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Topic: ABS: Current Research in Euclidean Geometry
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Kathie Barnes

Posts: 21
Registered: 12/3/04
ABS: Current Research in Euclidean Geometry
Posted: Sep 10, 1992 10:26 AM
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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

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.

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