Unit squares intersecting all secants of a square

Let S be a square of side length s > 0. We construct, for any sufficiently large s, a set of less than 1.994 s closed unit squares whose sides are parallel to those of S such that any straight line intersecting S intersects at least one square of S. It disproves L. Fejes Tòth's conjecture that, for integral s, there is no such configuration of less than 2s -1 unit squares

The opaque square

The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916. We call such a set {\em opaque} or a {\em barrier} for the square. The shortest known barrier has length $\sqrt{2}+ \frac{\sqrt{6}}{2}= 2.6389\ldots$. The current best lower bound for the length of a (not necessarily connected) barrier is $2$, as established by Jones about 50 years ago. No better lower bound is known even if the barrier is restricted to lie in the square or in its close vicinity. Under a suitable locality assumption, we replace this lower bound by $2+10^{-12}$, which represents the first, albeit small, step in a long time toward finding the length of the shortest barrier. A sharper bound is obtained for interior barriers: the length of any interior barrier for the unit square is at least $2 + 10^{-5}$. Two of the key elements in our proofs are: (i) formulas established by Sylvester for the measure of all lines that meet two disjoint planar convex bodies, and (ii) a procedure for detecting lines that are witness to the invalidity of a short bogus barrier for the square.Comment: 23 pages, 8 figure

On the convex hull of a space curve

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We express the degree of this surface in terms of the degree, genus and singularities of the curve. We present algorithms for computing their defining polynomials, and we exhibit a wide range of examples.Comment: 19 pages, 4 figures, minor change

Did Lobachevsky Have A Model Of His "imaginary Geometry"?

The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to provide a sound reading of Lobachevsky's geometrical works. Although the modern notion of model of a given theory has a counterpart in Lobachevsky's writings its role in Lobachevsky's geometrical theory turns to be very unusual. Lobachevsky doesn't consider various models of Hyperbolic geometry, as the modern reader would expect, but uses a non-standard model of Euclidean plane (as a particular surface in the Hyperbolic 3-space). In this paper I consider this Lobachevsky's construction, and show how it can be better analyzed within an alternative non-Hilbertian foundational framework, which relates the history of geometry of the 19th century to some recent developments in the field.Comment: 31 pages, 8 figure

Helpful Homework in Geometry: A Redesigned Circles Unit

Homework has been part of the educational system for many decades. During this time, public opinion has varied greatly on its usefulness in the classroom. Much of the more recent research has focused on the idea that homework can be valuable to students when the assignments are meaningful, as opposed to homework that is assigned with little or no purpose. This research was analyzed to find patterns in the various definitions of meaningful homework. It was found that meaningful homework generally contains the following qualities: brevity, choice, defined purpose, real-world connections, hands-on components, rigor through synthesis, the integration of technology or web-based activities, opportunities for family involvement, and the substantial incorporation of previously taught topics. Based on these findings, a Geometry unit on circles was redesigned to combine these elements into practical lessons, as an example to educators of how to begin making homework more constructive in the mathematics classroom