Science of Spherical Arrangements

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Science of Spherical Arrangements

• Peter Dragnev
• Mathematical Sciences, IPFW

• Well-distributed points on the sphere
• Motivation from Chemistry, Biology, Physics
• Survey of results in the literature
• New results for logarithmic points

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Well-distributed points on the sphere
Let ?N X1 , , XN? Sd-1 - the unit sphere in
d-dimensional space Rd X (x1 , x2 , , xd
) .
Sd-1 X x1 2 x2 2 xd 21
How do we distribute well the points?

• For d2, this problem is simple.The solution is
  up to rotations the roots of unity.

Reason - direction and order
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• For d?3 - no direction or order exists. Other
  methods and criteria are needed.

To well-distribute means to minimize some energy.
We distinguish - Best packing points -
Fekete points - Logarithmic points, etc.
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Minimum Energy Problem on the Sphere Given an
N-point configuration ?N X1 , , XN on the
sphere S2 we define its generalized energy as E?
(? N ) Si?j Xi - Xj?.

• Maximize E? (? N ) when ?gt0
• Minimize it when ?lt0
• When ?0 minimize the logarithmic energy
• E0 (? N ) Si?j log(1/Xi - Xj)
• or maximize the product P(? N )exp(-E0 (? N )).

Denote the extremal energy with E? (N,d) .
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? 1 Except for small N a long standing open
problem in discrete geometry (L. Fejes Toth -
1956)
? ? -? Tammes problem maximize the minimum
distance between any pair of points. Known for N
1-12 and 24.(Best packing points)
? -1 Thompson problem recent discovery of
fullerenes attracted the attention of researchers
in chemistry, physics, crystallography. Answer
known for N1-4, 6, 12. (Fekete Points)
? 0 The problem was posed by L. L. Whyte in
1952. Until recently the answer was known only
for N1-4. (Logarithmic Points)
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Motivation from Chemistry

• Fullerenes (Buckyballs)

Large carbon molecules discovered in 1985 by
Richard Smalley et. el.
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Nanotechnology
Nanowire -- a giant single fullerene molecule,
a truly metallic electrical conductor only a few
nanometers in diameter, but hundreds of microns
(and ultimately meters) in length, expected to
have an electrical conductivity similar to
copper's, a thermal conductivity about as high as
diamond, and a tensile strength about 100 times
higher than steel (R. Smalley).
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Motivation from Biology

• Problem of Tammes (? ? -? )
• Questions, raised by the Dutch botanist Tammes in
  1930 in connection with the distribution of pores
  on pollen grains.
• What is the largest diameter of n equal circles
  that can be packed on the surface of a unit
  sphere without overlap?
• How to arrange the circles to achieve this
  maximum, and when is the arrangement essentially
  unique?
• This is the same as to ask to maximize the
  minimum distance between the points in the
  arrangement.

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A disco ball in space
Starshine 3 satellite was launched in 2001 to
study Earths upper atmosphere. The satellite was
covered by 1500 small mirrors, which reflected
the sun light during its free fall, allowing a
large group of students nationwide to track the
satellite. Image credit Michael A.Savell and
Gayle R. Fullerton.
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Motivation from Physics
Electrons in Equilibrium
N electrons orbit the nucleus
Electrons repel
Equilibrium will occur at minimum energy
Problem If Coulombs Law is assumed, then we
minimize the sum of the reciprocals of the mutual
distances, i. e. Si?jXi - Xj-1 over all
possible configurations of N points X1 , , XN
on the unit sphere. This corresponds to the case
? -1.
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32 Electrons
122 Electrons
In Equilibrium
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Distribution of Dirichlet Cells (School Districts)
DjX? S2 X-Xjmink X-Xk j1,,N
The D-cells of 32 electrons at equilibrium are
the tiles of the Soccer Ball. Soccer Ball designs
occur in Nature frequently. The vertices of the
Soccer Ball form C60.
There has to be exactly 12 pentagons in a soccer
ball design.
Q. How are D-cells distributed for large N ?
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Buckyball under the Lions Paw
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Survey of results for small N

• Tammes problem (best packing) ? -?
• The solution is known for N1,2, , 12, 24.
• N4 - regular tetrahedron
• N5,6 - south, north and the rest on the equator
• N8 - skewed cube
• N12 - regular icosahedron (12 pentagons)

Thompson problem ? -1 (known for N4,6,12)
N4 - regular tetrahedron N 6 - regular
octahedron N12 - regular icosahedron
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Whytes problem ? 0 (Logarithmic points)
N4 - regular tetrahedron - 1,3 N12 -
regular icosahedron - 1,5,5,1 A 96 N6 -
regular octahedron - 1,4,1 KY 97 N5 -
D3h 1,3,1 - DLeggTownsend 01
Remark The method of proof is different from the
other two results - a mixture of analytical and
geometrical methods.
Definition A collection of points which
minimizes the logarithmic energy E0 (? N ) Si?j
log(1/Xi - Xj) is called optimal configuration.
The points are referred to as logarithmic points.
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Logarithmic points - new results
Let ?N X1 , , XN be an optimal
configuration on the sphere S2. Define dN
mini?j Xi - Xj.
Theorem 1 (Dragnev, Legg, Townsend - 01)
1,3,1 is the only optimal 5-point configuration.
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d2 configuration on Sd-1
Let ?NX1, X2 , XN ? Sd-1 be optimal, i.e.
E0 (? N ) E0 (N,d). Derivative conditions on
the energy functional (when N-1 points are fixed)
yield
Corollary The regular d-simplex is the only
optimal d1-configuration on Sd-1 for any ?? 0.
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• Definitions Configuration is called
• critical, if it satisfies (i) (recall (i)
  implies (ii) and (iii)).
• degenerate, if it does not span Rd.
• A vertex Xi is mirror related to Xj (Xi Xj) if
  Xi Xk Xj Xk for all k? i,j. Then ?N\ Xi , Xj
  lie in the orthogonal bisector hyperspace of
  XiXj.

Note Mirror relation is equivalence relation.
Theorem 3 (Dragnev 02) For fixed N and ??0 the
extremal energy E? (N,d) is strictly decreasing
for dltN, and for d?N, E? (N,d) E? (N,N-1).
Corollary Optimal configurations are
non-degenerate.
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Theorem 4 (Dragnev 02) If Nd2, then any
critical configurations satisfies at least one of
the following (a) it is degenerate (b) ? a
vertex with all edges stemming out equal (c)
every vertex has a mirror related partner.
Example Let N5, d3. By Theorem 3, optimal
configurations will satisfy at least one of the
following (a) degenerate ? 5 (b)
EAEBECED ? 1,4 (c) Every vertex has a
mirror related partner. In this case
we arrive at ABC and DE ? 1,3,1
Comparing 5, 1,4, and 1,3,1 proves
Theorem 2.
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• Examples
• N4, d2 diagonals of a square, form orthogonal
  simplexes.
• N5, d3 equilateral triangle on Equator and ?
  diameter.
• N6, d4 (cos k?/3, sin k?/3,0,0) (0,0,cos
  k?/3, sin k?/3), k0,1,2

In progress If Nd2 the optimal configurations
is unique up to rotations and consists of two
mutually orthogonal regular d/2- and
(d1)/2-simplexes. So,