Only 5 Platonic Solids

Why are there only five platonic solids? Well, in this video, I’m going to answer that question. The answer in brief is because only five are possible. This was pointed out by Euclid. In book 13, proposition 18 of the elements, which dates from 300 BC. If we start with a point on the plane, I want you to pretend that this point is one of the vertices of one of our platonic solids. And I’m going to add some regular polygons to that point. And I’ll start with equal lateral triangles. And I’ll add three of three by 60 degrees is only 180 degrees. So from this point I can form a solid. And that’s so what is the tetrahedron again, I’ll start with a point on the plane. And now I add four equal lateral triangles. And that’s 240 degrees, which is less than 360. I can form a solid and that’s solid is the octahedron. I start with a point on the plane. This time I add five equal lateral triangles for total 300 degrees, I can form a solid, and that’s solid is the icosahedron. Next I’ll try six equal lateral triangles. But now I find that equals 360 degrees. That’s a flat shape. No solid can be formed. So I’ve exhausted triangles, I think I’ll try squares. I’ll start with a point on the plane and add three squares. From here I can form a solid and that is a cube. I’ll try four squares then. But again, I’ve got 360 degrees. A dead flat shape. No solid can be formed from this point. Next I’ll try regular Pentagon’s. I’ll start with a point on the plane. And I’ll add three. Now I know from my formula insides, minus two by 190 equals the total interior angle sum of a polygon. And then I just divide by the number of vertices, in this case five. And I know each vertex is 108 degrees, three times 109 is 324. Less than 360 a solid is possible. And that’s all it is the dodecahedron Next, I’ll try for Regular Pentagon’s, but you can guess it’s way over 360 degrees it’s 432 degrees. And here no solid is possible. Well next time I try hexagons start from a point on the plane, add three hexagons. But of course, each angle of a hexagon interior angle is 120. So that makes 360. It’s a flat shape. No solid is possible. Seven sided polygons are no good. I’ve exhausted all the possibilities. And my conclusion is that only five platonic solids are possible. We have three equal lateral triangles of the vertex, which is a tetrahedron. For a collateral triangles, an octahedron, Avi collateral triangles, an icosahedron. three squares is a cube. And three regular Pentagon’s at each vertex will give me a dodecahedron. Next time we get around to making our model now you’ll have to download and print out the worksheets and you’ll need scissors and sticky tape to put them together. I’ll see you then.