Some external problems in geometry

AbstractThe question of how often the same distance can occur between k distinct points in n-dimensional Euclidean space En has been extensively studied by Paul Erdös and others. Sir Alexander Oppenheim posed the somewhat similar problem of investigating how many triangles with vertices chosen from among k points in En can have the same non-zero area. A subsequent article by Erdös and Purdy gave some preliminary results on this problem. Here we carry that work somewhat further and show that there cannot be more than ck3−ϵ triangles with the same non-zero area chosen from among k points in E5, where ϵ is a positive constant. Since there can be ck3 such triangles in E6, the result is in a certain sense best possible. The methods used are mainly combinatorial and geometrical. A main tool is a theorem on generalized graphs due to Paul Erdös

Spheres tangent to all the faces of a simplex

AbstractThere are at most 2n spheres tangent to all n + 1 faces of an n-simplex. It has been shown that the minimum number of such spheres is 2n − c(n, 12(n + 1)) if n is odd and 2n − c(n, 12(n + 1)) if n is even. The object of this note is to show that this result is a consequence of a theorem in graph theory

A Bichromatic Incidence Bound and an Application

We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between the k red points and m hyperplanes spanned by all n points provided that m = \Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal and Aronov. We use this incidence bound to prove that a set of n points, no more than n-k of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy's on the number of hyperplanes spanned by a set of points in dimensions higher than 3, and present new conjectures not subject to the counterexample.Comment: 12 page

Fabrication of micro-optical devices

We have fabricated a variety of micro-optic components including Fresnel and non-Frensel lenses, off-axis and dispersive lenses with binary stepped contours, and analog contours. Process details for all lens designs fabricated are given including multistep photolithography for binary fabrication and grayscale mask photolithography for analog fabrication. Reactive ion etching and ion beam milling are described for the binary fabrication process, while ion beam milling was used for the analog fabrication process. Examples of micro-optic components fabricated in both Si and CdTe substrates are given

Lines, Circles, Planes and Spheres

Let $S$ be a set of $n$ points in $\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k \binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2})$. For similar conditions and sufficiently large $n$, (inspired by the work of P. D. T. A. Elliott in \cite{Ell67}) we also show that the number of spheres determined by $n$ points is at least $1+\binom{n-1}{3}-t_3^{orchard}(n-1)$, and this bound is best possible under its hypothesis. (By $t_3^{orchard}(n)$, we are denoting the maximum number of three-point lines attainable by a configuration of $n$ points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles.Comment: 37 page