Conversations, puzzles, book reviews, conjectures solved and unsolved, mathematicians and beautiful mathematics. No math background required.
Join us every month, or check out our past episodes. Available on Apple Podcasts, Spotify, wherever you get your podcasts.
Most episodes are about 15-20 minutes, sometimes ending with a puzzle along with the solution to the puzzle from the previous episode.
I’m Carol Jacoby. Join me for the art of mathematics. You think math isn’t an art? It’s just numbers and formulas that need to be memorized? Join me for truth and beauty, music, art, infinity and intriguing questions. You don’t need a background in math. High school algebra at the most. We’ll converse, review books and do puzzles. We’ll talk about what mathematicians do and why, look at some mind-blowing results and deceptively simple concepts that open up all sorts of interesting questions. We welcome non-mathematicians and future mathematicians as well as any mathematicians who want to step back from their very specialized research and think about what they do and why. I look forward to sharing the art of mathematics with you.
Note that we are now on a once-a-month schedule. New episodes will appear on the fourth Wednesday of each month. Here’s the current and past ones.
Books for the Mathematical Tourist
Jan 23, 2024 • 20:41
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 here.
Reflecting on Kaleidoscopes
Dec 27, 2023 • 20:18
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. Learn more here. or here. Make your own kaleidoscope. Here’s a kaleidoscope society.
Meet the Young Davidson Fellowship Winners
Nov 22, 2023 • 14:11
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 enthusiastically discussed the remarkable accessibility of math. You can come up with interesting problems while staring out the window. You can invent your own tools.
Gödel’s Incompleteness, Fundamental Truths, and Reasoning in Math and Law
Oct 25, 2023 • 22:07
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.
Math and the Law
Sep 27, 2023 • 20:22
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.
Aug 23, 2023 • 20:32
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? Here are references and illustrations.
Vowels and Sounds and a Little Calculus
Jul 26, 2023 • 11:39
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 the links to hear these layers of pitches.
The Hat: A Newly Discovered “Ein-stein” Tessellation Tile
Jun 28, 2023 • 13:41
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. Learn more: Ein-stein tile resources.
Interfacing Music and Mathematics
May 24, 2023 • 21:13
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.
Fourier Analysis: It’s Not Just for Differential Equations
Apr 26, 2023 • 18:24
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 Fourier, the Heat Equation and the Age of the Earth
Mar 22, 2023 • 17:32
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?
The Ten Most Important Theorems in Mathematics, Part II
Feb 22, 2023 • 15:37
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.
The Ten Most Important Theorems in Mathematics, Part I
Jan 25, 2023 • 25:24
Jim Stein, Professor Emeritus of CSULB, presents his very subjective list of what he believes are the ten most important theorems, with several runners up. These theorems cover a broad range of mathematics–geometry, calculus, foundations, combinatorics and more. Each is accompanied by background on the problems they solve, the mathematicians who discovered them, and a couple personal stories. We cover all the runners up and numbers 10, 9 and 8. Next month we’ll learn about numbers 1 through 7.
Surprisingly Better than 50-50
Dec 28, 2023 • 18:15
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.
Nov 23, 2022 • 21:06
Jeanne Lazzarini joins us again to discuss fractals, a way to investigate the roughness that we see in nature, as opposed to the smoothness of standard mathematics. Fractals are built of iterated patterns with zoom similarity. Examples include the Koch Snowflake, which encloses a finite area but has infinite perimeter, and the Sierpinski Triangle, which has no area at all. Fractals have fractional dimension. For example, The Sierpinski Triangle is of dimension 1.585, reflecting its position in the nether world between 1 dimension and 2. Fractals are used in art, medicine, science and technology. Learn more about fractals.
Approximation by Rationals: A New Focus
Oct 16, 2022 • 20:56
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.
Sept 28, 2022 •
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). See more examples here.
Rational, Irrational and Transcendental Numbers
Aug 24, 2022 • 21:49
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.
Math as Art
Jul 25, 2022 • 18:42
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.
Exploration in Reading Mathematics
Jun 22, 2022 • 16:31
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.
Games for Math Learning
May 25, 2022 • 19:18
Jon Goga, of Brainy Spinach Math, is using the Roblox gaming platform to bring math learning to kids using something they already enjoy. Along the way, he teaches them some techniques that are useful for mathematicians at any level–breaking down and building up a problem. We also discuss the “inchworm” and “grasshopper” styles of learning.
The Power of Mathematical Storytelling
Apr 22, 2022 • 16:04
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.
The Mathematical World and the Physical World
Yusra Idichchou explores the question: Does math imitate life or does life imitate math? We touch on Oscar Wilde, philosophy of both math and language, how formal abstractions can describe the subjective physical world and various philosophies of mathematics.
Getting Athletes to Think Like Mathematicians
Feb 9, 2022 • 17:26
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.
The Art of Definitions
Jan 12, 2022 • 19:34
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?
12/8/21 Math Exploration for Kids
Mark Hendrickson, of Beast Academy Playground, talks about how to bring young kids into the joy, creativity and exploration that mathematicians experience. Kids enjoy art because they are free to try things and shun math for its apparent rigidness. He offers subtly mathematical games that invite even very young children to explore and question.
11/10/21 Is Mathematics an Art?
Joshua Sack, mathematics professor at California State University, Long Beach, explores the breadth of art and mathematics and finds much commonality in patterns, emotions and more.
10/13/21 Math as a Way of Thinking
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.
9/8/21 Symmetries in 3 and 4 Dimensions
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.
8/13/21 Symmetry, Shapes and Groups
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.
7/14/21 Freshmen and Sophomores Confront Unsolved Problems
Dana Clahane, Professor of Mathematics and Fullerton College, dispels some of the misconceptions about mathematics and discusses some famous unsolved problems that he has freshmen and sophomores working on, learning what math is really about.
6/16/21 Stereotypes of Mathematics and Mathematicians
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?
6/2/21 Prime numbers and their surprising patterns
Joseph Bennish talks about prime numbers, a simple concept with surprising characteristics. Are they regular or random? This takes us into unexpected realms–calculus, complex numbers and “the music of the primes.”
5/19/21 Creativity in Mathematics
Josh Hallam shares some of the ways he uses story writing and other creative endeavors in his math classes. He also discusses math in popular culture, including an original theorem in the animated show Futurama. [Here’s the part of the Futurama episode with the theorem https://www.youtube.com/watch?v=9ArMgY2dG0&list=PLx3skbat6Gw1oXSV2AIQc1YPkWJz8gs4O&index=9 If you freeze-frame at 1:12, you can see the proof on the board.]
5/5/21 The Unreasonable Effectiveness of Mathematics
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.
4/21/21 Alternative Proofs and Why We Seek Them
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.
4/7/21 Symmetry–It’s More Than You Think
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.
3/24/21 Is Math Discovered or Invented?
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?
3/10/21 That’s Impossible. Oh, Yeah? Prove It.
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?
2/24/21 The Joy of Mathematical Discovery
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.
2/10/21 The Monty Hall Problem
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.
1/27/21 What Is Mathematics? Some Surprising Answers
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.
1/13/21 Being a Mathematician
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.
12/30/20 Math Jokes and What They Say about Mathematicians
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.
12/16/20 The Most Famous (Formerly) Unsolved Problem
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.
12/2/20 The Mathematics of Art
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.
11/18/20 The Real World Is a Special Case
Abstract math is at once about nothing and about everything. The structures it builds may represent numbers, real world objects, music, or things we can barely imagine. Here we look at group theory for numbers, music, Rubik’s cubes and beyond.
11/4/20 How to Find Something You’ve Never Seen
Another seemingly easy problem that’s hard to solve. Find an odd perfect number or prove one doesn’t exist. The search involves “spoof” answers.
10/21/20 Beyond the Third Dimension
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?
10/7/20 One Theorem, 99 Proofs
Can you really approach one mathematical statement 99 different ways? We review the wonderful book 99 Variations on a Proof. The answer is yes.
9/30/20 A Beautiful Theorem with an Ugly Proof
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.
9/23/20 To Infinity…and Beyond
What is infinity, why does it seem so weird, and can you really go beyond it? (Shirt available on Amazon)
9/16/20 The Unsolved Is Solved…and Another
Two problems that had been unsolved for more than 50 years have both been conquered recently.
9/9/20 This Podcast is Lying
We explore the mind-blowing Liar and related paradoxes and how they changed mathematics
9/2/20 An Impossible Easy Question
Goldbach’s Conjecture and how a statement that is easy to understand may be difficult or impossible to resolve
8/26/20 Everything You Know About Math is Wrong
We explore some of the common misconceptions about mathematics and mathematicians. Here’s an example page of real math. No numbers.