Regular Polyhedra and Spheres
Some of the most specialized geometric surfaces are the regular polyhedra. In the special cases we've studied so far, the base or bases of a geometric surface is a special shape. In a regular polyhedron, all of the polygons that compose the polyhedron are special: they are all congruent regular polygons. Only five regular polyhedra exist. Their names and number of faces are as follows:
- A tetrahedron has four faces.
- A cube has six faces.
- An octahedron has eight faces.
- A dodecahedron has 12 faces.
- An isocahedron has 20 faces.
Another very specific geometric surface is the sphere. A sphere consists of all the points that are equidistant from a given fixed point in space. This fixed point is the center of the sphere; a segment with one endpoint at the center and one on the sphere is a radius. A sphere is basically like a three-dimensional circle. In a way, it is also like a regular polyhedron with an infinite number of faces, such that the area of each face approaches zero. This limit, however, does not exist because the set of regular polyhedra is finite--a regular polyhedron cannot have more than 20 faces.
Just as a semicircle is a 180 degree arc, or half a circle, a hemisphere is half a sphere. A hemisphere is drawn below.