Platonic Solids
Platonic solids
Alicia Armstrong + CJ Harrison
Personal Connection: Platonic solids
I really liked this project because it was fun to learn about how platonic solids relate to the different elements of the world. I also liked how fun it was to make the platonic solids mainly because we didn't have to take a long time to get the materials, we just had to get pizza boxes and it was a simple cutting out the shapes. In my group I think we really work well together because were both neat freaks so it kinda evened out in the end so it was nice to get things finished neatly. One thing I learned was that platonic solids have to be less that 360°. I could go on about different things but I think overall we like math a lot because there are so many different things you can do to make it fun but learn at the same time.
Math connection: Platonic solids
Ancient Greeks studied platonic solids and came to the conclusion that only five platonic solids can be made. These are the tetrahedron,cube or regular hexahedron, octahedron, dodecahedron, and the icosahedron. Geometric proof
The following geometric argument is very similar to the one given by Euclid in the Elements:
1. Each vertex of the solid must coincide with one vertex each of at least three faces.
2. At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be less than 360°.
3. The angles at all verticals of all faces of a Platonic solid are identical, so each vertex of each face must contribute less than 360°/3 = 120°.
4. Regular polygons of six or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. And for:
* Triangular faces: each vertex of a regular triangle is 60°, so a shape may have 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
* Square faces: each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.
* Pentagonal faces: each vertex is 108°; again, only one arrangement, of three faces at a vertex is possible, the dodecahedron.