Platonic Solids

Platonic Solids

Considering the Platonic Solids, there are five so named because they were known at the time of Plato circa (427-347 BC). These polyhedra are also called regular polyhedra because they are made up of faces that are all the same regular polygon. It is hoped that from the background information on polygons that the understanding of the sections on polyhedra will be made easier. Shown below are the only five platonic solids.

Regular Polyhedra

There are only five platonic solids. This is due to the fact that if you try to make a solid from all regular hexagonal faces or any regular polygon with more than five sides it is not possible to enclose the three dimensional space fully without using a second type of regular polygon. Take a solid such as a soccer ball, (truncated icosahedron) there are both regular hexagons and pentagons used to enclose this three dimensional space.

With Polyhedra as with polygons, two sides meet at a point called a vertex of the figure, so in a polyhedron two faces meet at or on a line (or in a line the mode of expression is variable). Thus each face shares each of its sides as lines in common with other faces. These lines are called the edges of the polyhedron. So each edge of a polyhedron belongs to exactly two faces and no more.

Identification Notation:

There is a certain method of describing polyhedra and identifying a set number of characteristics for each polyhedron usually does this. The first three characteristics to be identified are how many faces, edges and vertices has the polyhedron.

For the moment we will consider the platonic solids (regular polyhedra). It is convenient to identify the platonic solids with the notation (p,q) where p is the number of sides in each face and q is the number of faces that meet at each vertex. Thus the cube is (4,3) because it consists of squares (with 4 sides) meeting three to a vertex. The tetrahedron is then (3,3) and the octahedron (3,4).

Polyhedron:
No: of Edges of a Face:
No: of Faces at a Vertex: No: of Vertices:

Tetrahedron
3
3 4

Octahedron 3 4 6

Hexahedron 4 3 8

Icosahedron 3 5 12

Dodecahedron 5 3 20

From equilateral triangles you can make:

with 3 faces at each vertex, a tetrahedron

(p,q) = (3,3)

with 4 faces at each vertex, an octahedron

(3,4)

with 5 faces at each vertex, an icosahedron.

(3,5)

From squares you can make:

with 3 faces at each vertex a cube (hexahedron)

(4,3)

From pentagons you can make:

with 3 faces at each vertex, a dodecahedron.

(5,3)

Did you know?

In nature the cube, tetrahedron and octahedron appear in certain viruses and radiolaria. It is important to remember that when naming these polyhedra it is better to refer to them as regular. Sometimes it is taken for granted that a dodecahedron is a regular dodecahedron where as it could be describing any polyhedron with twelve faces.

The five platonic solids are the only polyhedra in three dimensions:

whose faces are identical regular polygons

whose vertices are all identical

which are convex

The tetrahedron which has four equilateral triangles for its faces, is the three-dimensional analogue of the two-dimensional equilateral triangle. It is the simplest polyhedron since it has the least number of faces possible to enclose a three dimensional space.

In the following sections designated to the five regular polyhedra a description of each polyhedron will be given. This description will contain the individual characteristics of each polyhedron. These summaries will contain elements such as which polygon is being used to make up the faces of the polyhedron, the angle between each of the faces and the net (or development) of each solid.

(CLICK ON EACH POLYHEDRON FOR INDIVIDUAL DISCRIPTION)