This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

Algorithms in Combinatorial Geometry

Injectivity Sets for the Radon Transform over Circles and Complete Systems of Radial Functions

The Three Color Problem is: Under what conditions can the regions of a planar map be colored in three colors so that no two regions with a common boundary have the same color? This paper describes the origin of the Three Color Problem and virtually all the major results and conjectures extant in the literature. I am deeply grateful to W.T. Tutte and Horst Sachs for thorough readings of an earlier version of the manuscript; each had subtle corrections and excellent suggestions. I would also like to thank: Noga Alon, Dan Archdeacon, Frank R. Bernhart, Avrim Blum, Vasek Chvatal, Michael Fellows, Steve Fisk, Herbert Fleischner, Branko Grunbaum, David Johnson, Felix Lazebnik, David Petford, Arvind Rajan, Kenneth H. Rosen, Richard E. Stone (who suggested the paper's title), Bjarne Toft, loan Tomescu, Craig A. Tovey, Peter Ungar, and two anonymous referees. The accuracy and presentation of the material in this paper is of course the responsibility of the author alone.

The State of the Three Color Problem

The allowable sequence associated to a configuration of points was first developed by the authors in order to investigate what combinatorial structure lay behind the Erdős-Szekeres conjecture (that any 2 n-2 + 1 points in general position in the plane contain among them n points which are in convex position) Though allowable sequences did not lead to any progress on this ancient problem, there did emerge an object that had considerable intrinsic interest, that turned out to be related to some other well-studied structures such as pseudoline arrangements and oriented matroids, and that had as well a combinatorial simplicity and suggestiveness which turned out to be effective in the solution of several other classical problems These connections and applications are discussed in Sections 2, 3, and 4 of this paper

Allowable Sequences and Order Types in Discrete and Computational Geometry

Semispaces of configurations, cell complexes of arrangements

2N Noncollinear points determine at least 2N directions

Freak Theorem About Functions on a Sphere

Advanced Problems: 5642,5665-5670

In a recent article in the Monthly [2] Peter Lax proved a mixing property of the pedal mapping P which maps the set of angles of a triangle into the angles of the pedal triangle. His result tells us that if we start with almost any triangle, the sequence of successive pedal triangles contains triangles with shapes arbitrarily close to all possible shapes. In this note we represent the pedal mapping as a shift. We could have stopped there and referred to books on ergodic theory for proofs of mixing properties of shift mappings but we thought it would help many readers to give brief derivations of what is needed here, using only the elements of measure theory and probability theory. Lax follows Kingston and Synge [1] in representing a triangle of given shape as a point x in an equilateral triangle E = A PQR. The barycentric coordinates of x are the angles of the triangle in units of r. In this representation the pedal mapping is the result of the following operations. Connect the midpoints of the sides of E. Denote the four congruent triangles into which E is subdivided by EP, EQ, ERand EM where EP is adjacent to P etc. On the interior of a corner triangle the pedal mapping P is the dilatation by a factor 2 which maps the interior of the corner triangle onto the interior of E. On the interior of EM, P is a reflection in the centroid of E combined with a dilatation by 2. The boundaries of the small triangles correspond to right angled triangles and degenerate triangles for which the pedal triangle is degenerate and P is undefined. We want to consider iterates of P. The mapping P is not defined on the boundaries of the four small triangles. Its square is not defined on the preimages of these, which are the boundaries of the subdivisions of EP1,..., EM, etc. We get rid of these loose ends by restricting P to the set E' obtained by removing from E its boundary points and the boundary points of all triangles obtained by successive subdivisions. The removed po,ints form a set of measure 0. Let us choose units so that the measure of E' is 1, so that we can employ the language of probability theory. The mixing property we are going to prove is the following.

http://www.jstor.org/stable/pdfplus/2324326.pdf

Peter Ungar

Papers

2N Noncollinear points determine at least 2N directions

Freak Theorem About Functions on a Sphere

Advanced Problems: 5642,5665-5670

Mixing property of the pedal mapping

Dissections and Intertwinings of Graphs