www.neubert.net - Dr. Neubert's Website The Entropy Reduction Laboratory Platonic Spheres

Introduction
• Platonic Bodies are regular polyhedra. Their faces are identical regular polygons. As shown by Euklid (300 BC), there exist 5 Platonic bodies.
• Platonic spheres are the central projections of the faces of the Platonic bodies onto the surface of the circumscribing unit sphere, R=1. The vertices are corners in the case of bodies and nodes in the case of spheres.
A systematic approach to generating the Platonic bodies is presented. The building stones of Platonic bodies are evaluated using Descartes' law of closure deficit. The symmetry of the Platonic spheres is discussed.
Tetrahedral
Sphere

The tetrahedral sphere has:
• 4 faces which are regular spherical triangles
• 4 vertices
• 6 arc-edges
The length of the arc-edges is given by 2*arcsin(sqrt(6)/3)
Octahedral
Sphere

The octahedral sphere has:
• 8 faces which are regular spherical triangles
• 6 vertices
• 12 arc-edges
The length of the arc-edges is given by 2*arcsin(sqrt(2)/2)
Hexahedral
Sphere

The hexahedral sphere has:
• 6 faces which are spherical squares
• 8 vertices
• 12 arc-edges
The length of the arc-edges is given by 2*arcsin(sqrt(3)/3)
Icosahedral
Sphere

The icosahedral sphere has:
• 20 faces which are regular spherical triangles
• 12 vertices
• 30 arc-edges
The length of the arc-edges is given by 2*arcsin(2*sqrt(2)/sqrt(5+sqrt(5)))
Dodecahedral
Sphere

The dodecahedral sphere has:
• 12 faces which are regular spherical pentagons
• 20 vertices
• 30 arc-edges
The length of the arc-edges is given by 2*arcsin(2*sqrt(3)/3*sqrt(1+sqrt(5)))
Rotate
a sphere
Click on the selected Platonic sphere (also the tiny ones) and drag the mouse.
Number
of
Platonic
bodies
and
spheres

### Number of Platonic bodies and spheres.

Here we derive the maximal number of Platonic bodies.

The angle at the corner of a regular polygon of n sides is given by
• pi*(n-2)/n.
If k polygons meet at a vertex, there results - as stated by René Descartes (1596-1650) - a planar deficit D,
• D = 2*pi - k*pi*(n-2)/n >= 0
which may be written as
• D = pi*w/n
where
• w = 2k+2n-kn >= 0      (x)
is the universal curvature parameter

If the curvature parameter w = 0, the corresponding geometrical structure is planar.

If w>0, the planar structure of k polygons may be tilted to form a corner in three dimensional space.

A collection of curvature parameters w for various tuples (k,n) can be calculated from the
formula (x) and is presented in the following

Table of w(k,n)
```            n = 2   3   4   5   6   7   8  ...      N
k
2      4   4   4   4   4   4   4  ...   4-0(N-3)
3      4   3   2   1   0  -1  -2  ...   3-1(N-3)
4      4   2   0  -2  -4  -6  -8  ...   2-2(N-3)
5      4   1  -2  -5  -8 -11 -14  ...   1-3(N-3)
6      4   0  -4  -8 -12 -16 -20  ...   0-4(N-3)
7      4  -1  -6 -11 -16 -21 -26  ...  -1-5(N-5)
8      4  -2  -8 -14 -20 -26 -32  ...  -2-6(N-3)
..    ........................... ... ..........
K      4  ..  .. ... ... ... ...  ...  2K+2N-KN
```
The table reflects the symmetry of the curvature parameter w in n and k.
It is interesting to note, that the curvature parameter w assumes not only values w>=0 but also w<0. An interpretation of these eccentric values w<0 will be suggested below.
Building
stones
of
Platonic
bodies
and
spheres

### Building stones of Platonic bodies and spheres.

Here we want to derive the building stones of Platonic bodies.

The key to solve this problem is provided by Descartes' Law of Closure Deficit as discussed by Kirby Urner. By this Law of Closure Deficit the sum of the planar deficits at the vertices of a closed convex surface adds up to the maximal deficit of two vertices, i.e. 4*pi:
• V*D = 4*pi
The number V of vertices of a regular polyhedron is thus given by V = 4*pi/D or
• V = 4*n/w
The number E of edges which meet at a vertex is k. Since each edge has a vertex at its two ends, E = V*k/2 or
• E = 2*n*k/w
Since each edge is shared by two faces, the number F of faces is given by F = 2*E/n or
• F = 4*k/w
Since for every tuples n,k the value of V turns out to be integer, there exist indeed 5 Platonic bodies and 5 Platonic spheres.

Now we can collect the building stones of Platonic bodies. For completeness we include also the properties of the planar lattices, w=0, and an example ( one of an "infinite" number ! ) for the eccentric case w<0 in the table:

```        w     n     k    D/pi     V     F     E      Name
-------------------------------------------------------
3     3     3      1      4     4     6   Tetrahedron
2     3     4     2/3     6     8    12   Octahedron
2     4     3     1/2     8     6    12   Hexahedron
1     3     5     1/3    12    20    30   Icosahedron
1     5     3     1/5    20    12    30   Dodecahedron
-------------------------------------------------------
0     3     6      0     oo    oo    oo   point group 6
0     6     3      0     oo    oo    oo   point group 3
0     4     4      0     oo    oo    oo   point group 4
-------------------------------------------------------
-1    7     3   -1/7 (-28    -12    -42 )
-1    3     7   -1/3 (-12    -28    -42 )
-2    ..................................
........................................
-------------------------------------------------------       ```

w>0:
Since w is symmetric in n and k, V and F are symmetric in n and k. Platonic bodies are called related, if the their k and n values are exchanged. Further, E depends on the product of n and k and therefore the number of edges is equal for related bodies. The octahedron and hexahedron are related as are the icosahedron and dodekahedron. The tetrahedron is self-related. The tetrahedron shows the highest, the dodecahedron the lowest curvature per vertex.

Leonhard Euler (1707 - 1783) derived the fundamental relation

E = F + V - 2
For completeness, we verify Euler's law in terms of n, k, and the curvature parameter w:
2*n*k/w = 4*k/w + 4*n/w -2
Multiplying by w and inserting w results in an identity.

The x,y,z coordinates of the vertices of the Platonic bodies may be calculated straightforward by use of the relationship between the edges a and the radius R of the circumscribing sphere.

Platonic spheres are the central projections of the faces of the Platonic bodies onto the surface of the circumscribing unit sphere, R=1. The polygons with edges a of the Platonic bodies are thus mapped onto spherical polygons with arc-edges b . The arc-edges of the spheres are given by b=2*arcsin(a/2) independent on the type of Platonic body. The edges a in units of R=1 depend, as mentioned before, on the type of Platonic body.

Thus, by this construction, the characteristics of Platonic spheres correspond to those of the Platonic bodies. For the spheres, the contribution to the curvature per vertex goes to zero, but the number of vertices goes to infinity such that the total curvature remains constant 4*pi.

w=0:
Here the angular deficit equals zero. Thus the allowed polygons lie in a plane. Analogue to the three-dimensional case, the polygons n,k = 3,6 and n,k = 6,3 are related, and the square, n,k = 4,4, is self-related.
Usually, the limited number of allowed cases is derived from the restrictions of point groups by translational symmetry reqirements.( see e.g. the classic book by Charles Kittel, Introduction to Solid State Physics, John Wiley). Indeed, Descartes' Law of Closure Deficit provides an independent prove on the limited number of allowed planar lattices..

Further a more than just remarcable curiosity: Euler's formula holds also for the limiting case of the planar lattices. After multiplying by w, with w=0 the term 2*w vanishes and the formula holds for each of the three tuples n,k = 3,6 , 6,3 , and 4,4. This indicates, that the "-2" term in Euler's formula indeed results from Descartes' closure deficit.

w<0:
After having seen that w>0 signifies bodies with convex curvature, w=0 signifies planar symmetric structures of zero curvature, we now extrapolate the eccentric cases w<0 as signifying structures with "opposite" curvature ( compared to the case w>0). A positive planar deficit D relates to the curvature of a convex body, thus the negative planar deficit D relates to an outward wrapped surface.

The 5-7-ring structure has also been shown in course of the mathematical excursion OPTIVERSE of John M. Sullivan, George Francis, Stuart Levy, and Camille Goudeseune. Here the 5-7-ring structure comprises an essential component in turning the sphere inside out.

History of
Platonic
spheres

### History of Platonic spheres.

Finally, we point to the rich history of Platonic bodies as collected by the Anderson Group, Max-Planck Institut fuer Festkoerperforschung, Stuttgart. Here also reference is given to the first picture of a truncated icosahedron which today has been realized as the fullerene C60. This picture is taken from Libellus de quinque corpibus regularibus, a book by the Italian renaissance painter and mathematician Pier della Francesca (1420 - 1492).

Symmetry of
Platonic
spheres

### Symmetry of Platonic spheres.

The symmetry of Platonic spheres is that of Platonic bodies. Symmetries of rigid rotations of 3D-space are represented by orthogonal 3x3 matrices, which are the elements of the Special Orthogonal matrix group SO(3). The symmetry groups related to the Platonic bodies are subgroups of SO(3). - Related Platonic bodies are described by identical subgroups and thus there are three subgroups of SO(3):
1. The tetrahedral group T consisting of 3- and 4-fold rotations. Including the unit element, there are 12 elements in this group.
2. The octahedral group O consisting of 2-, 3-, and 4-fold rotations. Including the unit element, there are 24 elements in this group. The octahedron and the hexahedron are of the same symmetry.
3. The icosahedral group Y consisting of 2-, 3-, and 5-fold rotations. Including the unit element, there are 60 elements in this group. The icosahedron and the dodecahedron are of the same symmetry.
If we add the rotation-reflection operation, we obtain the complete symmetry groups Th with 24 elements, Oh with 48 elements, and Yh with 120 elements.

Another point of view: There exist a group homomorphism between SO(3) and the Special Unitary group SU(2). The SU(2) group consists of 2x2 complex matrices which may be defined geometrically by using quaternions. Tony Phillips has introduced the Binary Tetrahedral Group which is the group Th considered as a subgroup of SU(2).