ALL ABOUT THE

REGULAR POLYHEDRA

©Christina Chang

A polyhedron is formed by enclosing a portion of 3-dimensional space with 4 or more plane polygons.

For example, a triangle is a polygon.

A tetrahedron is a polyhedron with 4 triangles as its faces.

Regular polyhedra are uniform and have faces of all of one kind of congruent regular polygon. There are five regular polyhedra. The regular polyhedra were an important part of Plato's natural philosophy, and thus have come to be called the Platonic Solids.

It's not hard to see that the cube is the simplest one to deal with. But when it comes to more complicated ones like the Icosahedron or the Dodecahedron. How do you find their volume, length or edges, etc.?

Cube	Tetrahedron	Octahedron	Dodecahedron	Icosahedron

These are the ONLY regular polyhedra. Proof:
Assume there exists a {p, q} regular convex polyhedron. Since every face has p edges there would be a total of pF edges in all except that every edge is shared by two faces. Therefore pF = 2E. On the other hand, q edges meet at every vertex. Since each edge connects two vertices, qV = 2E. Substituting this into the Euler's formula gives:
2E/p + 2E/q - E = 2 or 1/p + 1/q = 1/2 + 1/E
First of all, p3 and q3 since a polygon must have at least three vertices and three sides. p and q can't simultaneously be both greater than 3 because then the left hand side will be at most
1/4 + 1/4 = 1/2 < 1/2 + 1/E. Therefore, either p = 3 or q = 3.
If p = 3, 1/q - 1/6 = 1/E So that q can only be 3, 4 or 5. Solving the equation for E yields E equal 6, 12, or 30, respectively.
Similarly, if r = 3 q can only be 3, 4 or 5 with E equal 6, 12 and 30 respectively. Then it all comes to five possible pairs. All five actually represent realizable shapes.

Symbol	F	E	V	Name
{3,3}	4	6	4	Tetrahedron
{3,4}	8	12	6	Octahedron
{4,3}	6	12	8	Cube
{3,5}	20	30	12	Icosahedron
{5,3}	12	30	20	Dodecahedron