Science of Spherical Arrangements

Peter DragnevMathematical Sciences, IPFW

• Well-distributed points on the sphere

• Motivation from Chemistry, Biology, Physics

• Survey of results in the literature

• New results for logarithmic points

Let N ={X1 , … , XN} Sd-1 - the unit sphere in

d-dimensional space Rd : { X = (x1 , x2 , … , xd ) }.

Well-distributed points on the sphere

• For d=2, this problem is simple.The solution is up to rotations the roots of unity.

How do we distribute “well” the points?

Reason - direction and order

Sd-1 = { X : x1 2 + x2 2 + … + xd 2=1 }

• For d3 - 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.

Minimum Energy Problem on the SphereGiven an N-point configuration N ={X1 , … , XN} on

the sphere S2 we define its generalized energy as

E ( N )= Σij |Xi - Xj|.

• Maximize E ( N ) when >0;

• Minimize it when <0;

• When =0 minimize the logarithmic energy

E ( N )= Σij log(1/|Xi - Xj|)

or maximize the product P( N )=exp(-E ( N )).

Denote the extremal energy with E (N,d) .

= 1 Except for small N a long standing open problem in discrete geometry (L. Fejes Toth - 1956)

= -1 Thompson problem; recent discovery of fullerenes attracted the attention of researchers in chemistry, physics, crystallography. Answer known for N=1-4, 6, 12. (Fekete Points)

= 0 The problem was posed by L. L. Whyte in 1952. Until recently the answer was known only for N=1-4. (Logarithmic Points)

- Tammes problem; maximize the minimum distance between any pair of points. Known for N= 1-12 and 24.(Best packing points)

• Fullerenes (Buckyballs)Large carbon molecules discovered in 1985 by Richard Smalley et. el.

C60 C70

Motivation from Chemistry

NanotechnologyNanowire -- “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).

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.

Motivation from Biology

A “disco ball” in space

Starshine 3 satellite was launched in 2001 to study Earth’s 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.

N electrons orbit the nucleus

Electrons repel

Equilibrium will occur at minimum energy

Electrons in Equilibrium

Problem:If Coulomb’s Law is assumed, then we minimize the sum of the reciprocals of the mutual distances, i. e.

Σij|Xi - Xj|-1

over all possible configurations of N points X1 , … , XN on the unit sphere. This corresponds to the case = -1.

Motivation from Physics

32 Electrons

In Equilibrium

122 Electrons

Distribution of Dirichlet Cells (School Districts)

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.

Dj:={Xє S2: |X-Xj|=mink |X-Xk|} j=1,…,N

There has to be exactly 12 pentagons in a soccer ball design.

Q. How are D-cells distributed for large N ?

Buckyball under the Lion’s Paw

Survey of results for small N

Tammes problem (best packing) = -The solution is known for N=1,2, …, 12, 24.

N=4 - regular tetrahedron

N=5,6 - south, north and the rest on the equator

N=8 - skewed cube

N=12 - regular icosahedron (12 pentagons)

Thompson problem = -1 (known for N=4,6,12)

N=4 - regular tetrahedron N= 6 - regular octahedron N=12 - regular icosahedron

Whyte’s problem = 0 (Logarithmic points)

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 E ( N )= Σij log(1/|Xi - Xj|) is

called optimal configuration. The points are referred

to as logarithmic points.

N=4 - regular tetrahedron - {1,3} N=12 - regular icosahedron - {1,5,5,1} [A] ‘96 N=6 - regular octahedron - {1,4,1} [KY] ‘97 N=5 - D3h {1,3,1} - [D+Legg+Townsend] ‘01

Let N ={X1 , … , XN } be an optimal configuration on

the sphere S2. Define dN :=minij |Xi - Xj|.

Logarithmic points - new results

Theorem 2 (Dragnev ‘02)1

2

NdN

Remark: Rakhmanov, Saff, Zhou were first to show the

separation condition with constant 3/5. Dubickas obtained a

constant 7/4. It is not known whether exists.NN dNlim

Theorem 1 (Dragnev, Legg, Townsend - ‘01)

{1,3,1} is the only optimal 5-point configuration.

Let N={X1, X2 … , XN } Sd-1 be optimal, i.e.

E ( N )= E (N,d). Derivative conditions on the energy

functional (when N-1 points are fixed) yield:

d+2 configuration on Sd-1

Proposition (Properties of optimal configurations) Let N={X1, X2 … , XN } be an optimal configuration. Then

NiOXXX

XXi

ij ij

ij ,...,1,2

1N

|| 2

(i)

O is center of mass for {X1, X2,… , XN }. (ii)

(iii) NiNXXij

ij ,...,1,2|| 2

Corollary: The regular d-simplex is the only optimal d+1-configuration on Sd-1 for any 0.

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

d<N, and for dN, E (N,d) = E (N,N-1).

Corollary Optimal configurations are non-degenerate.

Theorem 4 (Dragnev ‘02) If N=d+2, 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 N=5, d=3. By Theorem 3, optimal configurations will satisfy at least one of the following: (a) degenerate {5}

(b) EA=EB=EC=ED {1,4}

(c) Every vertex has a mirror related partner. In this case we arrive at A~B~C and D~E {1,3,1}

Comparing {5}, {1,4}, and {1,3,1} proves Theorem 2.

In progress: If N=d+2 the optimal configurations is

unique up to rotations and consists of two mutually

orthogonal regular [d/2]- and [(d+1)/2]-simplexes.

So, 2/])2/)1)([(1]2/)1([(2/])2/)([1]2/([

2/)1)(2(

]2/)1[(

1]2/)1[(

]2/[

1]2/[2

dddd

dd

d

d

d

d

Examples • N=4, d=2; diagonals of a square, form orthogonal simplexes.• N=5, d=3; equilateral triangle on Equator and diameter.• N=6, d=4; (cos k/3, sin k/3,0,0); (0,0,cos k/3, sin k/3),

k=0,1,2;