to the Duffin-Schaeffer conjecture and this year's Fields Medal for James Maynard.

Jeanne Lazzarini, a math education specialist, returns to discuss tessellations and tiling in the works of Escher, Penrose, ancient artists and nature. We go beyond the familiar square or hexagonal tilings of the bathroom floor. M.C. Escher was an artist who made tessellations with lizards or birds, as well as pictures of very strange stairways. Roger Penrose is a scientist who discovered two tiles that, remarkably, can cover an area without repeat, as well as a strange stairway.

Joseph Bennish returns to take us beyond the rational numbers we usually use to numbers that have been given names that indicate they're crazy or other-worldly. The Greeks were shocked to discover irrational numbers, violating their geometric view of the world. But later it was proved that any irrational number can be approximated remarkably well by a relatively simple fraction. The transcendental numbers were even more mysterious and were not even proved to exist until the 19th century.

Jeanne Lazzarini, a math education specialist, shares the connections between math, such as fractals and the golden ratio, and art. These are everywhere--nature, architecture, film and more. She shares hands-on mathematical activities that helped her students see math as an exploration and an art.

Lara Alcock of Loughborough University shares what she learned, by tracking eye movements, about how mathematicians and students differ in the ways they read mathematics. She developed a 10-15 minute exploration training, that increases students' comprehension through self-explanation. We also discuss the transition between procedural math and proofs that many students struggle with early in their college careers.

Sunil Singh, the author of Chasing Rabbits and other books, shares fascinating stories that show mathematics as a universal place of exploration and comfort. Stories of mathematical struggle and discovery in the classroom help students connect deeply with the topic, feel the passion, and see math as multi-cultural and class-free.

Caron Rivera, a math teacher at a school for elite athletes, shares how she breaks through the myth of the "math person" and teaches athletes to think like mathematicians. Her problem solving technique applies to anything. Through it her students get comfortable with not knowing, with the adventure of seeking the answer. They build their brains in the process. 

Brian Katz of CSULB joins us once again to discuss mathematical definitions. Students often see them as cast in stone. Prof. Katz helps them see that they're artifacts of human choices. The student has the power to create mathematics through definitions. This is illustrated by the definitions of "sandwich" and "approaching a limit." What makes a good definition? How is mathematics like a dream?

Ian Stewart, prolific author of popular books about math, discusses how math is the best way to think about the natural world. Often math developed for its own sake is later found useful for seemingly unrelated real-world problems. A silly little puzzle about islands and bridges leads eventually to a theory used for epidemics, transportation and kidney transplants. A space-filling curve, of interest to mathematicians mainly for being counterintuitive, has applications to efficient package delivery. The mathematical theories are often so bizarre that you wouldn't find them if you started with the real-world problem.

Joseph Bennish joins us once again to continue his discussion of symmetry, this time venturing into higher dimensions. We explore the complex symmetry groups of the Platonic solids and the sphere and their relationships. We then venture into the 4th dimension, where we see that, with a change to the distance the symmetries are maintaining, we get Einstein's Theory of Relativity.

We are all born with an intuitive attraction to symmetry, through human faces and heartbeats. Joseph Bennish, of California State University Long Beach, explores the mathematical meaning of symmetry, what it means for one shape to be more symmetric than another, how symmetries form mathematical groups and groups form symmetries, and hints at implications for Fourier analysis, astronomy and relativity.

Will Murray, chair of the math department at California State University, Long Beach, discusses popular stereotypes of mathematicians and what they do when they do mathematics. Is it all lone geniuses generating big numbers? If so many people dislike mathematical thinking, why is Sudoku so popular?

Saleem Watson discusses the mysterious way math predicts the natural world. Much of math is invented, and yet there are many examples of cases in which purely abstract math, developed with no reference to the natural world, later is found to make accurate and useful models and predictions of the physical world.

Joseph Bennish discusses two famous theorems, proved long ago, and some modern alternative proofs. Why would we bother reproving something that was confirmed thousands of years ago? The answers are insight, aesthetics, and opening up surprising new areas of investigation.

Scott Crass, Professor of Mathematics at CSULB, expands our vague intuition about symmetry to look at transformations of various kinds and what they leave fixed. This approach finds applications in physics, biology, art and several branches of math.

Saleem Watson, Professor Emeritus of Mathematics, CSULB, confronts an ancient mathematical argument. Is math a body of eternal truths waiting for an explorer to uncover them, or an invention or work of art created by the human mind? Or some of each?

Paul Eklof, Professor Emeritus UCI, discusses the famous impossible straightedge-and-compass constructions of antiquity that have fascinated mathematicians and attracted cranks for centuries. There are infinitely many possible constructions. How can you prove not one of them will work?

Joseph Bennish, math professor at California State University, Long Beach, discusses how math is an exploration involving imagination and excitement. Kids get this. Adults can recapture this by generalizing and questioning. For example, a simple barnyard riddle leads to questions about optics.

You are a contestant on Let's Make a Deal, hosted by Monty Hall. There are 3 identical doors. Behind only one is the prize car. You make your choice, then Monty Hall opens one of the other doors to reveal a goat and asks whether you want to change your choice. Should you, or does it matter? Paula Sloan talks about the counterintuitive answer, and how she got the Duke MBA students in her math class to believe the answer.

Brian Katz, a professor at California State University, Long Beach, approaches math as a philosopher, a linguist and an artist. It is not a science, but a byproduct of consciousness, an expression of humanity and a way to make connections.

We talk with Kathryn McCormick, Assistant Professor at California State University, Long Beach, about why she got into this obscure field, what a mathematician really does, and where we can learn more about being a mathematician.

There are a lot of jokes that poke fun at mathematicians, how they think and how they fumble around in the real world. Many of them start, "A mathematician, an engineer and a physicist ..."  We'll look at what these jokes say about us. The most telling is a little joke that only a mathematician would enjoy, since it gives surprising insight into how mathematicians think through all this abstraction.

Fermat’s Last Theorem is easy to state but has taken over 300 years to prove. Fermat’s supposed “marvelous proof” has been a magnet for crackpots and obsessed mathematicians, leading through a treasure hunt across almost all branches of mathematics.

A surprising amount of art is inspired by mathematics. The book Fragments of Infinity describes many works of art and the mathematics behind them. Meet mathematicians who have become artists and artists who have become mathematicians, and some who have always straddled both worlds.

Another seemingly easy problem that’s hard to solve. In fact, it's unsolved. Find an odd perfect number or prove one doesn’t exist. The search involves “spoof” answers, trying to find the right answer (or prove it doesn't exist) by looking at wrong answers. Hey, nothing else has worked.

The fourth dimension is a staple of science fiction and the key to relativity. What exactly is it and how can we visualize it? What about higher dimensions?

Can you really approach one mathematical statement 99 different ways? We review the wonderful book 99 Variations on a Proof. The answer is yes.

The Four Color Theorem is a pretty little conjecture that has been intriguing mathematicians for more than a century. Too bad the proof stands as an example of really ugly mathematics.

What is infinity, why does it seem so weird, and can you really go beyond it?

We consider two problems, one in tiling and one in knots. They had each had been unsolved for over 50 years and their solutions hit the popular press in the same week. What kind of skills help people make surprising connections and new discoveries?

We explore some of the common misconceptions about mathematics and mathematicians.