Man, the static cling is ridiculous. (Off Camera) Yeah. Hey Crazies. You’ve probably heard before that you can fit 1.3 million Earths inside the Sun. It’s an easy enough calculation. Just take the volume of the Sun and divide it by the volume of the Earth. But that ignores a very important aspect of geometry: The Earth is a sphere and spheres aren’t space filling. There will be gaps. As a result, the number of Earths inside the volume of the Sun will be less than 1.3 million. But by how much? Let’s do an experiment and find out. Technically, there wouldn’t be gaps. That much mass would collapse under its own weight. Sure. A giant ball of 1.3 million Earths will have almost four times the mass of the Sun. It will collapse under its own gravity, turning into a giant ball of plasma. But only temporarily. There isn’t anywhere near enough hydrogen for sustained fusion. A bunch of the material will be blasted away into space from the sudden temperature change. What remains will collapse into a metal-heavy white dwarf with the mass of a star and the size of a planet. But, given the spirit of the original question, I think it’s safe to ignore that. Ok. [Title Screen: How do sphere's pack?] As I mentioned earlier, spheres don’t fill a volume. They leave gaps. How big those gaps are will depend on how the spheres are distributed. Let’s say we take a bunch of ping pong balls and just randomly pour them into a box. Only 88 balls fit in the box. Comparing the volume of the balls to the volume of the box, we get about 41.8% density, which means 58.2% of the box is still empty. That’s not very dense. Can we do better if we place them in the box carefully? Oh yeah and, believe it or not, that question is over 400 years old. To the timeline! Back in 1587, Sir Walter Raleigh wanted to know the best way to stack cannonballs on his ships. Who better to answer than Thomas Harriot, his personal mathematician? Rich people and their mathematicians, am I right? Anyway, the best way of stacking balls depends on the shape of the base and we have a few obvious options. First is the ever-popular square base. Each level will fit inside the corner gaps of the level underneath. Ah! Ok, let’s try that again. Level 1, check! Level 2… Steady… Don't you dare… [Censored]! Ok, one more try. Third time’s the charm. [Censored]! [Censored] it. Never mind. Let’s do this with C.G. With a square base, you get a standard pyramid. This one has a 3-by-3 square, then a 2-by-2 square, then one more for a total of 14 balls. You could do a triangle base instead, which gives us a tetrahedron. You could even do a hexagon base if you wanted. We’ve got lots of options. The best way to stack cannonballs is the one where they take up the least amount of space. So which shape is best? They’re all the same. What?! I mean, look closer at this hexagon stack. Boom! Square base. It’s just in a different orientation. Look closer at the triangle stack. Bam! Hexagon base. Well, at least most of it. They’re all the same, which means they all have the same packing density. So, Tom comes back to Sir Walter and says: “It doesn’t matter how you stack your cannonballs.” Shouldn’t you have done that in an English accent? Actually, back in 1587, the British accent was pretty close to a modern American accent. Where do you think the American accent came from? (Off Camera) The words would have been different though. Only a little! They were already speaking early modern English. Where was I? Right! Tom and Walter. Tom figured out that all the standard stacking techniques work equally well, but he didn’t prove they were the best way. In 1611, Johannes Kepler conjectured it was the best, but he didn’t prove it either. It wasn’t actually proven to be the best technique until 1998 by Dr. Thomas Hales while he was a professor at the University of Michigan. Oddly enough, that proof wasn’t 100% accepted until 2017. But this is about more than just cannonball stacking. Filling a volume with spears … Spears. Filling a volume with spears would be dangerous. Filling a volume with spheres is essentially what atoms do in solids. How atoms are packed can give you an understanding of why solids have the densities they do. So what’s the best packing density for spheres then? It’s about 74%. I’m going to explain how I got that. If you take the pattern and you slice it into unit cubes, you get something like this. There are six halves in the middle sections and eight eighths around the corners. That means we have four spheres all together. Some quick geometry on the cube's faces gives us the total volume of the cube. The volume of the four balls divided by the volume of the cube gives us a density of 74.05%. Matt Parker and Steve Mould did this same calculation using oranges back in 2017. If you haven’t seen it, it’s hilarious. So that’s it, right? 74.05% times 1.3 million Earths equals 962,624 Earths inside the Sun. Boom. (Off Camera) Wait a minute! Doesn’t that divide some of the Earths into eighths and halves? Ok, fine. I guess if we’re going to do this, let’s do it right. Stacking a bunch of these unit cubes together, you do build up a lattice of whole spheres. But the outside edges are still cut up. We don’t get to keep those in real life. That means the number of whole spheres in a box is necessarily less. This 74.05% is just an upper bound. It’s not the actual value. Real life ping pong balls, in a real life box, aren’t going to pack with that kind of density. Using this kind of packing with our original box, we’d be able to fit 107 balls in there, giving us a density of 50.8%. That’s definitely more than we got with a random pour, but it’s a far cry from 74%. Even with a box that’s perfectly-sized for these balls, the packing density is only 58%. Now, with a box twice the size, we can get it up over 65%. It depends on how the balls compare to the box. You can get pretty close to 74% if your balls are really small or the box is really big. On top of all that, the Sun isn’t a box. It’s a sphere. And that changes things. [Title Screen: How do sphere's pack inside sphere?] How sphere’s distribute inside another sphere isn’t necessarily going to be the nice lattice we got with the box. If we do a random pour again, we can start to see the problem. We can actually get 13 ping pong balls inside this hamster ball. For comparison, here are a couple different boxes perfectly-sized for 13 balls, and they both have a bigger volume than the hamster ball, which means the packing is better in the hamster ball. Which makes sense, I guess. This might maximize the density in this specific container, but the pattern is different in other spherical containers. Scaling this up is impractical, so let's do a physical model instead. My 120 pack of ping pong balls was $20, so that’s about 17 cents per ball. Multiply that by the number of Earths and we get $160,000?! Yeah, that’s not going to happen. I’m not on Mr. Beast’s level. Maybe if I scale it down a little more and buy tiny precision beads. Yeah, let’s check on this. How much would it cost for one million precision-molded one millimeter beads? Alright, let’s check on this. [Escalating Dramatic Music] Never mind. Never mind. Yeah, I still can’t justify it. It’s just too much for one video. But maybe one day. If you’d like to help out, I have a Patreon and my YouTube memberships are open. Pledging support either way gets you access to a private monthly livestream. In the meantime, we’ll have to settle for some code and maybe a model that isn’t to scale. First, it’s coding time! We’ll put in some standard header stuff. Hashtag Z Up for Life! Define the container and bead size. Give it some texture. We’ll want to procedurally draw the beads though. Let’s assume that as the container sphere gets larger, whatever pattern exists approaches the ideal box pattern from earlier. It's not perfect. This one is off by a couple balls, but there are 1122 balls in here. That's only an error of a tenth of a percent. It’ll be fine. Trust me. Anyway, the X and Y directions step forward by the bead diameter or 2 R, but the layers should sit inside each other, so the Z direction only steps up by root 2 R. We’ll need some code to shift the layers back and forth as Z increases. A counter for the beads would be good. Next, we’ll display the beads and then display our results. And go! Oops. It got stuck. This is probably way too many objects for the computer to handle. I suppose we could count them all, while only displaying the first few layers. Yeah, we’ll just add some display conditions. And go! Holy moly! Look at this thing! Dude, that’s so many Earths. Anyway, looks like we get 932,884 Earths for a packing density of 72.03%. Not bad. It’s less than the upper bound, but pretty close. Didn’t you say something about a model? Oh, right! I almost forgot. I managed to find a million tiny beads for a few hundred dollars. Packed with love! Eh, with some size variation. These are going to be perfect. I also found a container that’s roughly the right size. The scale is 100 to 1 instead of 109 to 1, but so be it. This will do. We can at least check my code with it. Now let’s fill it up and see what happens. First, we need a hole. [Censored]. It’s cracking. Ah! (Off Camera) You alright? Yep, it’s just freaking me out. (Off Camera) There you go. Man, the static cling is ridiculous! (Off Camera) Yeah. [Time-Lapse Music] The worst part of this is, when we finally get to that line, we’re halfway. (Off Camera) I know. [Time-Lapse Music] Yeah, ain’t no more getting in there. You want to give us a thumbs up? [Reveal Music] Now let’s calculate how many beads are inside. So 748 divided by the number of grams per bead should give us the number of beads in that ball. 397,375. That’s obviously way too small. When you get a result, always ask yourself if it makes sense. This does not make sense. Let’s double check it a different way. We bought about a million, right? Right. And how many did we use up? How many vials? Yeah. I’m just wondering if your calculation … Oh right, they're all over here. 49. 49 divided by 122 is right around the number of beads. That’s what I was wondering. What is going on?! I hate it when I don’t understand something. Let’s see what my code has to say about this. Just need to change a few numbers at the beginning. And go! It says I should get 718,394 beads with a packing density of 71.8%. That makes more sense. What could be wrong with the scale model? Oh... I think I figured it out. It has to be the manufacturer’s estimates. They say there are 8200 to 9000 beads per vial. I bet that’s wrong. The density of standard bead glass is about 2 grams per cubic centimeter. That means the mass per bead is a little over a thousandth of a gram. Divide the mass of all the beads by that number and we get 714,287 beads in the container, which is a lot closer to my code. There we go. Fixed. I guess that’s what I get for going cheap. So how many Earths fit inside the Sun? Well, according to my code that was double-checked with a real-life model: 932,884. The quick calculation says we should be able to get 1.3 million in there. But, when spheres pack, they leave gaps. That means some of the space isn’t filled. The real number of Earths will be necessarily less than 1.3 million. In fact, the math of sphere packing tells us the upper bound on density is 74.05%. That's means the number of Earths should actually be less than 962,624. To figure out exactly how much less, I wrote some code and I double-checked that code against a real life model. Those numbers are close enough, so it's safe to assume my code works. Using it on the Earth Sun problem gives us a total of 932,884. So there you have it. Less than a million Earths fit inside the Sun. The new question on my mind: What the heck am I supposed to do with all these extra beads? So have you ever thought this deeply about a simple question before? Please share in the comments. Thanks for liking and sharing this video. A special thanks goes out to all my generous supporters on Patreon and YouTube. Don’t forget to subscribe if you’d like to keep up with us. And until next time, remember: It’s ok to be a little crazy. What’s the mechanism that converts spacetime curvature into Hawking radiation? Well, we don't actually know. Using general relativity and quantum field theory at the same time can be a bit tricky. Hawking managed it using something called a Bogoliubov transformation, which is a non-local math tool. That means we don’t have a local mechanism for Hawking radiation. Yet. Anyway, thanks for watching.
