On the Steiner Ratio in 3-Space

| The \Steiner minimal tree" (SMT) of a point set P is the shortest network of \wires" which will suuce to \electrically" interconnect P. The \minimum spanning tree" (MST) is the shortest such network when only intersite line segments are permitted. The \Steiner ratio" (P) of a point set P is the length of its SMT divided by the length of its MST. It is of interest to understand which point set (or point sets) in R d have minimal Steiner ratio. In this paper , we introduce a point set in R d which we call the \d-dimensional sausage." The 1 and 2-dimensional sausages have minimal Steiner ratios 1 and p 3=2 respectively. (The 2-sausage is the vertex set of an innnite strip of abutting equilateral triangles. The 3-sausage is an innnite number of points evenly spaced along a certain helix.) We present extensive heuristic evidence to support the conjecture that the 3-sausage also has minimal Steiner ratio (0:784190373377122). Also: We prove that the regular tetrahedron minimizes among 4-point sets to at least 12 decimal places of accuracy. This is a companion paper to D-Z. Du and W.D.Smith: Three disproofs of the Gilbert-Pollak Steiner ratio conjecture in three or more dimensions, submitted Journal of Combinatorial Theory. We have tried to devote this paper more to 3D and the other paper more to general dimension, but the split is not clean.