2. 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

3. Let  N ={X 1, …, X N }  S d-1 - the unit sphere in d-dimensional space R d : { X = (x 1, x 2, …, x d ) }. 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 S d-1 = { X : x 1 2 + x 2 2 + … + x d 2 =1 }

4. 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.

5. Minimum Energy Problem on the Sphere Given an N-point configuration  N ={X 1, …, X N } on the sphere S 2 we define its generalized energy as E  (  N )= Σ i  j |X i - X j | . Maximize E  (  N ) when  >0; Minimize it when  <0; When  =0 minimize the logarithmic energy E  (  N )= Σ i  j log(1/|X i - X j |) or maximize the product P (  N )=exp(-E  (  N )). Denote the extremal energy with E  (N,d).

6.  = 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)

7. Fullerenes (Buckyballs) Large carbon molecules discovered in 1985 by Richard Smalley et. el. C 60 C 70 Motivation from Chemistry

8. 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).

9. 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

10. 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.

11. 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. Σ i  j |X i - X j | -1 over all possible configurations of N points X 1, …, X N on the unit sphere. This corresponds to the case  = -1. Motivation from Physics

12. 32 Electrons In Equilibrium 122 Electrons

13. 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 C 60. D j :={Xє S 2 : |X-X j |=min k |X-X k |} j=1,…,N There has to be exactly 12 pentagons in a soccer ball design. Q. How are D-cells distributed for large N ?

14. Buckyball under the Lion’s Paw

15. 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

16. 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 )= Σ i  j log(1/|X i - X j |) 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 - D 3h {1,3,1} - [D+Legg+Townsend] ‘01

17. Let  N ={X 1, …, X N } be an optimal configuration on the sphere S 2. Define d N :=min i  j |X i - X j |. Logarithmic points - new results Theorem 2 (Dragnev ‘02) 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. Theorem 1 (Dragnev, Legg, Townsend - ‘01) {1,3,1} is the only optimal 5-point configuration.

18. Let  N ={X 1, X 2 …, X N }  S d-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 S d-1 Proposition (Properties of optimal configurations) Let  N ={X 1, X 2 …, X N } be an optimal configuration. Then (i) O is center of mass for {X 1, X 2,…, X N }. (ii) (iii) Corollary: The regular d-simplex is the only optimal d+1- configuration on S d-1 for any  0.

19. Definitions: Configuration is called critical, if it satisfies (i) (recall (i) implies (ii) and (iii)). degenerate, if it does not span R d. A vertex X i is mirror related to X j (X i ~ X j ) if X i X k = X j X k for all k  i,j. Then  N \ {X i, X j } lie in the orthogonal bisector hyperspace of X i X j. 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 d  N, E  (N,d) = E  (N,N-1). Corollary Optimal configurations are non-degenerate.

20. 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.

21. 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, 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;