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Regular Polyhedra

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.

A few of these regular polyhedra are drawn below.

Figure %: The regular polyhedra

The tetrahedron, octahedron, and icosahedron are composed of congruent
triangles. The cube is composed of congruent
squares, and the dodecahedron is
composed of regular pentagons.

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Spheres

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.

Figure %: A hemisphere

Spheres are difficult to represent on a two-dimensional computer screen, so to
try to visualize a sphere, it may be best to study the hemisphere figure and
imagine two hemispheres joined together. There are also countless examples of
spheres or near-spheres in real life. Basketballs and bowling balls are
spherical. So are Earth and the other planets in this solar system. Luckily
for geometry students, the terms in which spheres are defined and the rules by
which spheres are governed are simple.