Regular Polyhedra
This worksheet describes the nine regular polyhedra that are part of the geom3d package.
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Introduction


First, we give a display of the nine regular polyhedra:
Some Definitions:
A polygon is said to be regular if it has equal sides and equal angles.
A solid angle (or polyhedron vertex) 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 called the vertex figure. In 1852, Schlafli invented the symbol {p,q} for the regular polyhedron whose faces are pgons, q meeting at each vertex, or the polyhedron with face {p} and vertex figure {q}.


The Five Platonic Solids



Enumeration of the Platonic Solids


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 , where , represent the number of vertices, edges, and faces, respectively.
For a regular polyhedron {p,q}, we also have . Therefore, we obtain . For the enumeration of such regular solids, we seek integers p and q, greater than 2, satisfying the inequality .
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 . 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 circumsphere that passes through all the vertices (and the circumcircles of the faces), the midsphere that touches all the edges (and contains the incircles of the faces), and the insphere that touches all the faces.
Let denote their respective radii, be the center of the face, the midpoint of a side of this face, one end of that side.
Since the triangle (i<j<k) is right angled at , Pythagoras' theorem gives:
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The Platonic Solids in Maple


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 circumsphere.
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, one uses:
 (2.2.1) 
or
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To access information relating to a Platonic solid, use the following function calls:
 (2.2.2) 
area(gon); returns the surface area of gon.
center(gon); returns the center of the circumsphere 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 inradius of gon.
MidRadius(gon); returns the midradius of gon.
radius(gon); returns the radius of the circumsphere 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:
 (2.2.3) 
and to visualize a Platonic, or a defined polyhedron in general, use the command draw:
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Kepler's Universe


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:



The KeplerPoinsot Solids (Star Polyhedra)


The possibility of further regular figures was first envisaged by Bredwardin, a 14thcentury Englishman, who extended the sides of an ordinary polygon to form a starpolygon.
Let {p} denote the regular polygon whose sides each subtend an angle at the center. This is an ordinary pgon when p is an integer, and a starpolygon for other rational values (greater than 2). In fact, {n/d} is an ngon of density d; for example, {5/2} is the pentagram whose sides go twice around the center.
Early in the 17th 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 starshaped corners.
Hence, besides the five Platonic solids, we now admit four more as regular polyhedra. They are:
The small stellated dodecahedron {5/2,5}
The great stellated dodecahedron {5/2,3}
The great dodecahedron {5,5/2}
The great icosahedron {3,5/2}
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 , where and are the densities of the face and vertex figure, while is the density of the whole polyhedron, or 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 nine and only nine finite regular polyhedra exist.
In Maple, as in the case of the five Platonic solids, one can define a KeplerPoinsot solid either by using the command RegularPolyhedron(gon,[m,n],o,r) or by using the command Polyhedron_Name(gon,o,r) where the Schlafli 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, or GreatDodecahedron. The information about these solids are the same as for the Platonic solids. For instance, to define a great dodecahedron of center (0,0,0), radius 2, one uses:
 (3.1) 

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