Ivan Niven

Convex Polygons That Cannot Tile the Plane

The seemingly straightforward task of assigning seats to states according to population runs into several politically unacceptable complications. It is shown that this is a serious problem in the sense that these complications will occur for most population densities. The mathematical reasons for these complications are discussed, and one of them is related to flows on a higher dimensional torus. Finally, a simple apportionment method is suggested.

Lattice points in regions

1. Let S be a bounded set of points in the Euclidean plane with a unit distance defined. If a rectangular coordinate system is imposed, a certain number of points of S are lattice points, i.e. points with integer coordinates. Let m(S) be the minimum number of lattice points of S under all possible choices of the axis system, and M(S) the maximum number. For example if S is a closed disk of diameter one, then mr(S) = 0 and M(S) =2. The definitions of mr(S) and M(S) could be given in terms of a fixed rectangular coordinate system, with the set S being freely rotated and translated in the plane. It will be convenient in the proofs to use sometimes one and sometimes the other of these two formulations. Although the definitions and theorems of this paper are given for 2-dimensional Euclidean space, the generalization to higher dimensions involves no difficulties whatsoever. It is apparent that mr(S) < M(S) for any set S. R. M. Robinson suggested that the strict inequality holds for a nonempty bounded closed set, which is a more general result than we had formulated.

Lattice Point Coverings by Plane Figures

(1967). Lattice Point Coverings by Plane Figures. The American Mathematical Monthly: Vol. 74, No. 4, pp. 353-362.

Special Mathematical Topics

One of the purposes of the mini-conferences at the meeting is to introduce new areas of mathematics which have found their way into the curriculum at some universities and colleges but deserve more attention. At the same time one can consider the question whether it concerns a development in mathematics that could in some way be incorporated in the high school curriculum. In the present case the topic is algebraic coding theory, an area between information theory, combinatories and applied algebra, which has only been around for about 30 years. More than likely at most of the world’s universities there has never been a course in this subject. On the other hand at a few it has been taught for at least 15 years and it is usually received with enthusiasm by the participating students.