For a simply connected polyhedron in which every simple closed curve drawn on the surface can be shrunk, the number of elements satisfy Euler's formula $N_0-N_1\+N_2\=2$, where $N_0\,N_1\,N_2$, represent the number of vertices, edges, and faces, respectively.

For a regular polyhedron {p,q}, we also have $p{N}_2\&equals;\left(2N_1\&equals;q{N}_0\right)$. Therefore, we obtain $\frac{1}{p}\&plus;\frac{1}{q}\&equals;\frac{1}{2}\&plus;\frac{1}{N_1}$ .  For the enumeration of such regular solids, we seek integers p and q, greater than 2, satisfying the inequality $\frac{1}{2}<\frac{1}{p}\&plus;\frac{1}{q}$ .
We thus obtain the tetrahedron {3,3}, the octahedron {3,4}, the cube {4,3}, the icosahedron {3,5}, and the dodecahedron {5,3}.

These are the five Platonic solids. The first three occur in nature and have been studied for at least 2500 years. A dodecahedron made by the Etruscans was found near Padua. Plato tells how Timaeus of Locri thought of the Universe as being enveloped by a gigantic dodecahedron, while the other four solids represented the "elements" of fire, air, earth, and water. Euclid's monumental treatise, the Elements, begins with the equilateral triangle and culminates in the five Platonic solids, which are again the subject of the extra books XIV and XV (added a few centuries later). Sir D'Arcy W. Thompson once remarked that Euclid never dreamed of writing an Elementary Geometry. What Euclid really did was to write a very excellent account of the regular solids, for the use of Initiates.

Consider a regular polyhedron {p,q} of sides $2l$. We see that the perpendicular to the plane of a face at its center will meet the perpendicular to the plane of a vertex figure at its center in a point O, which is the center of three important spheres: the circum-sphere that passes through all the vertices (and the circum-circles of the faces), the mid-sphere that touches all the edges (and contains the in-circles of the faces), and the in-sphere that touches all the faces.

Let $R_0\&comma;R_1\&comma;R_2$ denote their respective radii, ${\mathrm{O}}_2$ be the center of the face, ${\mathrm{O}}_1$ the midpoint of a side of this face, ${\mathrm{O}}_o$ one end of that side.
Since the triangle ${\mathrm{O}}_i{\mathrm{O}}_j{\mathrm{O}}_k$ (i<j<k) is right angled at ${\mathrm{O}}_j$, Pythagoras' theorem gives:

$R_0^2\&equals;\left(l^2\&plus;R_1^2\&equals;l{\csc \left(\frac{\mathrm{\pi }}{p}\right)}^2\&plus;R_2^2\right)$ , $R_1^2\&equals;l{\cot \left(\frac{\mathrm{\pi }}{p}\right)}^2\&plus;R_2^2$

To define a Platonic solid  in Maple, one can use the command RegularPolyhedron(gon,[m,n],o,r) where gon is the name of the polyhedron to be defined, [m,n] the Schlafli symbol, o the center of the polyhedron, and r the radius of the circum-sphere.

The values of [m,n] can be one of the following:

         Schlafli symbol           Maple's Schlafli symbol          Polyhedron type
             {3,3}                              [3,3]                                           tetrahedron
             {3,4}                              [3,4]                                           octahedron
             {4,3}                              [4,3]                                           hexahedron (cube)
             {3,5}                              [3,5]                                           icosahedron
             {5,3}                              [5,3]                                           dodecahedron

Another way to define a Platonic solid is to use the command Polyhedron_Name(gon,o,r) where Polyhedron_Name is one of tetrahedron, octahedron, hexahedron (or cube), icosahedron, or dodecahedron. For example, to define a tetrahedron with center (0,0,0), radius 1, use:

or

To access information relating to a Platonic solid, use the following function calls:   

area(gon);      returns the surface area of gon.
center(gon);    returns the center of the circum-sphere of gon.
faces(gon);     returns the faces of gon, in which each face is represented as a list of coordinates of its vertices.
form(gon);      returns the form of gon (tetrahedron3d, octahedron3d, ...)
InRadius(gon);  returns the in-radius of gon.
MidRadius(gon); returns the mid-radius of gon.
radius(gon);    returns the radius of the circum-sphere of gon.
schlafli(gon);  returns the Schlafli symbol of gon.
sides(gon);     returns the sides of gon.
vertices(gon);  returns the vertices of gon.
volume(gon);    returns the volume of gon.

For example:

and to visualize a Platonic, or a defined polyhedron in general, use the command draw:

Since the five Platonic solids may be nested together with six concentric spheres, it was an obvious prescription for a solar system of six planets (all that were known at the time), with relative distances from the Sun as given by the enclosed spheres.

  
To emphasize his theory, Kepler envisaged an impressive model of the universe which shows a cube, with a tetrahedron inscribed in it, a dodecahedron inscribed in the tetrahedron, an icosahedron inscribed in the dodecahedron, and finally an octahedron inscribed in the dodecahedron. The following is an illustration of the model: