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.
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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:
The number V of vertices of a regular polyhedron is thus given
by V = 4*pi/D or
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
Since each edge is shared by two faces, the number
F of faces is given by F = 2*E/n or
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.
By example, the convex curvature produced by a vertex of 5-fold symmetry may be partially
compensated by the concave curvature due to an adjacent vertex of 7-fold symmetry.
The occurrence of 5-7-ring structures for reducing the strain energy
of droplet structures has recently experimentally observed by international cooperation
of many renowned institutions, see
Topic XV " Droplet Surface-Structure " in the mini-anthology
on Fullerenes.
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.
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Symmetry of
Platonic
spheres
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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):
- The tetrahedral group T consisting of 3- and 4-fold rotations.
Including the unit element, there are 12 elements
in this group.
- 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.
- 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).
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