Claim OwnershipThe Art of Mathematics
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© Carol Jacoby
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Conversations, explorations, conjectures solved and unsolved, mathematicians and beautiful mathematics. No math background required.
66 Episodes
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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.
1Lee 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?
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