Euclidean Geometry: a summary of some current research
In his article in the January 1991 issue of SIAM News, "Euclidean Geometry Alive and Well," Barry Cipra describes three recent results in Euclidean geometry.
The first, an optimization problem called the Steiner ratio conjecture, was solved by Ding Zhu Du and Frank Hwang. The Steiner ratio conjecture involves objects called trees, which are line segments that connect a given (finite) set of points, without ever looping back on itself. Thus, in a tree, if you follow a path, you can never get back to your starting point, or even to a point you have already moved through. A tree can be thought of as an optimization problem: how can you connect all the dots with the shortest length of string? Obviously, one thing you don't want to do is double back on yourself (this would be redundant), and that is exactly what having a tree insures will not happen. The question then becomes: What is the shortest tree one can have connecting all the dots?
But this is not exactly the question that the Steiner ratio conjecture answers. The Steiner ratio conjecture has to do with the ratio of the shortest tree connecting a set S of points, and the shortest Steiner tree connecting the same set S of points. Steiner trees are different from "normal" trees because in connecting the dots, you can add extra dots. Obviously, normal trees are just special cases (no extra points) of Steiner trees. Why add the extra points? They can make the tree length shorter, as this diagram illustrates.
a a | /\
| / \ | d / \ / \ / \ / \ / \ / \ / \ b c b c
Let's say that abc is an equilateral triangle with sides of length 1. The Steiner tree on the right, formed by adding the center of the triangle, d, as a point, is shorter than the tree using only points a, b, and c. The normal tree has length 2; the Steiner tree has length Sqrt(3). It turns out that this ratio, Sqrt(3)/2, is the smallest that the ratio can be, provided that you are measuring the shortest Steiner tree and the shortest regular tree possible (these do exist).
The second result that Cipra discusses is an algorithm for triangulating large polygons (large meaning many (n) vertices). Bernard Chazelle has found an algorithm that will triangulate (cut up into triangles, with certain rules about what kind of intersections the triangles can have) a polygon in an amount of time proportional to the number of vertices that the polygon has. Since a more naive approach yields a time proportional to the square of the number of vertices, this algorithm makes it much less time consuming for a computer to triangulate a polygon.
The third result Cipra summarizes is the long-held conjecture that the best way to stack spheres is the "face centered cubic lattice packing" or, in layman's terms, the way grocers stack oranges. Wu-Yi Hsiang proves this problem that Kepler proposed in 1611. However,as the problem has such a long history, mathematicians will be going over Hsiang's proof with a fine tooth comb, and, right now at least, the jury is not in on whether or not Hsiang's proof is valid.
For further details see "Euclidean Geometry Alive and Well," by Barry A. Cipra, in the January 1991 issue of SIAM News.