14.4 - Regular Polyhedra
Links and useful resources
- gr10's Geometry >>>START HERE<<<
- AoPS Online Textbook
- AoPS Alcumus
- Big Ideas Geometry textbook
- GeoGebra Online Geometry Constuction Tool
- Two-column math templates
- Proofs unit slides from mathgiraffe.com
- Proofs unit printables for two-column proofs
Regular Polyhedra
- The Platonic solids
- polyhedron
- Why aren't there any polyhedra with regular hexagonal faces?
You might think that you could just produce a regular polyhedron with faces of any kind of regular polygon. However, there's a problem! To see what it is, we have to think about angles and corners of a polyhedron.
To make a polyhedron, you have to be able to arrange the polygons around a single point that will become a vertex in the final shape. That means that when the polygons have their edges coincident, their end points all connect at the same place. That in turn means that the sum of the interior angles of all of the polygon corners you're using has to be less than 360, otherwise they won't be able to "bend outward" to make a vertex.
here's an example of a dodecahedron:
If you try to do that with a hexagon, you'll find out that hexagons tile together into a plane! Any any polygon with more than six sides has interior angles are too large to allow the vertices to meet in a single point no matter how you arrange them! So that means... There are only five regular polyhedra!
How weird is it that there would be some arbitrary finite number of these things? Sometimes reality is surprising. You'll recognize what they are from D&D:
- tetrahedron (4 faces, equilateral triangles, a.k.a d4)
- cube (6 faces, squares, a.k.a d6)
- octahedron (8 faces, equilateral triangles, a.k.a. d8)
- dodecahedron (12 faces, pentagons, a.k.a. d12 -- the best die type)
- icosahedron (20 faces, equilateral triangles, a.k.a. d20)