The Art of Mathematics

Carol Jacoby

Conversations, explorations, conjectures solved and unsolved, mathematicians and beautiful mathematics. No math background required.

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79 episodes
Updated July 22

Episodes79

Newest first

July 22 · 15 min
Zeno's Paradox: Is Motion Possible?
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.
June 27 · 16 min
The Most Beautiful Formula
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.
May 27 · 17 min
Math as it Should Be
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.
April 22 · 12 min
Crocheting Mathematics
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.
March 25 · 20 min
Pythagorean Triples and Some New Conjectures
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.
February 25 · 15 min
Proofs and Buckets of Fish
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.
January 28 · 14 min
Fractals: Simple rules, complex shapes
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.
Apr 23, 2025 · 15 min
The Many Facets of Math
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.
Mar 26, 2025 · 20 min
Will AI Replace Mathematicians?
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.
Feb 26, 2025 · 17 min
The National Museum of Mathematics
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!"
Jan 22, 2025 · 14 min
Contemporary Math Research for Artistic Undergrads
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.
Dec 25, 2024 · 20 min
Where do Math Concepts Come From?
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."
Nov 27, 2024 · 14 min
A Clockmaker, an Egg, and a Cathedral
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.
Oct 23, 2024 · 12 min
What is a Pattern?
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.
Sep 25, 2024 · 17 min
What's the Big Deal about Pi?
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.
Aug 28, 2024 · 22 min
Turning Math-Hating Prisoners into Mathematicians
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.
Jul 24, 2024 · 17 min
Stop Overselling Mathematics
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.
Jun 26, 2024 · 21 min
Math for Kids: It's not a Spectator Sport
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.
May 22, 2024 · 17 min
Egyptian Fractions
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.
Apr 24, 2024 · 16 min
Da Vinci's Math Teacher: Merging the Practical and Theoretical
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."

The Art of Mathematics

Zeno's Paradox: Is Motion Possible?

The Most Beautiful Formula

Math as it Should Be

Crocheting Mathematics

Pythagorean Triples and Some New Conjectures

Proofs and Buckets of Fish

Fractals: Simple rules, complex shapes

The Many Facets of Math

Will AI Replace Mathematicians?

The National Museum of Mathematics

Contemporary Math Research for Artistic Undergrads

Where do Math Concepts Come From?

A Clockmaker, an Egg, and a Cathedral

What is a Pattern?

What's the Big Deal about Pi?

Turning Math-Hating Prisoners into Mathematicians

Stop Overselling Mathematics

Math for Kids: It's not a Spectator Sport

Egyptian Fractions

Da Vinci's Math Teacher: Merging the Practical and Theoretical