(85)

Science

Natural Sciences

Science

Natural Sciences

Average Ratings

66

5

6

4

4

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I've been blazing through this podcast's back episodes at 1.5 speed, it's really fun!!

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This is a very intense science podcast. It goes deeply into science issues.

Average Ratings

66

5

6

4

4

Read more

I've been blazing through this podcast's back episodes at 1.5 speed, it's really fun!!

Read more

This is a very intense science podcast. It goes deeply into science issues.

Read more

According to our best theories of physics, the universe is a fixed block where time only appears to pass.

Jul 19 2016

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The ancient Greeks argued that the best life was filled with beauty, truth, justice, play and love. The mathematician Francis Su knows just where to find them.

Feb 02 2017

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Can a fluid analogue of a black hole point physicists toward the theory of quantum gravity, or is it a red herring?

Nov 08 2016

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The ancient creatures who first crawled onto land may have been lured by the informational benefit that comes from seeing through air.

Mar 07 2017

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By folding fractals into 3-D objects, a mathematical duo hopes to gain new insight into simple equations.

Jan 03 2017

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Two leading candidates for a “theory of everything,” long thought to be incompatible, may be two sides of the same coin.

Jan 12 2016

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An eminent mathematician reveals that his advances in the study of millennia-old mathematical questions owe to concepts derived from physics.

Dec 01 2017

19 mins

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A previously unnoticed property of prime numbers seems to violate a long-standing assumption about how they behave.

Mar 14 2016

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A recently proposed experiment would confirm that gravity is a quantum force.

Mar 06 2018

20 mins

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New findings are fueling an old suspicion that fundamental particles and forces spring from strange eight-part numbers called “octonions.”

Jul 20 2018

26 mins

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Is the brain a blank slate, or is it wired from birth to understand the world?

Jan 10 2017

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A quickly closed loophole has proved that the “great smoky dragon” of quantum mechanics may forever elude capture.

Jul 25 2018

18 mins

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Psychedelic drugs can trigger characteristic hallucinations, which have long been thought to hold clues about the brain’s circuitry. After nearly a century of study, a possible explanation is crystallizing.

Jul 30 2018

23 mins

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Researchers have discovered that simple “chemically active” droplets grow to the size of cells and spontaneously divide, suggesting they might have evolved into the first living cells.

Jan 19 2017

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A new idea is helping to explain the puzzling success of today’s artificial-intelligence algorithms — and might also explain how human brains learn.

Sep 21 2017

15 mins

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Researchers find evidence that neural systems actively remove memories, suggesting that forgetting may be the default mode of the brain.

Jul 24 2018

19 mins

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Spiders appear to offload cognitive tasks to their webs, making them one of a number of species with a mind that isn’t fully confined within the head.

May 23 2017

12 mins

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A different kind of dark matter could help to resolve an old celestial conundrum.

Jun 13 2017

14 mins

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Physicists theorize that a new “traversable” kind of wormhole could resolve a baffling paradox and rescue information that falls into black holes.

Oct 23 2017

16 mins

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A satellite spotted a burst of light just as gravitational waves rolled in from the collision of two black holes. Was the flash a cosmic coincidence, or do astrophysicists need to rethink what black holes can do?

Mar 02 2016

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Elizabeth Kolbert: We have locked in centuries of climate change. Elizabeth Kolbert covers climate change for the New Yorker. She's the Pulitzer prize-winning author of The Sixth Extinction. And she recently wrote a paragraph I can't stop thinking about. "The problem with global warming—and the reason it continues to resist illustration, even as the streets flood and the forests die and the mussels rot on the shores—is that experience is an inadequate guide to what’s going on. The climate operates on a time delay. When carbon dioxide is added to the atmosphere, it takes decades—in a technical sense, millennia—for the earth to equilibrate. This summer’s fish kill was a product of warming that had become inevitable twenty or thirty years ago, and the warming that’s being locked in today won’t be fully felt until today’s toddlers reach middle age. In effect, we are living in the climate of the past, but already we’ve determined the climate’s future."Kolbert lives, to an unusual degree, in the planet's future. She travels to the places around the world where the climate of tomorrow is visible today. She has watched glaciers melting, and seen species dying. And she is able to convey both the science and the cost with a rare lucidity. Talking with Kolbert left me with an unnerving thought. We look back on past eras in human history and judge them morally failed. We think of the Spanish Inquisition or the Mongol hordes and believe ourselves civilized, rational, moral in a way our ancestors weren't. But if the science is right, and we do unto our descendants what the data says we are doing to them, we will be judged monsters. And it will be all the worse because we knew what we were doing and we knew how to stop, but we decided it was easier to disbelieve the science or ignore the consequences. Kolbert and I talk about the consequences, but also about what would be necessary to stabilize the climate and back off the mass extinction event that is currently underway. We discuss geoengineering, political will, the environmental cost of meat, and what individuals can and can't do. We talk about Trump's cabinet, about whether technological innovation will save us, and if pricing carbon is enough. We talk about whether hope remains a realistic emotion when it comes to our environmental future.Books:-Edward Abbe’s “Desert Solitaire”-Rachel Carson’s “Silent Spring”-David G. Haskell’s “The Forest Unseen”-Bill McKibben’s “The End of Nature”Learn more about your ad choices. Visit megaphone.fm/adchoices

#877 - Jordan Peterson. Jordan Peterson is a clinical psychologist and tenured professor of psychology at the University of Toronto. https://www.youtube.com/user/JordanPetersonVideos http://www.selfauthoring.com/ 100% off the Future Authoring Program code: "ChangeYourself" - The offer is valid until the end of Nov 30th.

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Your brain hallucinates your conscious reality | Anil Seth. Right now, billions of neurons in your brain are working together to generate a conscious experience -- and not just any conscious experience, your experience of the world around you and of yourself within it. How does this happen? According to neuroscientist Anil Seth, we're all hallucinating all the time; when we agree about our hallucinations, we call it "reality." Join Seth for a delightfully disorienting talk that may leave you questioning the very nature of your existence.

The riddle of experience vs. memory | Daniel Kahneman. Using examples from vacations to colonoscopies, Nobel laureate and founder of behavioral economics Daniel Kahneman reveals how our "experiencing selves" and our "remembering selves" perceive happiness differently. This new insight has profound implications for economics, public policy -- and our own self-awareness.

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Rank #1: Quantum mechanics in popular-science books. As usual, the podcast is hosted by James Dacey, who is joined by Physics World‘s editor Matin Durrani and the magazine’s reviews editor Margaret Harris. The first part of the podcast addresses the question of why so many authors decide to write these books. The Physics World hosts are joined by physicist Chad Orzel, author of the bestselling book How to Teach Quantum Physics to Your Dog, which was released in 2010. The middle section of the podcast looks in more detail at the process of writing these books. It features the established popular-science writer Marcus Chown, who describes his experience of writing the book Quantum Theory Cannot Hurt You, which was published in 2007. Chown admits that he found the Pauli exclusion principle to be the most challenging aspect of quantum mechanics to explain in everyday language. This leads on to an interesting debate about the pros and potential pitfalls of using metaphors to describe complex science and mathematics. If scientists and science writers go through such pain to describe these features of the quantum world, then surely somebody without a scientific background should run a mile. But they don’t, instead they keep buying these books. In the final section of the podcast, the historian and philosopher Robert P Crease shares his thoughts on why the counterintuitive nature of quantum physics holds such a fascinating appeal for readers.

Rank #2: The masters of antimatter. Physics World reporter Tushna Commissariat recently visited the ALPHA antimatter experiment at CERN and caught up with its spokesperson Jeffrey Hangst. In this podcast, they talk about the perfect recipe for making antihydrogen, they discuss dealing with the fact and fiction that surrounds the field, and reveal the everyday realties of being an antimatter architect. Housed within CERN’s Antimatter Factory, which includes the Antiproton Decelerator (AD) (the source that provides low-energy antiprotons), ALPHA and the other antimatter experiments – ACE, AEGIS, ATRAP and ASACUSA – all study the many puzzling facets of antimatter. From its interaction with regular matter to the biological effects of antiprotons to how it falls under gravity, the various experimental teams hope that all will be revealed about antimatter’s true nature in the coming years. In particular, the ALPHA experiment – which won the Physics World Breakthrough of the Year in 2010 for trapping 38 antihydrogen atoms for about one-fifth of a second – is gearing up to scrutinize the stuff, as it will begin an experimental run this summer with the newly updated ALPHA2 device, which uses lasers to spectroscopically study the internal structure of the antihydrogen atom. In addition to finding out how exactly one makes and holds a few thousand atoms of the most volatile stuff in the universe, listen to this podcast to find out why Hangst thinks he has the coolest job in the world and what it is like to visit the one place in the universe where, as far as we know, antimatter is actively being produced.

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Rank #1: Episode 43: Electric Forces and Fields. An overview of the basics of electric charges, electric fields, and electric potential energy. I also discuss how objects become charged, how charged particles interact via Coulomb’s Law, how electroscopes work, and how batteries generate voltage. Recommended prerequisites are Episode 9: Matter and Molecules, and Episode 17: Energy, Work, and Momentum.

Rank #2: Episode 33: Disturbing Social Psychology Experiments. A discussion of three of the most chilling experiments in the field of social psychology: the Ash Conformity Experiment, the Stanford Prison Experiment, and the Milgram Obedience Experiment. In each case I discuss the motivation and setup of the experiment, outline the results, discuss replications and variations of the original experiment, and end with a look at the implications of the experiment for understanding the darker side of human nature.

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Rank #1: Scaling Down the Solar System. Meg interviews Wylie Overstreet, whose recent viral hit video "To Scale: The Solar System" gives us a glimpse of what the solar system would look like from the outside.

Rank #2: A Time Capsule of the Universe. The DASCH initiative, "Digital Access to a Sky-Century at Harvard", seeks to scan and upload tens of thousands of photographic plates, creating a comprehensive record of what the night sky has looked like over the past hundred years.

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Rank #1: Episode 3: Time Travel. Dave, Jocelyn and I sit down with dr mcninja creator Christopher Hastings. We teach him about time travel.

Rank #2: Episode 64: E and N (The edges of Einstein). Maya Inamura Joins Me, Katie Mack and Leo Stein and we talk aboutTEST OF GENERAL RELATIVITY

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Rank #1: Gravitoelectromagnetism. Randy talks to Jim about gravitoelectromagnetism. Based on the similarity between Newtonian gravity and electrostatics, there should be a second gravitational field,the gravitomagnetic field. What are the implications of the existence of such a field, and how large are those effects? Show notes: http://frontiers.physicsfm.com/3

Rank #2: The Dimensionality of Space-Time. Jim discusses why the world we observe is 4-dimensional with Randy. We discuss anthropic and fundamental reasons why we need 3 dimensions and no more than one time dimension for reasons of complexity, predictability and stability. Show Notes: http://frontiers.physicsfm.com/38

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Rank #1: f(θ)=1-sin(θ). f(θ)=1-sin(θ) If you ever want to conduct a quick social experiment on the status of mathematics in the world just get yourself a dating profile and mention on it that you are a mathematician. The messages you get will be quite illuminating: “I hate to break it to you, but while I appreciate math for its logic and beauty, I don’t think I’ll ever like it. lol TOO many formulas.” “I got up to AP Calc during my senior year of high school, cheated off my best friend on all the tests and still got 70s in the class, and swore off math from thereon.” Even when people do not say outright that they despise math the contents can leave a bit to be desired: “I’m awful at math but it fascinates me–much like historical linguistics and conjugating Russian.” It is not like it was all bad though. Samuel did once get this message: “I also really like math and spend a lot of time trying to figure out how to get people to like it more!” But come to think of it he doesn’t think they actually ended up actually going on a date. We really shouldn’t be so negative about all of this. Samuel has been told by more than one person that being a mathematician makes him sexy, really he has and it is so validating for him, and he doubts anyone ever turned me down for a date just because he loves mathematics. But given all the times he has received messages with gloomy words about math and how often on a first date some of the first words out of his companion’s mouth is how much they hate math he couldn’t help but wonder if mathematics has impacted my dating life negatively, if only a little bit. Of course mathematics has never let us down in the past, doubt it is going to start now. Download the Episode Subscribe: iTunes or RSS Support the Kickstarter An Economist Cupid Andrea Silenzi was the host of Why Oh Why, a radio show about where love and sex meets technology and she was looking for a date. So when Planet Money called her up and asked if she would be interested in getting some dating advice from economist Tim Harford she definitely said yes. Samuel spoke with Andrea about what it was like to follow an economist’s advice on dating, why we should not treat dating like a job, and where to draw the line when it comes to formulaic dating. Helping you Math your way to Someone Special Back in 2009 for the podcast Strongly Connected Components Samuel interviewed Sam Yagan then the CEO and co-founder of an upstart online dating site which was differentiating itself from the competition by putting a real focus on the data side of dating. That little upstart was OKCupid and Sam is now the CEO of Match Group, which includes Match.com, OkCupid and Tinder. They talked about why OKCupid puts such a focus on math and data, how the OKCupid algorithm relies on its users, and why you shouldn’t stress out on having the perfect dating profile photo. Full Strongly Connected Components Interview: Optimal Date Stopping Mathematics communicator and comedian Matt Parker tells Samuel about the optimal stopping problem, and how it could help him date more effectively. Masters and Disasters of Relationships John Gottman is a psychologist, therapist, mathematician, and co-founder, with his wife Julie Schwartz Gottman, of the Gottman Institute where they do research in order to better understand relationships. For our purposes we are most interested in the work John has done in mathematically modelling marriage, in particular the factors which lead to divorce. John tells Samuel about his research, how he transitioned from mathematics to psychology, and what, mathematically, is the biggest predictor of a lasting relationship. Social Network Leveraged Speed Dating Andrea and Samuel had so much fun talking about her economist advised dating experiments that they continued chatting for quite a while. This is eventually where they eventually landed.

Rank #2: Their Favorite Theorem. Have you ever wondered what mathematicans’ favorite theorems were? How about what food or music pairs perfectly with those theorems? Well whether your answer to those questions was yes or no or what are you talking about there is a new mathematics podcast on the scene you need to check out called My Favorite Theorem. My Favorite Theorem is the brain child of Kevin Knudson and Evelyn Lamb. You may recognize those names as a writer who contributes to The Conversation, Forbes, and is a mathematics professor at the University of Florida and as freelance mathematics journalist who runs the Scientific American blog Roots of Unity. They were kind enough to talk to me early in the morning about where the idea for the show came from, why the pairings are so cool, and how mathematical audio can help humanize mathematicians. Oh, and I make them come up with a pairing for our conversation. Plus, as a super special bonus they were kind enough to let me share episode 3 of My Favorite Theorem with Emille Davie Lawrence as part of the episode. I know you will soon have another podcast added to you subscription list. Don’t forget to support Relatively Prime on Patreon and make sure Samuel can afford to make rent next month. Download the EpisodeSubscribe: Apple Podcasts or RSS Music SUPERMILK

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Rank #1: Episode #9: From Boredom to 'Sporedom'. The science of boredom is actually quite stimulating, as Carl makes clear in an utterly meta segment of the Boredom Bin. And they're old, they're fast, they're huge, they're magical ... they're mushrooms. From "hat" tossing to cleaning up nuclear waste, Alison explores the fantastic capabilities of fungi. The brain is a mysterious thing, and the few scientists who have studied the "Tetris Effect" have revealed intriguing quirks about memory, and as Bill explains, there are plenty of questions remaining. And finally, Nate explores the linguistic nature of words that have no translation in English, such as tartle, Iktsuarpok or komorebi. Can we truly experience a feeling or idea if we don't have a word for it?

Rank #2: Episode #10: We're Talkin' Poop and Urinal Cakes. Yes, we're going there. In this episode of the podcast, Alison, our latest Boredom Bin victim, gets stuck explaining the science of urinal cakes. She's the one person in the room who hasn't actually seen one in person, but her stream of knowledge on the topic is on point. Carl follows up "topic #1" with "topic #2": poop. He spoke with David Waltner Toews, author of "The Origin of Feces," to see why we might want to consider polishing a turd, and rethinking excrement's stinky reputation. And as a nice palate cleanser, Bill busts out the Banach-Tarski paradox. In a nutshell, you can take one sphere, cut it up in a most precise way, and end up with two spheres identical to the first one. In fact, you could, theoretically, take a pea, cut it up just right, and make a sphere the size of the size. Believe it or not, it'll sort of make sense by the end of the episode...sort of. And finally, Nate wraps this whole thing up with bacon—that is, why meteoroids streaking through the sky can sound just like sizzling bacon. A very cool experiment reveals the fascinating physics behind this phenomena.

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Rank #1: Darkness Visible: The Hunt For Dark Matter . Members of the Rudolf Peierls Centre for Theoretical Physics hosted the 3rd morning of Theoretical Physics covering the connections between cosmology and particle physics.

Rank #2: Networked Quantum Information Technologies . This talk reviews the developments in quantum information processing.

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Rank #1: AaS! 77: Seriously, what is gravity? (Part 1). Part 1! How did Einstein develop General Relativity? What does it mean for different kinds of masses to be equivalent? How does gravity do what it does? Why is curvature so important in understanding gravity? I discuss these questions and more in today’s Ask a Spaceman! Support the show: http://www.patreon.com/pmsutter All episodes: http://www.AskASpaceman.com Follow on Twitter: http://www.twitter.com/PaulMattSutter Like on Facebook: http://www.facebook.com/PaulMattSutter Watch on YouTube: http://www.youtube.com/PaulMSutter Go on an adventure: http://www.AstroTours.co Keep those questions about space, science, astronomy, astrophysics, physics, and cosmology coming to #AskASpaceman for COMPLETE KNOWLEDGE OF TIME AND SPACE! Big thanks to my top Patreon supporters this month: Robert R., Justin G., Matthew K., Kevin O., Chris C., Helge B., Tim R., Steve P., Lars H., Khaled T., Chris L., John F., Craig B., Mark R., and David B.! Music by Jason Grady and Nick Bain. Thanks to WCBE Radio for hosting the recording session, Greg Mobius for producing, and Cathy Rinella for editing. Hosted by Paul M. Sutter, astrophysicist at The Ohio State University, Chief Scientist at COSI Science Center, and the one and only Agent to the Stars (http://www.pmsutter.com).

Rank #2: AaS! 14: What is spacetime? (Part 1). What is spacetime? What was the breakthrough idea that led Einstein to develop Special Relativity? Who was right: Newton or Maxwell? I discuss these questions and more in today's Ask a Spaceman! Follow all the show updates at askaspaceman.com. Keep those questions about space, science, astronomy, astrophysics, and cosmology coming to #AskASpaceman on Twitter@PaulMattSutter and Facebook/PaulMattSutter for COMPLETE KNOWLEDGE OF TIME AND SPACE! Music by Jason Grady and Nick Bain. Thanks to Lynn Stevenson and Gideon Koekoek for their generous contributions to support this podcast!

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Rank #1: 14 August, 2019 – Episode 734 – What’s that Noise?. Free Will Falters?, First Cells Collaborated, Underwater Neanderthals, Monkey Worry Molecule, Big Frog Nests, Picking Baby Sex, Justin's Airship Future, Supernova Dust, Plastic Snow, Black Hole Rising, Daddy Longlegs Venom, Feeling Foreshocks, And Much More... The post 14 August, 2019 – Episode 734 – What’s that Noise? appeared first on This Week in Science - The Kickass Science Podcast.

Rank #2: 13 February, 2019 – Episode 708 – This Week in Science Podcast (TWIS). Interview w/ Dr. Brian Keating, Bug-pocalypse, Heart-Shaped Tail, Sandy Footprints, Slimy Rocks, Love is for Losers, Emotional Alterations, Valentine's Drinking Plans, Ultima Pancake, Mars Of Lost Loves, And Much More... The post 13 February, 2019 – Episode 708 – This Week in Science Podcast (TWIS) appeared first on This Week in Science - The Kickass Science Podcast.

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Rank #1: Jordan Ellenberg, “How Not To Be Wrong: The Power of Mathematical Thinking” (Penguin Press, 2014). The book discussed in this interview is How Not To Be Wrong: The Power of Mathematical Thinking (Penguin Press, 2014), by Jordan Ellenberg. This is one of those rare books that belong on the reading list of every educated person, especially those who love mathematics, but more importantly, those who hate it. Ellenberg succeeds in explaining the value of mathematical reasoning without ever needing to go into technical detail, which makes the book ideal for those who want to learn why mathematics is so important. What makes the book doubly delightful is Ellenberg’s writing style; he intersperses the math with amusing anecdotes, dispensed with a sense of humor rarely found in books such as this. The book is chock-full of OMG moments; the introductory anecdote about Abraham Wald and the missing bullet holes absolutely whets the appetite for more and Ellenberg never fails to deliver.Learn more about your ad choices. Visit megaphone.fm/adchoices

Rank #2: Vicky Neale, “Closing the Gap: The Quest to Understand Prime Numbers” (Oxford UP, 2017). Today I talked to Vicky Neale about her new book Closing the Gap: The Quest to Understand Prime Numbers (Oxford University Press, 2017). The book details one of the most exciting developments to happen in the last few years in mathematics, a new approach to the Twin Primes Conjecture. The story involves mathematicians from five different centuries and probably every continent except Antarctica. Vicky does a great job of telling not only what the problem is and how work on it has proceeded, but also how mathematical research has evolved given the resources available in the twenty first century. If you like numbers, you’ll love this book—and if you don’t like numbers, maybe this book can help you appreciate them.Learn more about your ad choices. Visit megaphone.fm/adchoices

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Rank #1: Episode 2 - Dave Richeson. This transcript is provided as a courtesy and may contain errors. Evelyn Lamb: Welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I am a freelance math writer usually based in Salt Lake City but currently based in Paris. And this is your other host. KK: I’m Kevin Knudson, professor of mathematics at the University of Florida. EL: Every episode we invite a mathematician on to tell us about their favorite theorem. This week our guest is Dave Richeson. Can you tell us a little about yourself, Dave? Dave Richeson: Sure. I’m a professor of mathematics at Dickinson College, which is in Carlisle, Pennsylvania. I’m also currently the editor of Math Horizons, which is the undergraduate magazine of the Mathematical Association of America. EL: Great. And so how did you get from wherever you started to Carlisle, Pennsylvania? DR: The way things usually work in academia. I applied to a bunch of schools. Actually, seriously, my wife knew someone in Carlisle, Pennsylvania. My girlfriend at the time, wife now, and she saw the list of schools that I was applying to and said, “You should get a job at Dickinson because I know someone there.” And I did. KK: That never happens! EL: Wow. DR: That never happens. KK: That never happens. Dave and I actually go back a long way. He was finishing his Ph.D. at Northwestern when I was a postdoc there. DR: That’s right. KK: That’s how old-timey we are. Hey, Dave, why don’t you plug your excellent book. DR: A few years ago I wrote a book called Euler’s Gem: The Polyhedron Formula and the Birth of Topology. It’s at Princeton University Press. I could have chosen Euler’s Formula as my favorite theorem, but I decided to choose something different instead. KK: That’s very cool. I really recommend Dave’s book. It’s great. I have it on my shelf. It’s a good read. DR: Thank you. EL: Yeah. So you’ve told us what your favorite theorem isn’t. So what is your favorite theorem? DR: We have a family joke. My kids are always saying, “What’s your favorite ice cream? What’s your favorite color?” And I don’t really rank things that way. This was a really challenging assignment to come up with a theorem. I have recently been interested in π and Greek mathematics, so currently I’m fascinated by this theorem of Archimedes, so that is what I’m giving you as my favorite theorem. Favorite theorem of the moment. The theorem says that if you take a circle, the area of that circle is the same as the area of a right triangle that has one leg equal to the radius and one leg equal to the circumference of the circle. Area equals 1/2 c x r, and hopefully we can spend the rest of the podcast talking about why I think this is such a fascinating theorem. KK: I really like this theorem because I think in grade school you memorize this formula, that area is π r2, and if you translate what you said into modern terminology, or notation, that is what it would say. It’s always been a mystery, right? It just gets presented to you in grade school. Hey, this is the formula of a circle. Just take it. DR: Really, we have these two circle formulas, right? The area equals π r2, and the circumference is 2πr, or the way it’s often presented is that π is the circumference divided by the diameter. As you said, you could convince yourself that Archimedes theorem is true by using those formulas. Really it’s sort of the reverse. We have those formulas because of what Archimedes did. Pi has a long and fascinating history. It was discovered and rediscovered in many, many cultures: the Babylonians, the Egyptians, Chinese, Indians, and so forth. But no one, until the Greeks, really looked at it in a rigorous way and started proving theorems about π and relationships between the circumference, the diameter, and the area of the circle. EL: Right, and something you had said in one of your emails to us was about how it’s not even, if you ask a mathematician who proved that π was a constant, that’s a hard question. DR: Yes, exactly. I mean, in a way, it seems easy. Pi is usually defined as the circumference divided by the diameter for any circle. And in a way, it seems kind of obvious. If you take a circle and you blow it up or shrink it down by some factor of k, let’s say, then the circumference is going to increase by a factor of k, the diameter is going to increase by a factor of k. When you do that division you would get the same number. That seems sort of obvious, and in a way it kind of is. What’s really tricky about this is that you have to have a way of talking about the length of the circumference. That is a curve, and it’s not obvious how to talk about lengths of curves. In fact, if you ask a mathematician who proved that the circumference over the diameter was the same value of π, most mathematicians don’t know the answer to that. I’d put money on it that most people would think it was in Euclid’s Elements, which is sort of the Bible of geometry. But it isn’t. There’s nothing about the circumference divided by the diameter, or anything equivalent to it, in Euclid’s Elements. Just to put things in context here, a quick primer on Greek mathematics. Euclid wrote Elements sometime around 300 BCE. Pythagoras was before that, maybe 150 years before that. Archimedes was probably born after Euclid’s Elements was written. This is relatively late in this Greek period of mathematics. KK: Getting back to that question of proportionality, the idea that all circles are similar and that’s why everybody thinks π is a constant, why is that obvious, though? I mean, I agree that all circles are similar. But this idea that if you scale a circle by a factor of k, its length scales by k, I agree if you take a polygon, that it’s clear, but why does that work for curves? That’s the crux of the matter in some sense, right? DR: Yeah, that’s it. I think one mathematician I read called this “inherited knowledge.” This is something that was known for a long time, and it was rediscovered in many places. I think “obvious” is sort of, as we all know from doing math, obvious is a tricky word in math. It’s obvious meaning lots of people have thought of it, but if you actually have to make it rigorous and give a proof of this fact, it’s tricky. And so it is obvious in a sense that it seems pretty clear, but if you actually have to connect the dots, it’s tricky. In fact, Euclid could not have proved it in his Elements. He begins the Elements with his famous five postulates that sort of set the stage, and from those he proves everything in the book. And it turns out that those five postulates aren’t enough to prove this theorem. So one of Archimedes’ contributions was to recognize that we needed more than just Euclid’s postulates, and so he added two new postulates to those. From that, he was able to give a satisfactory proof that area=1/2 circumference times radius. KK: So what were the new postulates? DR: One of them was essentially that if you have two points, then the shortest distance between them is a straight line, which again seems sort of obvious, and actually Euclid did prove that for polygonal lines, but Archimedes is including curves as well. And the other one is that if you have, it would be easier to draw a picture. If you had two points and you connected them by a straight line and then connected them by two curves that he calls “concave in the same direction,” then the one that’s in between the straight line and the other curve is shorter than the second curve. The way he uses both of those theorems is to say that if you take a circle and inscribe a polygon, like a regular polygon, and you circumscribe a regular polygon, then the inscribed polygon has the shortest perimeter, then the circle, then the circumscribed polygon. That’s the key fact that he needs, and he uses those two axioms to justify that. EL: OK. And so this sounds like it’s also very related to his some more famous work on actually bounding the value of π. DR: Yeah, exactly. We have some writings of his that goes by the name “Measurement of a Circle.” Unfortunately it’s incomplete, and it’s clearly not come down to us very well through history. The two main results in that are the theorem I just talked about and his famous bounds on π, that π is between 223/71 and 22/7. 22/7 is a very famous approximation of π. Yes, so these are all tied together, and they’re in the same treatise that he wrote. In both cases, he uses this idea of approximating a circle by inscribed and circumscribed polygons, which turned out to be extremely fruitful. Really for 2,000 years, people were trying to get better and better approximations, and really until calculus they basically used Archimedes’ techniques and just used polygons with more and more and more and more sides to try to get better approximations of π. KK: Yeah, it takes a lot too, right? Weren’t his bounds something like a 96-gon? DR: Yeah, that’s right. Exactly. KK: I once wrote a Geogebra applet thing to run to the calculations like that. It takes it a while for it to even get to 3.14. It’s a pretty slow convergence. DR: I should also plug another mathematician from the Greek era who is not that well known, and that is Eudoxus. He did work before Euclid, and big chunks of Euclid’s Elements are based on the work of Eudoxus. He was the one who really set this in motion. It’s become known as the method of exhaustion, but really it’s the ideas of calculus and limiting in disguise. This idea of proving these theorems about shapes with curved boundaries using polygons, better and better approximations of polygons. So Eudoxus is one of my favorite mathematicians that most people don’t really know about. KK: That’s exactly it, right? They almost had calculus. DR: Right. KK: Almost. It’s really pretty amazing. DR: Yes, exactly. The Greeks were pretty afraid of infinity. KK: I’m sort of surprised that they let the method of exhaustion go, that they were OK with it. It is sort of getting at a limiting process, and as you say, they don’t like infinity. DR: Yeah. KK: You’d think they might not have accepted it as a proof technique. DR: Really, and maybe this is talking too much for the mathematicians in the audience, but really the way they present this is a proof by contradiction. They show that it can’t be done, and then they get these polygons that are close enough that it can be done, and that gives them a contradiction. The final style of the proof would, I think, be comfortable to them. They don’t really take a limit, they don’t pass to infinity, anything like that. EL: So something we like to do on this podcast is ask our guest to pair their theorem with something. Great things in life are often better paired: wine and cheese, beer and pizza, so what’s best with your theorem? DR: I have to go with the obvious: pie, maybe pizza. KK: Just pizza? OK? EL: What flavor? What toppings? KK: What goes on it? DR: That’s a good question. I’m a fan of black olives on my pizza. KK: OK. Just black olives? DR: Maybe some pepperoni too. KK: There you go. EL: Deep dish? Thin crust? We want specifics. DR: I’d say thin crust pizza, pepperoni and black olives. That sounds great. EL: You’d say this is the best way to properly appreciate this theorem of Archimedes, is over a slice of pizza. DR: I think I would enjoy going to a good pizza joint and talking to some mathematicians and telling them about who first proved that circumference over diameter is π, that it was Archimedes. Actually, I was saying to Kevin before we started recording that I actually have a funny story about this, that I started investigating this. I wanted to know who first proved that circumference over diameter is a constant. I did some looking and did some asking around and couldn’t really get a satisfactory answer. I sheepishly at a conference went up to a pretty well-known math historian, and said, “I have this question about π I’m embarrassed to ask.” And he said, “Who first proved that circumference over diameter is a constant?” I said, “Yes!” He’s like, “I don’t know. I’d guess Archimedes, but I really don’t know.” And that’s when I realized it was an interesting question and something to look at a little more deeply. EL: That’s a good life lesson, too. Don’t be afraid to ask that question that you are a little afraid to ask. KK: And also that most answers to ancient Greek mathematics involve Archimedes. DR: Yeah. Actually through this whole investigation, I’ve gained an unbelievable appreciation of Archimedes. I think Euclid and Pythagoras probably have more name recognition, but the more I read about Archimedes and things that he’s done, the more I realize that he is one of the great, top 5 mathematicians. KK: All right, so that’s it. What’s the top 5? DR: Gosh. Let’s see here. KK: Unordered. DR: I already have Archimedes. Euler, Newton, Gauss, and who would number 5 be? KK: Somebody modern, come on. DR: How about Poincaré, that’s not exactly modern, but more modern than the rest. While we’re talking about Archimedes, I also want to make a plug. There’s all this talk about tau vs. pi. I don’t really want to weigh in on that one, but I do think we should call π Archimedes’ number. We talk about π is the circumference constant, π is the area constant. Archimedes was involved with both of those. People may not know he was also involved in attaching π to the volume of the sphere and π to the surface area of the sphere. Here I’m being a little historically inaccurate. Pi as a number didn’t exist for a long time after that. But basically recognizing that all four of these things that we now recognize as π, the circumference of a circle, the area of a circle, the volume of a sphere, and the surface area of a sphere. In fact, he famously asked that this be represented on his tombstone when he died. He had this lovely way to put all four of these together, and he said that if you take a sphere and then you enclose it in a cylinder, so that’s a cylinder that’s touching the sphere on the sides, think of a can of soda or something that’s touching on the top as well, that the volume of the cylinder to the sphere is in the ratio 3:2, and the surface area of the cylinder to the sphere is also the ratio 3:2. If you work out the math, all four of these versions of π appear in the calculation. We do have some evidence that this was actually carried out. Years later, the Roman Cicero found Archimedes’ tomb, and it was covered in brambles and so forth, and he talks about seeing the sphere and the cylinder on Archimedes’ tombstone, which is kind of cool. EL: Oh wow. DR: Yeah, he wrote about it. KK: Of course, how Archimedes died is another good story. It’s really too bad. DR: Yeah, I was just reading about that this week. The Roman siege of Syracuse, and Archimedes, in addition to being a great mathematician and physicist, was a great engineer, and he built all these war devices to help keep the Romans at bay, and he ended up being killed by a Roman soldier. The story goes that he was doing math at the time, and the Roman general was apparently upset that they killed Archimedes. But that was his end. KK: Then on Mythbusters, they actually tried the deal with the mirrors to see if they could get a sail to catch on fire. DR: I did see that! Some of these stories have more evidence than others. Apparently the story of using the burning mirrors to catch ships on fire, that appeared much, much later, so the historical connection to Archimedes is pretty flimsy. As you said, it was debunked by Mythbusters on TV, or they weren’t able to match Archimedes, I should say. KK: Well few of us can, right? DR: Right. The other thing that is historically interesting about this is that one of the most famous problems in the history of math is the problem of squaring the circle. This is a famous Greek problem which says that if you have a circle and only a compass and straightedge, can you construct a square that has the same area as the circle? This was a challenging and difficult problem. Reading Archimedes’ writings, it’s pretty clear that he was working on this pretty hard. That’s part of the context, I think, of this work he did on π, was trying to tackle the problem of squaring the circle. It turns out that this was impossible, it is impossible to square the circle, but that wasn’t discovered until 1882. At the time it was still an interesting open problem, and Archimedes made various contributions that were related to this famous problem. EL: Yeah. KK: Very cool. DR: I can go on and on. So today, that is my favorite theorem. KK: We could have you on again, and it might be different? DR: Sure. I’d love to. KK: Well, thanks, Dave, we certainly appreciate you being here. DR: I should say if people would like to read about this, I did write an article, “Circular Reasoning: Who first proved that c/d is a constant?” Some of the things I talked about are in that article. Mathematicians can find it in the College Math Journal, and it just recently was included in Princeton University Press’s book The Best Writing on Mathematics, 2016 edition. You can find that wherever, your local bookstore. EL: And where else can our loyal listeners find you online, Dave? DR: I spend a lot of time on Twitter. I’m @divbyzero. I blog occasionally at divisbyzero.com. EL: OK. DR: That’s where I’d recommend finding me. KK: Cool. EL: All right. Well, thanks for being here. DR: Thank you for asking me. It was a pleasure talking to you. Show notes (click here)

Rank #2: Episode 3 - Emille Davie Lawrence. This transcript is provided as a courtesy and may contain errors. EL: Welcome to My Favorite Theorem. I’m one of your hosts, Evelyn Lamb. I’m a freelance math and science writer currently based in Paris. And this is my cohost. KK: Hi, I’m Kevin Knudson, professor of mathematics at the University of very, very hot Florida. EL: Yeah. Not so bad in Paris yet. KK: It’s going to be a 96-er tomorrow. EL: Wow. So each episode, we invite a mathematician to come on and tell us about their favorite theorem. Today we’re delighted to welcome Emille Davie Lawrence to the show. Hi, Emille. EDL: Hello, Evelyn. EL: So can you tell us a little bit about yourself? EDL: Sure! So I am a term assistant professor at the University of San Francisco. I’m in the mathematics and physics department. I’ve been here since 2011, so I guess that’s six years now. I love the city of San Francisco. I have two children, ages two and almost four. EL: Who are adorable, if your Facebook is anything to go by. EDL: Thank you so much. You’ll get no arguments from me. I’ve been doing math for quite a while now. I’m a topologist, and my mathematical interests have always been in topology, but they’ve evolved within topology. I started doing braid groups, and right now, I’m thinking about spatial graphs a lot. So lots of low-dimensional topology ideas. EL: Cool. So what is your favorite theorem? EDL: My favorite theorem is the classification theorem for compact surfaces. It basically says that no matter how weird the surface you think you have on your hands, if it’s a compact surface, it’s only one of a few things. It’s either a sphere, or the connected sum of a bunch of tori, or the connected sum of a bunch of projective planes. That’s it. EL: Can you tell us a little bit more about what projective planes are? EDL: Obviously a sphere, well, I don’t know how obvious, but a sphere is like the surface of a ball, and a torus looks like the surface of a donut, and a projective plane is a little bit stranger. I think anyone who would be listening may have run into a Möbius band at some point. Basically you take a strip of paper and glue the two ends of your strip together with a half-twist. This is a Möbius band. It’s a non-orientable half-surface. I think sometimes kids do this. They pop up in different contexts. One way to describe a projective plane is to take a Möbius band and add a disc to the Möbius band. It gives you a compact surface without boundary because you’ve identified the boundary circle of the Möbius band to the boundary of the disc. EL: Right, OK. EDL: Now you’ve got this non-orientable thing called a projective plane. Another way to think about a projective plane is to take a disc and glue one half of the boundary to the other half of the boundary in opposite directions. It’s a really weird little surface. KK: One of those things we can’t visualize in three dimensions, unfortunately. EDL: Right, right. It’s actually hard to explain. I don’t think I’ve ever tried to explain it without drawing a picture. EL: Right. That’s where the blackboard comes in hand. KK: Limitations of audio. EL: Have you ever actually tried to make a projective plane with paper or cloth or anything? EDL: Huh! I am going to disappoint you there. I have not. The Möbius bands are easy to make. All you need is a piece of paper and one little strip of tape. But I haven’t. Have you, Evelyn? EL: I’ve seen these at the Joint Meetings, I think somebody brought this one that they had made. And I haven’t really tried. I’d imagine if you tried with paper, it would probably just be a crumpled mess. EDL: Right, yeah. EL: This one I think was with fabric and a bunch of zippers and stuff. It seemed pretty cool. I’m blanking now on who is was. KK: That sounds like something sarah-marie belcastro would do. EL: It might have been. It might have been someone else. There are lots of cool people doing cool things with that. I should get one for myself. EDL: Yeah, yeah. I can see cloth and zippers working out a lot better than a piece of paper. EL: So back to the theorem. Do you know what makes you love this theorem? EDL: Yeah. I think just the fact that it is a complete classification of all compact surfaces. It’s really beautiful. Surfaces can get weird, right? And no matter what you have on your hands, you know that it’s somewhere on this list. That makes a person like me who likes order very happy. I also like teaching about it in a topology class. I’ve only taught undergraduate topology a few times, but the last time was last spring, a year ago, spring of 2016, and the students seemed to really love it. You can play these “What surface am I?” games. Part of the proof of the theorem is that you can triangulate any surface and cut it open and lay it flat. So basically any surface has a polygonal representation where you’re just some polygon in the plane with edges identified in pairs. I like to have this game in my class where I just draw a polygon and identify some of the edges in pairs and say, “What surface is this?” And they kind of get into it. They know what the answers, what the possibilities are for the answers. You can sort of just triangulate it and find the Euler characteristic, see if you can find a Möbius band, and you’re off to the races. KK: That’s great. I taught the graduate topology course here at Florida last year. I’m ashamed to admit I didn’t actually prove the classification. EDL: You should not be ashamed to admit that. It’s something at an undergraduate level you get to at the end, depending on how you structure things. We did get to it at the end of the course, so I don’t know how rigorously I proved it for them. The combinatorial step that goes from: you can take this polygonal representation, and you can put it in this polygonal form, always, that takes a lot of work and time. EL: There are delicacies in there that you don’t really know about until you try to teach it. I taught it also in class a couple years ago, and when I got there, I was like, “This seemed a little easier when I saw it as a student.” Now that I was trying to teach it, it seemed a little harder. Oh, there are all of these t’s I have to cross and i’s I have to dot. KK: That’s always the way, right? EDL: Right. KK: I assigned as a homework assignment that my students should just compute the homology of these surfaces, and even puncture them. Genus g, r punctures, just as a homework exercise. From there you can sort of see that homology tells you that genus classifies things, at least up to homotopy invariants, but this combinatorial business is tricky. EDL: It is. EL: Was this a love at first sight kind of theorem, or is this a theorem that’s grown on you? EDL: I have to say it’s grown on me. I probably saw it my first year of graduate school, and like all of topology, I didn’t love it at first when I saw it as a first-year graduate student. I did not see any topology as an undergrad. I went to a small, liberal arts college that didn’t have it. So yeah, I have matured in my appreciation for the classification theorem of surfaces. It’s definitely something I love now. KK: You’re talking to a couple of topologists, so you don’t have to convince us very much. EDL: Right. KK: I had a professor as an undergrad who always said, “Topology is analysis done right.” EDL: I like that. KK: I know I just infuriated all the analysts who are listening. I always took that to heart. I always took that to heart because I always felt that way too. All those epsilons and deltas, who wants all that? EDL: Who needs it? KK: Draw me a picture. EL: I was so surprised in the first, I guess advanced calculus class I had, a broader approach to calculus, and I learned that all these open sets and closed sets and things actually had to do with topology not necessarily with epsilons and deltas. That was really a revelation. KK: So you’re interested in braids, too, or you were? You moved on? EDL: I would say I’m still interested in braids, although that is not the focus of my research right now. KK: Those are hard questions too, so much interesting combinatorics there. EDL: That’s right. I think that’s sort of what made me like braid groups in the first place. I thought it was really neat that a group could have that geometric representation. Groups, I don’t know, when you learn about groups, I guess the symmetric group is one of the first groups that you learn about, but then it starts to wander off into abstract land. Braid groups really appealed to me, maybe just the fact that I liked learning visually. EL: It’s not quite as in the clouds as some abstract algebra. KK: And they’re tied up with surfaces, right, because braid groups are just the mapping class group of the punctured disc. EDL: There you go. KK: And Evelyn being the local Teichmüller theorist can tell us all about the mapping class groups on surfaces. EL: Oh no! We’re getting way too far from the classification of surfaces here. KK: This is my fault. I like to go off on tangents. EDL: Let’s reel it back in. EL: You mentioned that you’ve matured into true appreciation of this lovely theorem, which kind of brings me to the next part of the show. The best things in life are better together. Can you recommend a pairing for your theorem? This could be a fine wine or a flavor of ice cream or a favorite piece of music or art that you think really enhances the beauty of this classification theorem. EDL: I hate to do this, but I’m going to have to say coffee and donuts. KK: Of course. EDL: I really tried to say something else, but I couldn’t make myself do it. A donut and cup of coffee go great with the classification of compact surfaces theorem. EL: That’s fair. KK: San Francisco coffee, right? Really good dark, walk down to Blue Bottle and stand in line for a while? EDL: That’s right. Vietnamese coffee. KK: There you go. That’s good. EL: Is there a particular flavor of donut that you recommend? EDL: Well you know, the maple bacon. Who can say no to bacon on a donut? KK: Or on anything for that matter. EDL: Or on anything. KK: That’s just a genus one surface. Can we get higher-genus donuts? Have we seen these anywhere, or is it just one? EDL: There are some twisted little pastry type things. I’m wondering if there’s some higher genus donuts out there. EL: If nothing else there’s a little bit of Dehn twisting going on with that. EDL: There’s definitely some twisting. EL: I guess we could move all the way over into pretzels, but that doesn’t go quite as well with a cup of coffee. EDL: Or if you’re in San Francisco, you can get one of these cronuts that have been all the rage lately. EL: What is a cronut? I have not quite understood this concept. EDL: It is a cross between a croissant and a donut. And it’s flakier than your average donut. It is quite good. And if you want one, you’re probably going to have to stand on line for about an hour. Maybe the rage has died down by now, maybe. But that’s what was happening when they were first introduced. EL: I’m a little scared of the cronut. That sounds intense but also intriguing. EDL: You’ve got to try everything once, Evelyn. Live on the edge. EL: The edge of the cronut. KK: You’re in Paris. We’re not too concerned about your ability to get pastry. EL: I have been putting away some butter. KK: The French have it right. They understand that butter does the heavy lifting. EDL: It’s probably a sin to have a cronut in Paris. EL: Probably. But if they made one, it would be the best cronut that existed. EDL: Absolutely. KK: Well I think this has been fun. Anything else you want to add about your favorite theorem? EDL: It’s a theorem that everyone should dig into, even if you aren’t into topology. I think it’s one of those foundational theorems that everyone should see at least once, and look at the proof at least once, just for a well-rounded mathematical education. KK: Maybe I should look at the proof sometime. EL: Thanks so much for joining us, Emille. We really enjoyed having you. And this has been My Favorite Theorem. EDL: Thank you so much. KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Del Mitchell, and Bao-xian Lin. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics. Show notes (click here)

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