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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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