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This is Radiolab. I'm Soren Wheeler. I'm sitting in for Lulu and Latif today. They will be back next week. But for today's show, we got a good one. And I brought along a friend. Hello. Our editor, Alex Neeson. Yeah. All right. So tell us what we're, tell everybody what we're up to today. Yeah. So we're here because a little while back, we got a question from a listener. It seemed like a pretty simple question.
But the more we got into it and tried to figure out how to answer it,
the more it just dragged us into the middle of everything. Yeah. It's like one big, gigantic spiral. Like, what are we? Who are we? I'm sorry. That's okay. So the question came from a woman named Laura Andrews. Yeah. My name is Laura and I'm a student. Laura is an undergraduate student at the University of Missouri. Mizzou. Mizzou. Yeah. But when she sent us this question almost a year ago now, she was standing kind of on the precipice
at a very particular transition moment in her life. Yeah. I think you had just finished high school, right? Yes, I was a couple months into my gap year, but I hadn't applied to any colleges I had planned to, but
But some personal things went on that just kept me from doing that. So I was living with my parents. The most that I was ever doing was dog sitting once a month or something like that. That was it. And I didn't have any plans afterwards.
So Laura is just graduated from high school. She's at her parents' house. She's feeding some dogs from time to time. Reflecting on myself and what I'm meant to be doing. Staring into her future. And she starts asking herself a lot of questions. Where do I belong?
Sort of philosophical questions. What is my place in the universe? About herself. Where are we in relation to everything ever? About everything. What is everything in relation to everything? And Laura told us that when she started thinking about everything... The vastness, the bigness. She just felt small. But then she thought, am I small?
And that led her to the question that she sat down, wrote out, and sent to us. And her question turned out to be weirdly mathematical. What is the most average size that a thing could be in the universe?
If one were to take the size of the largest singular thing, be it a star or a black hole or something, rather than a cluster of stuff like a galaxy, and the size of the smallest thing, my guess would be an electron or something of the sort, and found the exact median size, how big would that be? I've tried to work it out myself, but I'm good at neither math nor science, and my answers always seem to be entirely too large. Where is the midpoint between big and small in relation to...
Literally everything ever. Okay. Okay. Can I ask, do you have a guess about what the answer might, what an answer might be? One of my thoughts was like an apple would be a good size. I mean, you can hold it in your hand. You could eat the size of the middlest thing. Yeah. When I asked my wife, she said toaster. And then we had like somebody, I think Annie's friend was like definitely a watermelon. He was convinced this would be a watermelon.
The first thing I thought of was an atom. A grain of sand. So we actually put a call out to our listeners to see what they thought the middle thing was. A small rock. A proton. And we got... Maybe Jupiter? A huge variety of answers. The palm of my hand. The sun. But mostly what people talked about was...
How do you even start to try to answer the question? I've been really thinking about this question a lot. Thinking about it more and more. What are the boundaries? I don't know. I don't know. How do you choose the littlest thing? Quarks and atoms. Particles. Neutrinos. How do you choose the biggest thing? Massive black holes. Trillions of stars. Giant supernovas. The Dalit sea. The universe itself. Actually, what even is a thing? Like, I don't know. All the things are still made up of atoms, I think. That's what I think.
And I have to say, I spiraled out in exactly the same way. So much so that I wasn't actually sure if we could answer this question at all. Yeah. I actually weirdly just immediately thought of a particular person. Hi, Soren. Hi, how are you? I can't believe that I feel like the... Steve Strogatz, a mathematician at Cornell, an old friend of the show. Oh, my God.
In the past, we've called him up to help us untangle impossible logic puzzles. And that was fun. Or understand statistics and probabilities. Yeah, time flies. But this time, I just got him into the studio. Well, let's jump in without even telling him what I wanted to talk about. I mean, I do feel a little off as far as our usual thing, because usually I've had something to think about hard before we talk. We're just winging it today. Well, we could see what happens. No, but I mean, I have a very specific thing I want to talk about.
about actually oh oh really yeah well this was sort of the spark here let me just let me just tell you and so i was like let me just hit you with this question okay laura andrews wrote in and said what is the most average i literally read him the text of laura's question where is the midpoint between big and small in relation to literally every single thing ever great
What a great question. It reminds me a lot. He was into it, so I was just like, I don't know, do you think we could just right here on the fly right now rough it out? Huh. Okay. What's in the middle? Yeah. Well, first of all, I did use... Right away, Steve was like, okay, there's a couple things we got to do just to get a grip on this question. I think we should measure everything with a common yardstick. Let's say...
Okay. And a meter is approximately the scale of a person, of Laura herself. Depending how tall she is, she's between one and two meters tall, probably. Right. And also, to simplify a bit and make the math doable, we're going to do some rounding. We don't care about numbers like one or two. We're only interested in plus, you know, up to factors of ten. And in particular, Steve said, like, if we're going to talk about really big and really little numbers, the easiest way to do that is to talk about powers of ten. Right.
Remind me what powers of 10 means? I mean, really, it's just like a mathy way of saying numbers like 10, 100, 1,000. Like you talk about how many zeros come after the one. Okay. So 10 to the 2 has two zeros after the one, which is just like that's 100. And then 10 to the 3, power of 3, has three zeros after, which is just 1,000. Sort of like when people talk about salaries. Are you making a four-figure salary or a five-figure salary? Right. So each step is just going up times 10, 10 times each step. Gotcha. And then you can do this.
In the other direction, like I'm getting smaller, so you just do divided by 10, divided by 10. So if you take 1 and divide it by 10, you get a tenth. That's 10 to the negative 1. Okay.
And then in that case, you're just talking about the number of zeros that are on the other side of the decimal point. Okay. I believe you. So that's what we're going to do. We're going to think about what the biggest and smallest things are using powers of 10. But so back to Laura's question, though. Which immediately took us into some very weird spaces. People say that the smallest conceivable thing, the physicists will tell us nowadays, the smallest conceivable thing is life.
A unit of the size of space at which space is thought to lose its integrity. Integrity? Something called the Planck length. This is a pretty far out thing. No one has experience with this in their daily life. But emptiness, the ordinary space between your hands when you hold your hands apart before you clap them together. Emptiness itself has a fabric to it. And at the scale of the Planck length, space would be made of grains of plastic.
Just dots. Yeah, dots. Kind of pixelated. And what we don't know is are they neat little pixels like checkerboard squares? Or is it that space itself starts to kind of rip apart? We have reason to think that because in quantum theory, everything gets very jittery. Things pop into and out of existence. Okay. Okay.
Yeah. Anyway, the Planck length. That's 10 to the minus 35 meters. Okay. So that's just a decimal point and then 34 zeros and a one. It's about a trillionth of a trillionth of a trillionth? Of a meter. Of a meter. So if we start with a meter, which is roughly a person, we have to zoom in to a freckle on that person's cheek, like 10 to the negative two.
into tiny blood vessels, then a cell in the blood, then the coiled molecules of DNA inside that cell, then down to an atom. Yeah, much smaller than an atom. How big is an atom? Around 10 to the minus 10. Oh, we're not even close. We're not even close. Way, way smaller than that. Apparently, if an atom was the size of the Earth, then the Planck length would be the size of an atom on that Earth. So...
We have to keep going into the tiny bits of the nucleus of the atom, the protons and the neutrons, and then to the smallest fundamental particles that we know of. And it's still like a billionth of that. Wow.
But anyway, that's what we currently think is the smallest conceivable thing. Now, what's the biggest thing? Right. And then we're going to get to what's the middle thing. So for the biggest thing, we have to, of course, zoom back up through protons and neutrons and up to the atom, then out to molecules and dust mites, dolphins, soccer fields, which are like 10 to the 2, oceans, etc.
Earth, about 10 to the 7th. The solar system, then galaxies and clusters of galaxies. And then out, out, out to include all the vast empty spaces between everything. Wow. The size of the whole universe measured from one end to the other. Now, what does that mean?
Okay, we don't know if there's an end to the universe. It's possible the universe itself is spatially infinite, but all we can really observe is how far can light travel since the beginning of the universe. Right. So if you use that estimate, you'd say the universe is something on the order of 14 or so billion light years in diameter. Okay.
Which, okay, now that's not what we were going to do things in meters. Meters, right. So how do you go from light years to meters? I think if I do it right, I think in meters, that's about 10 to the 25th meters. We could quickly ask our cell phones. Okay. You could say, hey, Siri, should I do it? Yeah, sure. All right. Hey, Siri, how big is the diameter of the universe measured in meters? Okay. I found this on the web for What's the Matter? What's the diameter of the universe measured in meters? Check it out.
Oh, she's just going to send you to a web page. She's sending me somewhere. She's like, here's the internet, Steve. Oh, man. Well, all right. I'm going to try using, without asking her, I'm going to type into my phone. Diameter of universe in meters. Okay. This says it's about...
8.8 times 10 to the 26 meters. Wait, so is that 14 billion light years and that's just changing the meters? Because does that come out right? It's 14 billion light years would be... Well, you may want to get a physicist or an astrophysicist because they say that's not actually the diameter of the universe. They're now quoting a number that is much bigger than that, 93 billion light years. And
And they say it has to do with the expansion of the universe at the very beginning in this process called inflation.
Okay. So that's how you get your number, which is 8.8 times 10 to the 26, which I guess with the 8, it's really just 10 to the 27 for our purposes. Yeah, that gave you a 27. I mean, if that's what the smarties are saying. Let's go with that. Let's go with that. So that's a 1 with 27 zeros behind it, which just means that we've taken one meter and times it by 10, 27 times. All right, so we're ready to do it. Yeah, okay. So we just have to take the big and little and figure out what the average is.
thing is. Well, I think we should be careful about the word average because there are lots of kinds of averages. You know, kids are taught median, mean. Can you remind me what those are? Well, the mean, the way you compute it is you add up all the numbers and then divide by how many of them there are. So we need to know how many big or little or medium things there are. Right.
For a median, we'd have to count up all the objects in the universe and then put them in a line from smallest to biggest. And like all the quarks that there's so many of, they'd all be in line. So there'd be a lot of quarks lined up. Yeah, because every big thing is made of little things. So if you add a big thing, you've also added a bunch of little things, I guess. Yeah.
Yeah, so I don't know. It seems like it would drag it to the little stuff. So I'm just saying there's a lot of different concepts of middle, and depending on the context, one is more appropriate or convenient or useful than another. But I sort of, when I hear Laura's question about what's in the middle of the universe—
I think of, this might be an off-putting word, what we would call the geometric mean. So Steve said he thought the most intuitive and simplest thing we could do, because we now had the biggest and smallest numbers as powers of 10, is just figure out what the average of those two numbers is in a way that would tell us from the middle, it would be the same number of times 10s up as it would be like divided by 10s down.
So that's the one that we're going to go for. Got it. Okay. Okay. We're ready to answer now. Go time. Yeah, except we're actually going to take a quick little break. But when we come back, Steve and I actually get to an answer that honestly felt to me, I mean, sort of freaked me out, but also I felt like maybe we had actually landed in the center of everything.
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Hey, I'm Soren Wheeler. I'm Alex Neeson. This is Radiolab, and we're back doing the math my eighth grade algebra teacher always swore we would use. They were right, though. They were right. I don't think they knew what we'd be using it for, but we are here with mathematician Steve Strogas using that math to figure out what the middle-est
sized thing in the universe is all right so we're ready to do it we're ready to answer now so we've got so before the break we had decided the smallest thing you could measure is the teeny tiny beyond comprehension plank length which is 10 to the negative 35 meters and then the biggest thing is the unknowable enormity of the universe itself which is 10 to the 27 meters wide
And to find the middle, Steve actually does this very simple bit of math, almost to the point of being anticlimactic. Okay. Just a good old-fashioned mean of two numbers. So we got 27 on the upside. He just takes the two powers. And negative 35, so I should add them together. Adds them up, 27 plus negative 35. That gives me negative 8. And then because we want the average of just two things, he divides them by 2. Divided by 2 is negative 4. So what?
So I'm saying four below zero is four orders of magnitude below zero is the middle. So that's 10 to the negative four, which is a decimal point and then three zeros? Yeah. So wait, is that, that's a millimeter? That's a tenth of a millimeter. It's a tenth of a millimeter. Yeah. So like a millimeter would be like a grain of sand or? Yeah. So it's a tenth of a grain of sand. It's a very little tiny dusty, a little piece of dust. Dust part of it. Very little piece of dust. Yeah.
Now, if I take a bacterium, say, a cell, you know, that's a single-celled organism, a big bacterium is something like 10 to the minus 5 meters. That's a little small. So— Maybe a particularly large cell might get close to— Yes, maybe. It would probably still be a couple steps. It's a good question. I think a eukaryotic, let's see. That's a cell with a—
A nucleus. The kind of cells we're made of. Right. I'm going to look that one up. Diameter of a eukaryotic cell. Look at that, Soren. It says here diameter of eukaryotic cell 10 to 100 microns. So micron is 10 to the minus 6 meters. And 100 of those is 10 to the minus 4 meters. So the biggest eukaryotic cell is our happy place in the middle.
Hmm. So it's a small bit of us. Yeah. Now, some people would say this is just an exercise in circular reasoning on our part. That's what I keep on wondering. That it's going to come out that, yeah, because of our perceptual limitations, we're going to tend to see things centered on us. Like a perceiving thing of us.
is going to see out in each direction about the same and thus call itself an average? That's sort of plausible, isn't it? Yeah. But I feel a certain amount of confidence in all of this. I don't think it's just anthropocentric. Yeah. I mean, we are using science that stretches our senses as much as we know how to. Yeah. Yeah. I mean, yeah. It still makes you wonder. But if you, but like, what if you like set that aside for just a beat? It,
if you can manage to set that aside. What we have here is that the idea is that the middle-est thing is the most fundamental unit of life. Right. Of complicated life. Yeah. It's a big eukaryotic cell. Which I mean, I think it's kind of, I think that's kind of cool. But, but, you know, I'm not actually sure if
We've answered Laura's question, though, because she was asking about things. What's the middle-sized thing? And what you and Steve are talking about is space, I guess. Yeah. I'm just not sure if the universe counts as a thing. Yeah, I was actually thinking the same thing when I was talking to Steve. Well, this—now, so to be fair, Laura, I think, might have asked a question that we were scared to do.
Okay. All right. Which is fine because I think I like what we did too. But we have now figured out the middle of all measurables or something like that. That's right. She did seem to like, let me just return to the text. If one were to take the size of the largest singular thing, and she says rather than a cluster of stuff like a galaxy. So she really is trying to like, what's the largest thing that you could consider its own object? Right.
That is such a peculiar framing. Because I don't think there—what is a singular thing? Well— Isn't that a fiction? Is anything a singular thing? Aren't we all multitudes? A star is made of electrons. Electrons are made of superstrings. But we do certainly walk around objecting things all the time. And we could say— We do. That's the sun. That's the sun. Yeah. Right. Okay. Well, let's go with it. But a galaxy is not a thing either. Maybe that's like a sort of—
So then we just start trying to figure out, like, what is the biggest thing? But do we call on... Okay, it's in. If you just think of a normal idea of thing, do you know pulsar or black hole? I mean, black holes, like, they have a lot of mass, but they're actually sort of small. Yeah. I sort of think a big star, under her definition, is the biggest thing. Right. You know, like a red giant. Do you have a guess about...
Well, let me just look it up. You can look it up. You've got the whole world right there in front of you. This show is just going to be like, Stephen Soren, Google, what's the biggest single object? Cosmic record holders. Largest exo... No. Largest empty spot. That's weird. Largest star. Yeah. It's called UY Scooty.
Really? I'm sure I'm saying that wrong. Did not know that. So that's 10 to the 12 meters, which, whoa, is apparently so big you could fit almost 5 billion of our suns in it. Okay. So I guess...
For the smallest thing. Currently the smallest physical size. And after a lot more Googling. Particle accelerator. And Googling. Quarks are smaller than that. Turns out the idea of measuring sizes of things that are that small gets really dicey. But we eventually ended up settling on 2,000 times smaller than a proton or five. A rough estimate of the size of like a quark. 10 to the minus 20 meters? Yeah. Yeah, that sounds to me in the right ballpark for Laura's question. So.
So we had our littlest thing thing and our biggest thing. Right. Okay. So the middle. So we've got our same thing, 12 and negative 20, and we add them up and get negative 8. Divide by 2, we get negative 4. It's the same damn answer. We're back to our big, big eukaryotic cell.
That might be the first time I've been spooked in a while. I'm looking at Sorin. He's folding his arms. I had to lean back. He's leaning back. His mouth is hanging open. It seems very odd to me that we got the same. I mean, like, well, okay. I don't know what to make of it either. Sort of. I think it's interesting. Yeah. All right. So the basic unit of complicated life is the middle. Yeah.
That's nice. It's the thing that we have in common with all the life on this planet, whether plant or animal. Yeah, like complex life. Uh-huh. I think this is the answer to Laura. It's smaller than I expected. Does it feel satisfying for the answer to be a cell that's in us? It feels like it should be profound. I mean, I'm not having like a...
A mind-blown kind of reaction like I thought I would. But it feels like I should have more of a reaction to it. What if we were to put some music underneath this? I mean, it'll be more dramatic for sure.
I totally get that. I totally get that. Yeah. But let me see if I can at least offer you this. Okay. Because there's a little bit of a feeling that like, I don't know, being small or average or a little bit whatever in the middle, it makes you feel sort of insignificant or something. But size is actually like a weirdly interesting thing because when something gets bigger, it's not just a bigger version of the same thing. Right.
like when you get bigger and bigger and bigger, the physics, the physical stuff actually works differently. Mostly because...
A really large thing has more volume compared to its surface area. And a small thing has more surface area compared to its volume. That's why like you can get salt or sugar to dissolve in water if it's in little grains. But if you had a big cube of sugar, it would take forever. So there are certain kind of physical events that happen differently if you're small than if you're big. And there's an argument out there that like cells being the basic unit of the way life functions, which has to do with like
making energy and getting out waste and doing all the things that a body needs to do you can't be much bigger than the cells are because you have to have the right amount of surface where you're like sending things out and bringing things in and interacting with the world given like what you've got going on inside so so it might be that this average middle size is actually ideal but
to allow this very rare, precious thing, which is life, to even happen in a cold, cold bit of empty space. Well, when you put it like that. I mean, yeah, when you put it like that, it was more profound for sure. Okay, we're making progress. Yeah. I mean, I guess the most comforting part of it is that we're bigger than we seem. We're maybe not so kind of tiny as we sometimes feel. Yeah. Yeah. Yeah.
Well, thank you, Alex, for sticking with me and bringing me this one and going on the journey with me and not giving up. And thank you to Laura for sending us on the journey. We love getting questions from our listeners. And Laura spent so much time talking to us about the question, about the method to find the answer, and again, about the answer itself. So thank you.
We appreciate you. Thank you. Yeah. Laura is not small to us. For this episode, she was the center of our universe. Yes. Yeah, exactly. This episode was reported by me and Alex Neeson. It was produced by Annie McKeown with mixing help from Arianne Wack.
And I got to say, if you're going to talk about math and space on a podcast, get yourself a Steve Strogatz. Steve, by the way, in addition to being a great mathematician, is also a great writer. And his books on math are gorgeous. And yes, they have math, but they're also easy to read, fun to read, funny and full of humanity.
He also now has a podcast that he does called The Joy of Why, where he talks to some big name scientists of all kinds about their work, but also about their lives. You can find a link to that on our website at Radiolab.org. That's it for us today. Thanks for listening. Lulu and Latif will be back next week. Radiolab was created by Jad Abumrad and is edited by Soren Wheeler.
Lulu Miller and Latif Nasr are our co-hosts. Dylan Keefe is our director of sound design. Our staff includes: With help from Andrew Vinales.
Our fact-checkers are Diane Kelly, Emily Krieger, and Natalie Middleton. Hi, this is Susanna calling from Washington, D.C. Leadership support for Radiolab's science programming is provided by the Gordon and Betty Moore Foundation, Science Sandbox, a Simons Foundation initiative, and the John Templeton Foundation. Foundational support for Radiolab was provided by the Alfred P. Sloan Foundation. ♪
