The Nine Regular Polyhedra: (VRML)

Definition

A solid angle is said to be regular if it has equal face angles and equal dihedral angles.

A polyhedron is said to be regular if its faces are regular polygons and its corners are regular solid angles; it then necessarily has equal faces and equal angles.

The shape of the solid angle is conveniently described in terms of the section by a plane perpendicular to the axis of symmetry through the vertex. This is call the vertex figure . Schläfli in 1852 invented the symbol {p,q} for the regular polyhedron whose faces are p-gons, q meeting at each vertex, or the polyhedron with face {p} and vertex figure {q} .

The Five Platonic Solids

For a regular polyhedron {p,q} , we also have p*N2=(2*N1=q*N0) . Hence, we obtain 1/p+1/q=1/2+1/N1 .

For the enumeration of such regular solids, we seek integers p and q , greater than 2, satisfy the inequality 1/p+1/q>1/2 . Thus, we 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 Etruscas 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 represent 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 Eulid 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 2*l , we easily 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 which passes through all the vertices (and the circum-circles of the faces), the mid-sphere which touches all the edges (and contains the in-circles of the faces), and the in-sphere which touches all the faces.

Let R0, R1, R2 denote their respective radii, O2 be the center of the face, O1 the midpoint of a side of this face, O0 one end of that side. Since the triangle Oi*Oj*Ok (i < j < k) is right angled at Oj , Pythagorias' theorem gives R0^2 = (l^2+R1^2=(l*csc(Pi/p)^2+R2^2 , R1^2=(l*cot(Pi/p)^2+R2^2.

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 Schl&aum;lfli 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:

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 (cube), icosahedron, or dodecahdron . For example, to define a tetrahedron with cente r (0,0,0) , radius 1, one uses:

> with(geom3d):
> RegularPolyhedron(te1,[3,3],point(o,0,0,0),1);
                                      te1

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

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

> draw(te1,cutout=7/8,lightmodel=light4);

We'll close this section on a historical note. In 1596 Kepler published a tract called The Cosmic Mystery . Here is an extract:

... Before the universe was created, there were no numbers except the Trinity, which is God himself ... For, the line and the plane imply no numbers: here infinitude itself reigns. Let us consider, therefore, the solids. We must first eliminate the irregular solids, because we are only concerned with orderly creation. There remains six bodies, the sphere and the five regular polyhedra. To the sphere corresponds the heaven. On the other hand, the dynamic world is represented by the flat-faces solids. Of these there are five: when viewed as boundaries, however, these five determine six distinct things: hence the six planets that revolve about the sun. This is also the reason why there are but six planets ...

... I have further shown that the regular solids fall into two groups: three in one, and two in the other. To the larger group belongs, first of all, the Cube, then the Pyramid, and finally the Dodecahedron. To the second group belongs, first, the Octahedron, and second, the Icosahedron. That is why the most important portion of the universe, the Earth - where God's image is reflected in man - separates the two groups. For, as I have proved next, the solids of the first group must lie beyond the earth's orbit, and those of the second group within...Thus I was led to assign the Cube to Saturn, the Tetrahedron to Jupiter, the Dodecahedron to Mars, the Icosahedron to Venus, and Octahedron to Mercury ...

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:

Illustration that the earth is the center of the Universe: (VRML)

Let {p} denote the regular polygon whose sides each subtend an angle 2*Pi/p at the center. This is an ordinary p-gon when p is an integer, and a star-polygon for other rational values (greater than 2). In face, {n/d} is an n-gon of density d ; eg, {5/2} is the pentagram, whose sides go twice around the center.

Early in the seventeenth century, Kepler conceived the idea of admitting pentagrams as faces of regular polyhedra. He constructed the two stellated dodecahedra, {5/2,5} and {5/2,3} . Two hundred years later Poinsot rediscovered these and added their reciprocals, {5,5/2} and {3,5/2} , which have star-shaped corners.

Hence, besides the five Platonic solids, we now admit four more as regular polyhedra, they are:

Note that Euler's formula is not valid for the regular star polyhedra. Calley ( Collected Mathematical Papers, vol. 4, pp 81-85 ), who gave the star polyhedra the above names, extended Euler's formula to dp*N2+dq*N0 = N1+2*D where dp and dq are the densities of the face and vertex figure, while D is the density of the whole polyhedron, i.e., the number of intersections that the faces make with a ray drawn from the center in a general direction, counting two intersections for penetration of the core of a pentagram.

In 1811, Cauchy used the principle that every regular polyhedron must have the same face planes as a Platonic solid to prove that there exists nine and only nine finite regular polyhedra.

In Maple, as in the case of the five Platonic solids, one can define a Kepler-Poinsot solid by either using the command RegularPolyhedron(gon,[m,n],o,r); or Polyhedron_Name(gon,o,r); where the Schl&aum;lfli symbol is one of [5/2,5], [5/2,3], [5,5/2], [3,5/2], and Polyhedron_Name is one of GreatStellatedDodecahedron, SmallStellatedDodecahdron, GreatIcosahedron, GreatDodecahedron . The information about these solids are the same as for the Platonic solids.