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  • 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

Today let's clothe our minds with knowledge on Michael's toys

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ミステリウムコスモグラフィックム (Mysterium Cosmographicum)

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    林宜悉 に公開 2021 年 01 月 14 日
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