Rachel Feltman: For Scientific AmericanIt is Science quickly, My name is Rachel Feltman.
If you like math, you probably already subscribe to Scientific AmericanThe weekly Proof Positive newsletter. But if you feel like you don’t do it I love math, proof positive can prove you wrong.
Here to give us a taste of some of the surprising and delightful stories you’ll find in Proof Positive is Manon Bischoff. Manon is a theoretical physicist and editor at spectrum of science, the German-language sister publication of Scientific American.
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Thank you very much for coming to chat with us today.
Manon Bischoff: Thank you for inviting me.
Feltman: So one of the things you cover in your newsletter is the impact of mathematics on our daily lives. A recent example is that mathematicians have figured out why waiting for the elevator can seem to take forever, which is very relevant to my life: my building has two elevators, and one of them is currently out of service. [Laughs.] So can you tell us more about how this experiment works?
Bischoff: Yeah, so you just described it: you press the elevator button, and you’re hoping to go down or up or whatever, and the first elevator that comes along, it’s going in the wrong direction, right?
Feltman: Yeah.
Bischoff: And it almost feels personal, like the building is plotting against you. [Laughs.] I know this feeling. [Laughs.] But in reality, it’s not just bad luck or Murphy’s Law; this is actually happening – the building is actually plotting against you. [Laughs.] And it’s for mathematical reasons.
This phenomenon was studied by physicists in the 1950s, by George Gamow and Marvin Stern, and they worked in the same building but on different floors, and they only had one elevator, like you do today. And they noticed that the elevator that arrived first was going in the wrong direction.
And Gamow even tracked it, so much so that he noticed that five times out of six, the first elevator went in the opposite direction. And he talked about it with his colleague, and they started thinking about it. And it seems paradoxical because the elevator goes up and down so often, right? So why does it go wrong most of the time for you?
And there is a fairly simple explanation for this. So imagine you’re right at the top of a building, or near the top, and every elevator that reaches you must first come from below, and then shortly after, it comes back down. So on your floor there [is] just a tiny moment when the elevator goes down, and there’s a much longer period of time when it first goes up. And if you get to the elevator at a random time, it’s much more likely that you’ll catch it on the way up and not on the way down. And that’s the explanation.
Feltman: Wow, this is so interesting. How did you discover this study?
Bischoff: I was just doing some research, and then I read this study that Gamow and Stern were doing and… or it’s, like, a little report, and they were, like, making jokes about it. And it was really fun to read it, and [I] I wrote it down, and while… yeah, you can make, like, a little diagram, and then you really notice, “Ah, it makes sense, actually, that he’s plotting against you.” [Laughs.]
Feltman: Another daily math example that you had in the newsletter was that, you know, math can help us live more deliciously. What have mathematicians said about the optimal cutting of a pizza? You know, I would say that any pizza that you cut is optimally cut pizza because you can eat pizza…
Bischoff: [Laughs.]
Feltman: But mathematically, what is the answer?
Bischoff: If we’re sharing a pizza, we’d both like to have the same amount, and we’d also like not only to have the same amount of dough, but also the same amount of topping. So where I get my pizza, they don’t put the topping evenly, but usually it’s just crumbled in one place, and the rest is, like, bare. [Laughs.] And if we split it and you got all the pepperoni, for example, I’d be a little mad at you, I guess. [Laughs.]
So mathematicians were wondering, “Okay, so how can we divide it evenly so that not only is the dough the same amount on both sides, but also the filling?” » And they understood that there was actually always a way to do it fairly.
So if you think about it, you would naturally do it: cut it in half in the middle of the pizza, right? But if you [don’t] cut it straight, but just rotate your knife, then the amount of filling varies on both sides. So on one half, there’s, let’s say, more pepperoni on the right half, and if you rotate it slightly, then you’ll have less pepperoni on one half and a little more on the other half. And the amount of filling changes smoothly and not just abruptly. This is the important point mathematically. So you can really show that there’s always a point while you’re rotating your knife where there’s the same amount of filling on both sides.
Feltman: So maybe the takeaway for regular people who aren’t, you know, sitting there with a bunch of graphics tools is to just keep spinning and observe it and know that there’s a fair solution and you just have to feel it. [Laughs.]
Bischoff: Exactly. I mean, it’s kind of mean because the mathematicians just proved that there’s a solution, but they didn’t tell you how to get it – I mean, you rotate it, but they didn’t tell you how to find the best angle. [Laughs.] So you have to figure it out for yourself, and that may take some time. But yes, there is a fair solution. [Laughs.]
Feltman: Well, that’s a reason to study mathematics, I suppose, if people need inspiration. And mathematicians also thought about cutting ham sandwiches, right?
Bischoff: You may have noticed: I like to associate mathematics with food. [Laughs.] I really like this connection. Mathematicians therefore like to generalize things. So when they did this pizza theorem, it was like a 2D version of the theorem. So you had a two-dimensional disk, which was the pizza, and you had two objects, so it was the pizza dough and the topping, and you wanted to cut it evenly.
And then they thought, “What happens if you go to three dimensions and with three objects instead of two?” “So they looked at this ham sandwich, and you had a slice of bread, a slice of ham and another slice of bread. And there, once again, the person who prepares the sandwich doesn’t pay too much attention, and it’s not just [Laughs] superimposed finely, just on top of each other, but spread out a little.
And if we want to share this sandwich fairly, we would then have to find the perfect cut, which cuts everything in two: therefore the upper part of the bread, the ham and the lower part of the bread. And mathematicians could show that, similar to pizza, if you make a cut and just vary the angles continuously and smoothly, then you will always find a cut that is perfect and will just split the sandwich evenly.
Feltman: Besides food, what are some of your favorite practical applications of math?
Bischoff: Yes, this is a difficult question, because mathematics turns out to be everywhere in our lives. Of course you can describe everything mathematically. But I have a fact that I Really as with everyday mathematics. He has to shuffle the cards. So I don’t know if you like card games.
Feltman: Mm, yeah.
Bischoff: And every time you shuffle a deck of cards, you’re actually writing the story – I don’t know if you know that.
Feltman: No. [Laughs.]
Bischoff: [Laughs.] So that means that if you take a deck of 52 cards and shuffle them carefully – so that you shuffle them really well – then it’s almost certain that you’ve created an arrangement of cards that no human on Earth has ever created before.
Feltman: Oh, wow. [Laughs.]
Bischoff: [Laughs.] It’s amazing, isn’t it? SO …
Feltman: Yeah.
Bischoff: And the reason is that the number of all possible [arrangements] is simply huge. So if you have 52 cards, the number of arrangements is 52!, or 52 x 51 x 50 x 49, and so on, up to 2 x 1. And it’s a number, I won’t read it because it would be necessary [Laughs] way too time consuming and it’s quite annoying, but it’s a 68 digit number.
Feltman: Wow, that’s a really fun thing to think about.
Bischoff: Yes, and if you create just one example of this 68-digit number, you can guess that the probability that another human being has created the same arrangement is so low that you probably just created [for] this is the first time this arrangement has existed on Earth.
Feltman: What would you like people to know about mathematics? What do you think they misunderstood about this?
Bischoff: I think a big misunderstanding is that you have to be really smart or, like, a genius to understand or like math, and I think that’s completely false. So, as long as you are interested – and there are so many interesting stories about mathematics or facts about mathematics that anyone can be fascinated by them.
Plus, this great myth that everyone has – like mathematicians are unbeatable and never make mistakes – is completely false. So one of my favorite stories is about Alexander Grothendieck. He was one of the most influential mathematicians of the 20th century, and he did, for example, Really complicated stuff.
But a colleague once asked him: “Here, Alex, tell me a prime number, please,” meaning a number that is simply divisible by 1 and itself. And he said 57, which looks like a prime number, but it’s not. [Laughs.] So it’s divisible by 3. And I mean, it’s so easy to check that 57 is not a prime number, and that shows you that even a genius like Grothendieck [Laughs] I could be wrong about such simple things. This shows that mathematics is all about ideas and not just calculations.
Feltman: Well, thank you very much for coming to share these fun math stories with us, and I’m sure our listeners will enjoy reading more in your newsletter.
Bischoff: I hope so. [Laughs.] Thank you for inviting me.
Feltman: That’s all for today’s episode. Check out Proof Positive for more surprising math stories s. You can also subscribe to SciAm newsletters focused on health, space, parenting and more. Go to ScientificAmerican.com/newsletters to subscribe.
We’ll be back Friday to learn more about the multi-year international effort to rename the disease formerly known as PCOS.
Science quickly is produced by me, Rachel Feltman, with Fonda Mwangi, Sushmita Pathak and Jeff DelViscio. This episode was edited by Alex Sugiura. Shayna Posses and Aaron Shattuck check in on our show. Our theme music was composed by Dominic Smith. Subscribe to Scientific American for more recent and in-depth scientific news.
For Scientific American, This is Rachel Feltman. See you next time!