The Art of Mathematics

Latest Episodes:

Vowels and Sounds and a Little Calculus Jul 26, 2023

Show notes

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

The Hat: A Newly Discovered "Ein-stein" Tessellation Tile Jun 28, 2023

Show notes

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.

Interfacing Music and Mathematics May 24, 2023

Show notes

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

Show notes

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

Show notes

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

Show notes

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

Show notes

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, 2022

Show notes

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.

Fascinating Fractals Nov 23, 2022

Show notes

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.

Approximation by Rationals: A New Focus Oct 26, 2022

Show notes

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.

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