Today let's clothe our minds with knowledge on Michael's toys We're gonna be talking about this shirt. I designed this shirt along with the brilliant designer John laser It depicts Kepler's model of our solar system what he called the mysterium cosmografica If you want this shirt or any of the cool science and math toys that Jake Kevin, and I love the most There's always the Vsauce curiosity box a quarterly shipment of our favorite math and science stuff right to your door this shirt Comes in the latest box that are still some left But to fully understand why this shirt matters we need to first talk about regular polytopes A polytope is a shape with straight or flat sides. In two dimensions, we call them polygons in three dimensions We call them polyhedra, but polytope is the general term that encompasses all of them. A polytope is regular if all of its elements are alike. For instance side lengths, angles vertices, faces cells. In two dimensions there are an Infinite number of regular polytopes you can just keep adding sides: equilateral triangle, square, regular pentagon, regular hexagon, regular heptagon, regular octagon, Regular tillyagon, regular megagonn. There's no end But in three dimensions there are only five Those five are very special. They are called the Platonic solids. They're named after Plato who hypothesized that these five three-dimensional regular polytopes must be what make up the elements of our universe. A regular three-dimensional polytope must be constructed out of regular Polygons as faces, and at every Vertice there must be the same meeting of faces. In the case of a cube, we have three squares that meat at every vertice. So it is nice and regular. It's a platonic solid. But, to see why there are only five possible in three dimensions. Let's start building. We'll begin with the least sided regular polygon, the equilateral triangle. Now, if I'm going to build a shape in dimensions out of equilateral triangles, I'll notice that I've got a limitation. I need to make sure that where all of their vertices meet I haven't covered more than 360 degrees. If I do that there won't be room for them to come together and meet in three dimensions. Oh look at that! I've already started building the first platonic solid, the tetrahedron. Plato believed that the tetrahedron must be what fire is made out of because tetrahedra are... sharp? And fire is sharp I guess?Anyway, that's what Plato thought. Now, you might be wondering Why can't you just take three more equilateral triangles, connect them all like this and call this a platonic solid? You can't because the vertices aren't all exactly the same. Here three triangles meet. But here, and here, and here, four meet. Pfft. Don't waste my time with that. (Get out of here). Oh, platonic. Okay, now with these three equilateral triangles, we still have room. Um, I can put another Triangle right here, and I still don't have a full 360 in the middle. I still have room for this shape to fold up into the third dimension to fold it. I'm gonna tape it first Okay now that it's taped I can fold it together. Beautiful! What I have begun to build is the second platonic solid the octahedron. Here we have eight faces. Octahedron. Plato figured that air must be made of octahedra. Because it's not quite as sharp as fire? Anyway, let's move on because we still have room for more triangles. I can fit another one in and I've still got room for three-dimensional folding. I will tape this fifth Equilateral triangle in place fold the whole thing up into three dimensions. Oh I now have the beginnings of the third platonic solid, The icosahedron. This beautiful 20 sided shape is quite wonderful Because it is so round Plato believed that the icosahedron must be what makes up water. Because water is Roley and rolls and falls out of your hands? Look point is we can't go any further. If I take a sixth equilateral triangle and put it in I now have a full 360 degrees occupied here in the middle. There are no gaps left for this shape to fold up on itself without there being some Overlap. Nope not good. We can't move forward So we will skip the triangles and move right on to a four-sided shape, the square. With squares I can build a shape like this. Sure, I don't have any more than 360 degrees taken up in the middle. And I can fold the squares up into three dimensions. aha! And I am beginning to build the fourth platonic solid, the cube. The cube to Plato must be what made up earth because it can be stacked and Balanced nice and rigidly with itself. The cube is also The only platonic solid that can tessellate Euclidean space completely Which perhaps earthed it if it didn't have a beginning or end. But that's it. That's all I can do with squares if I brought in a fourth square It would go right there And then I'd have 360 degrees filled here in the middle and there wouldn't be room For the shape to be folded into three dimensions, so let's move on to five sides the Pentagon. When it comes to Pentagon's let's see how many I can use as faces. Well if I have three, I've got myself ah I've got myself a problem. When three meet I have this tiny little triangle left. There's not enough room for a fourth to go in so three is the most I can use the most that can meet at one vertice if I fold them up until they meet oh Yes, I have begun to build the fifth and final platonic solid, the dodecahedron. This wonderful 12 sided shape Didn't really have a place in the classical elements. Because we've already covered fire, air water, and earth. Plato said that, perhaps the dodecahedron was used by the gods to form the constellations. Aristotle said that maybe it was what made up the ether. He was wrong. But, these shapes are still, incredible, The Platonic solids. These are the only regular polytopes possible in three dimensions. We can't keep adding sides to our faces because, well, I'll show you why. Let's move on from the Pentagon to a six-sided shape, the hexagon. If I try to put hexagons together, I'll notice that Oh crud. I got a problem! They have to be completely flat to all need up because this is a complete 360 degrees. If I try to Fold these up into three dimensions they start overlapping, each other. We cannot have more than five sides on a face for a shape to be a regular polytope in three dimensions. So these are the only five we have. But let's now Fast-forward to this shirt. Well actually not that far. Let's go back few hundred years from when this shirt was created to Kepler. Kepler was born at a time when the Sun was not believed to be the center of our solar system. Clearly, the earth was the middle of everything he was the most important planet! But Kepler put forward a beautiful argument that perhaps the Sun was in the middle, and the Platonic solids could explain everything. He found that if you Inscribed a sphere, inside an octahedron, Circumscribed a sphere around that, placed the whole thing inside an icosahedron, Circumscribed a sphere around that, put the whole thing inside a dodecahedron, Threw in another sphere put the whole thing in a tetrahedron, another sphere, and then put the whole thing in a cube, surrounded by yet another sphere, you would have yourself, six spheres. At the time there were six known planets, and the ratios between the radii of these six spheres, matched the distances of those six known planets from the Sun: Mercury, Venus, Earth, Mars, Jupiter and Saturn. All explained with the beauty of regular polytopes Now, modern measurements have shown that this model Isn't how the solar system works. But, Kepler was able to make a convincing argument that a heliocentric universe was beautiful, that it was harmonic, that it made sense, and he made a fundamental step in this model of marrying physics and math to the real world into observations that had ginormous consequences on scientific reasoning down the road if you try to look for a Model or a diagram of Kepler's mysterium cosmagraphicum, you will mainly just be confused. Most of what's out there have Detail that's really hard to decipher or they're just literally incorrect That's why I knew that this shirt had to be born. It comes in the latest curiosity box which is full of Awesome science and math toys curated by myself, Jake, and Kevin. This comes to your house four times a year and a portion of all The proceeds from the Curiosity Box goes towards Alzheimer's research. This is the part of the box That's the most important to Jake Kevin and I. Alzheimer's has affected people that we have loved. So the box isn't just good for your brain, it's good for, well, everyone's brain. This is actually, I actually use this in one of my previous videos. I won't give you too many details Otherwise it wouldn't be a curiosity box. It would be a I know what's in it box, he am I right? Okay, look, so there's a steam game, there's... I'm not gonna tell you too much, but the t-shirt that comes in the current box, we still have some left if you subscribe soon, is the wonderful Mysterium Cosmographicum. By wearing this shirt, you turn your body into a walking monument to the marriage of physics and math to the universe we find ourselves in. And as always, Thanks for watching
I just noticed I typed Astrology instead of Astronomy. FFS.
Sorry about that.
Great video. Now I need a Curiosity Box.