Min-Energy Configurations of Electrons On A Sphere
The vertices of the five Platonic Solids give the only perfectly
symmetrical distributions of points on the surface of a sphere.
Therefore, if N positively charged particles are constrained to
the surface of a sphere, and N is not a Platonic number, the
particles must have an equilibrium configuration that is not
perfectly symmetrical. It turns out that, even if N is a Platonic
number, the equilibrium configuration is not necessarily the
corresponding Platonic solid. This article describes one
equilibrium configuration for values of N up to 32.
The sphere is normalized to a radius of 1. Since the force law
between the particles is assumed to be of the inverse-square type,
the equilibrium configuration of n particles is the set of n(n-1)/2
separations s[i,j] such that
n-1 n 1
Q = SUM SUM ------
i=1 j=i+1 s[i,j]
is a minimum.
N = 0: This leaves the sphere empty.
N = 1: The case of a single particle.
N = 2: Two particles go to opposite points on the sphere, and the
line between them is a diameter of the sphere, having length 2.
Thus, Qmin = 0.5.
N = 3: Three particles produce an equalateral triangle on a great
circle of the sphere. The three separations between particles are
each of length sqrt(3). For this configuration we have Qmin = sqrt(3).
N = 4: Four points arrange themselves at the vertices of a tetrahedron.
The six separations between particles are each of length 2sqrt(2/3).
N = 5: Five points give a north/south pole and an equlateral triangle
at the equator. Of the ten separations, one is of length 2, six are
of length sqrt(2), and three are of length sqrt(3). Thus, the
configuration of 5 contains the configurations of 2 and 3 as
subsets. This is the first non-symmetrical arrangement.
N = 6: Six particles arrange themselves as the vertices of an
octahedron. Of the 15 separations between particles, three are
mutually perpindicular diameters of length 2, and the remaining
twelve are of length sqrt(2). Thus, we can say that the configuration
of 6 consists of three orthogonal configurations of 2. It can also
be regarded (in three different directions) as a north/south poly
with an equilateral square at the equator.
N = 7: Seven points give north/south poles and an equilateral
pentagon at the equator. Of the 21 separations between particles,
one is a diameter of length 2, ten are of length sqrt(2), and five
each of lengths a and b, where ab = sqrt(5) and a/b = (1+sqrt(5))/2.
This gives
_____________ _____________
/ 5 + sqrt(5) / 2sqrt(5)
a = / ----------- b = / -----------
\/ 2 \/ 1 + sqrt(5)
N = 8: Eight particles arrange themselves into two squares on
parallel planes, with the squares rotated by 45 degrees relative to
each other. The 28 separations between particles come in the
following four lengths:
a = 1.2876935 b = 1.8968930 c = 1.6563945 d = 1.1712477
We observe that c = sqrt(2)d, so the two parallel square faces have
edge lengths d. This configuration has Q = 19.675..., as opposed
to Q = 22.485... for the verticies of a cube inscribed in a sphere.
This shows that the verticies of Platonic solid are not necessarily
in an equlibrium configuration. In other words, perfect symmetry
does not imply stable equilibrium.
N = 9: Nine points on the surface of a sphere will configure
themselves as three equilateral triangles on parallel planes.
One triangle is at the equator of the sphere, while the other
two are located at latitudes above and below, and oriented at
60 degrees relative to the equator triangle. Looking down from
above the north pole, the three triangles would appear as shown
below:
The separations between points have 6 distinct magnitudes, as
summarized below.
1.8496207 (1,6) (7,6) (2,4) (8,4) (3,5) (9,5)
1.1355402 (1,4) (1,5) (7,4) (7,5)
(3,4) (3,6) (9,4) (9,6)
(2,5) (2,6) (8,5) (8,6)
1.8695243 (1,8) (1,9) (2,7) (2,9) (3,7) (3,8)
1.4072969 (1,7) (2,8) (3,9)
1.2307059 (1,2) (1,3) (2,3) (7,8) (7,9) (8,9)
sqrt(3) = 1.7320509 (4,5) (4,6) (5,6)
N = 10: Ten points on the surface of a sphere will arrange themselves
with one north-south polar axis and two squares at equal and opposite
latitudes, oriented at 45 degrees relative to each other. When viewed
from above the north pole the points would appear as shown below.
The separations come in 7 distinct lengths, summarized as follows:
1.8758698 (1,6) (1,7) (1,8) (1,9) (10,2) (10,3) (10,4) (10,5)
1.0745353 (1,2) (1,3) (1,4) (1,5) (10,6) (10,7) (10,8) (10,9)
1.8125511 (3,5) (2,4) (6,8) (7,9)
1.2816593 (2,3) (3,4) (4,5) (5,2) (6,7) (7,8) (8,9) (9,6)
1.0935212 (2,7) (7,3) (3,8) (8,4) (4,9) (9,5) (5,6) (6,2)
1.6868248 (2,8) (2,9) (3,6) (3,9) (4,6) (4,7) (5,7) (5,8)
2.0000000 (1,10)
N=11: The equilibrium configuration of 11 charged particles on the
surface of a sphere has one particle at the north pole (level 0).
At a depth of 0.484692 (level 1) down from the north pole are two
particles on opposite sides.
Then at a depth of 0.6714 (level 2) are four particles in an oblong
rectangle oriented at right angles to the previous two particles.
The dimensions of this rectangle are 0.9778448 by 1.7118527, and it
has diagonals of 1.9714513.
Next, at a depth of 1.552681 (level 3) below the pole, are two
particles on opposite longitudes, oriented parallel to the first
two points below the pole. Interestingly, these two points are
very nearly whole diameters away from the two particles on level 1.
They are each 1.9995113 away from the opposite particle on the
other level.
At the lowest level, 1.805954 (level 4) below the pole, are two
particles on opposite longitudes, oriented at right angles to the
previous two.
The arrangement of particles on each level is illustrated below:
This configuration has separations of 20 distinct magnitudes, as
summarized below.
separation number example separation number example
---------- ------ ------- ---------- ------ -------
1.6323310 4 (1,2) 0.9940217 4 (1,3)
1.2896824 4 (1,4) 1.8123901 4 (1,5)
1.7118527 2 (1,6) 0.9778448 2 (1,7)
1.1709697 4 (1,8) 1.1216075 4 (1,9)
1.9714513 2 (1,10) 1.7323473 4 (1,11)
1.7140111 1 (2,3) 0.9845739 2 (2,4)
1.6824472 4 (2,5) 1.9995113 2 (2,8)
1.0682496 2 (2,11) 1.9005025 2 (4,5)
1.7622040 2 (4,8) 1.0531511 4 (5,11)
1.1839539 1 (5,9) 1.6667852 1 (8,11)
N=12: The equilibrium configuration of 12 charged particles on the
surface of a sphere is as the verticies of an icosahedron, one of
the five Platonic solids. The 66 separations come in 3 distinct
sizes, as summarized below:
separation number
---------- ------
1.0514622 30
1.7013016 30
2.0000000 6
N=16: This is the first occurrance of more than 1 distinct local
equilibrium configuration. The two configurations have energies
C16.1: 92.91165530254497
C16.2: 92.92035396234466
These two configurations are shown in the figure below.
Notice that C16.2 has two opposing "square faces", whereas C16.1 has
only triangular faces.
N=20: The equilibrium configuration of 20 charged particles on the
surface of a sphere has a north and south pole.
Below the north pole is an equilateral triangle of three particles.
On the next lower level is another equilateral triangle of three
particles, rotated by 60 degrees.
On the next lower level, the equator, there are six particles,
arranged in two equilateral triangles. These triangles are squewed,
but not by 60 degrees.
Then there are two more levels of equilateral triangles, and then
the south pole. The configuration at each level is illustrated below.
The separations have 26 distinct magnitudes, as summarized below:
separation number example separation number example
---------- ------ ------- ---------- ------ --------
0.8331592 12 (1,2) 0.7894919 12 (2,8)
1.8657231 6 (1,3) 1.4142136 12 (2,9)
1.2478910 6 (1,4) 1.3615374 12 (2,11)
1.4578665 12 (1,6) 1.8769439 12 (2,14)
1.3869718 3 (1,7) 1.9924151 3 (2,15)
0.8016163 12 (1,8) 1.7320508 6 (2,16)
0.7829612 6 (1,9) 0.8455036 3 (2,19)
1.8403727 6 (1,10) 0.9568466 6 (8,9)
1.4651562 12 (1,11) 1.7562587 6 (8,10)
1.5680184 6 (1,14) 1.8149394 6 (8,11)
1.7833875 12 (1,15) 1.4553299 6 (8,12)
1.9906722 6 (1,17) 1.0844446 3 (8,18)
1.1469114 3 (2,6) 2.0000000 1 (9,10)
N = 22: The equilibrium configuration of twenty-two particles on the
surface of a sphere is with 6 of the points arranged as the vertices
of an octahedron. Then, in the northern hemisphere, two opposing
faces have a particle located at the center of the triangular face.
Similarly in the southern hemisphere, the two orthogonal opposing
faces have a central particle. Then from each of these four face-
centered particles you can imagine proceeding in three directions
straight toward the edge of the respective triangle, and then a
little beyond, where another particle is located. Thus, for each
of the face-centered particles there are three other particles,
giving 4 + (3)(4) + 6 = 22 total.
In the drawing below, the six particles 22, 17, 6, 8, 3, and 19 are
configured as the verticies of an octahedron. The four face-centered
particles are 11, 1, 2, and 13.
top view of upper top view of lower
hemisphere hemisphere
The 231 separations between particles come in 17 distinct lengths,
summarized as follows:
separation number example
--------- ------- -------
1.6329932 6 (1,2)
0.9194017 12 (1,3)
0.7379041 12 (1,4)
1.3559300 24 (1,5)
1.9438121 12 (1,7)
1.7761477 12 (1,17)
1.5098498 12 (3,4)
0.7743894 24 (3,5)
1.4142136 12 (3,6)
1.8439959 24 (3,7)
1.3116225 12 (3,12)
2.0000000 3 (3,19)
1.7263978 24 (4,5)
1.1879164 12 (4,9)
1.9803530 6 (4,11)
1.4279710 12 (4,12)
0.7924366 12 (4,18)
N=24: The convex solid formed by these verticies has several interesting
symmetry properties. It has 6 square faces and 32 triangular faces, and
a total of 60 edges. Here is one specific side view, aligned so that four
of the square faces are seen as edges of the outer perimeter of the
outline:
SUMMARY OF N-PARTICLE CONFIGURATIONS
N seps dist poles description
-- --- --- ---- -------------------------------------------
0 0 0 0
1 0 0 0 North pole.
2 1 1 1 North and south poles.
3 3 1 0 Triangle at equator.
4 6 1 0 Tetrahedron.
5 10 3 1 North and south pole with triangle at equator.
6 15 2 3 Octahedron.
7 21 4 1 North and south poles with pentagon at equator.
8 28 4 0 Twisted cube; opposite squares at 45 degrees.
9 36 6 0 Three parallel triangles, one at equator, at 60 deg.
10 45 7 1 North and south pole with two sqewed squares.
11 55 20 0 Pole, line, rectangle, line, line.
12 66 3 6 Icosahedron.
13 78 28 0
14 91 9 1
15 105 21 0
16 120 12,13 0 Two distinct equilibrium configs. One contains N=4
as a subset.
17 136 14 1 Contains N=7 as a subset.
18 153 18 1
19 171 52 0
20 190 26 1 Pole, tri, tria, double tri, tri, tri, pole.
21 210 69
22 231 17 3 Contains N=6 (octahedron) as a subset.
23 253 48 1 North and south pole, with 7 parallel triangles, one
at equator and 3 on either side. Contains N=5 as a
subset.
24 276 16
25 300 160
26 325 168
27 351 30
28 378 35
29 406 75
30 435 118
31 465 93
32 496 11,43
Following are links to summaries and images of the minimal-energy
configurations of N particles for N up to 32:
N = 4 N = 13 N = 21 N = 30
N = 5 N = 14 N = 22 N = 31
N = 6 N = 15 N = 23 N = 32 (1)
N = 7 N = 16 (1) N = 24 N = 32 (2)
N = 8 N = 16 (2) N = 25
N = 9 N = 17 N = 26
N = 10 N = 18 N = 27
N = 11 N = 19 N = 28
N = 12 N = 20 N = 29
Of all these solutions, the most appealing in some ways is the case
n=4!=24, which is a polyhedron whose surface includes 24/4 perfect
squares, 24/3 perfect triangles, and 24/2 additional "bi-gons",
arranged so that each vertex is identical. In addition, the
verticies have a "handedness", so there are really two distinct
versions of this solid. If your browser supports Java applets
you can see a rotating image of this solid here.
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