• The faces are congruent regular polygons.

• The same number of faces meet at each vertex.

• Platonic solids exhibit rotational symmetry

• Platonic Solids are convex.

• If you put a dot in the center of each face of a Platonic solid, and connect adjacent dots with line segments, you create another Platonic solid. This new Platonic solid is called the dual of the original(see "dual" section below for more details). If you form the dual of this dual, you get a third Platonic solid that is similar to the original.

Despite how many complicated patterns can be found in Platonic solids, however, the definition is pretty simple. Any polyhedron in which:

1) The faces are congruent regular polygons

and

2) The same number of faces meet at each vextex

is called a Platonic solid. Any such polyhedron must be similar to one of the five polyhedra shown in the picture, and it therefore must exhibit all the other types of symmetry described above and be convex.

. In a cube, each face has four edges and three edges meet at each vertex, so the Schläfli symbol for a cube is {4,3}.

• Because {4,3} is the Schläfli symbol for a cube, it can't be the Schläfli symbol for any other Platonic solid.
• No Platonic Solid has the Schläfli symbol {3,6}.

Schläfli symbols are enough information to uniquely specify a Platonic Solid, but any given Platonic Solid has many properties that one might find interesting. The table below summarizes that information for each of the five platonic solids.

What are the five Platonic Solids

The five Platonic Solids are:

• The tetrahedron (Schläfli symbol = {3,3}). A tetrahedron has 4 faces that are each equilateral triangles.
• The cube (Schläfli symbol = {4,3}). A cube has 6 faces

Advanced Explanation

From Tom Gettys - Platonic Solids:

The simplest polygon is the triangle, so P [the number of edges is on each face] is at least 3, and the number of faces at each vertex, Q, is also at least 3. Therefore the simplest possible solid is {3,3}, and in fact it exists; it is called the Tetrahedron.

embed picture here

Keep the equilateral triangle for the face, but now put four of them around each vertex. This yields the Octahedron, having eight faces and 6 vertices.

Image:Octa.gif Octahedron {3,4}

Putting five equilateral triangles around each vertex yields the Icosahedron, which has 20 faces and 12 vertices. Six such triangles lay flat in a plane, so this is the last possible solid with P=3. Annotation: For information about why this is true, see: need good source here

[icosa.gif] Icosahedron {3,5}

We next try square faces, and succeed in creating the familiar Cube. However, four squares again lay flat, so there are no others possible with P=4.

[cube.gif] Cube {4,3}

Three pentagons joining at each vertex yields the Dodecahedron.

End quoted webpage.

Dual of a platonic solid

Advanced Explanation

If you draw a line segment from the center of each face in a Platonic solid to the center of each adjacent face, you form the outline of a new Platonic solid. This new Platonic solid is called the dual of the original (Note: we need to insert images here).

• The dual of a tetrahedron is always another tetrahedron.
• The dual of a cube is always an octahedron.
• The dual of an octahedron is always a cube.
• The dual of a dodecahedron is always an icosahedron.
• The dual of an icosahedron is always a dodecahedron.