36 research outputs found
Experimental study of energy-minimizing point configurations on spheres
In this paper we report on massive computer experiments aimed at finding
spherical point configurations that minimize potential energy. We present
experimental evidence for two new universal optima (consisting of 40 points in
10 dimensions and 64 points in 14 dimensions), as well as evidence that there
are no others with at most 64 points. We also describe several other new
polytopes, and we present new geometrical descriptions of some of the known
universal optima.Comment: 41 pages, 12 figures, to appear in Experimental Mathematic
Optimality and uniqueness of the (4,10,1/6) spherical code
Linear programming bounds provide an elegant method to prove optimality and
uniqueness of an (n,N,t) spherical code. However, this method does not apply to
the parameters (4,10,1/6). We use semidefinite programming bounds instead to
show that the Petersen code, which consists of the midpoints of the edges of
the regular simplex in dimension 4, is the unique (4,10,1/6) spherical code.Comment: 12 pages, (v2) several small changes and corrections suggested by
referees, accepted in Journal of Combinatorial Theory, Series
Optimal Discrete Riesz Energy and Discrepancy
The Riesz -energy of an -point configuration in the Euclidean space
is defined as the sum of reciprocal -powers of all mutual
distances in this system. In the limit the Riesz -potential
( the Euclidean distance) governing the point interaction is replaced with
the logarithmic potential . In particular, we present a conjecture
for the leading term of the asymptotic expansion of the optimal
\IL_2-discrepancy with respect to spherical caps on the unit sphere in
which follows from Stolarsky's invariance principle [Proc.
Amer. Math. Soc. 41 (1973)] and the fundamental conjecture for the first two
terms of the asymptotic expansion of the optimal Riesz -energy of points
as .Comment: 8 page
Point configurations that are asymmetric yet balanced
A configuration of particles confined to a sphere is balanced if it is in
equilibrium under all force laws (that act between pairs of points with
strength given by a fixed function of distance). It is straightforward to show
that every sufficiently symmetrical configuration is balanced, but the converse
is far from obvious. In 1957 Leech completely classified the balanced
configurations in R^3, and his classification is equivalent to the converse for
R^3. In this paper we disprove the converse in high dimensions. We construct
several counterexamples, including one with trivial symmetry group.Comment: 10 page
Point configurations that are asymmetric yet balanced
A configuration of particles confined to a sphere is balanced if it is in
equilibrium under all force laws (that act between pairs of points with
strength given by a fixed function of distance). It is straightforward to show
that every sufficiently symmetrical configuration is balanced, but the converse
is far from obvious. In 1957 Leech completely classified the balanced
configurations in R^3, and his classification is equivalent to the converse for
R^3. In this paper we disprove the converse in high dimensions. We construct
several counterexamples, including one with trivial symmetry group.Comment: 10 page
The contact polytope of the Leech lattice
The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits