Extremal Properties of 0/1-Polytopes

We provide lower and upper bounds for the maximal number of facets of a d-dimensional 0/1-polytope, and for the maximal number of vertices that can appear in a two-dimensional projection (``shadow') of such a polytope. Received June 14, 1996, and in revised form September 30, 1996.     

Visage—Visualization of algorithms in discrete mathematics

Which route should the garbage collectors' truck take? Just a simple question, but also the starting point of an exciting mathematics class. The only “hardware” you need is a city map, given on a sheet of paper or on the computer screen. Then lively discussions will take place in the classroom on how to find an optimal routing for the truck. The aim of this activity is to develop an algorithm that constructs Eulerian tours in graphs and to learn about graphs and their properties. This teaching sequence, and those stemming from discrete mathematics, in particular combinatorial optimization, are ideal for training problem solving skills and modeling—general competencies that, influenced by the German National Standards, are finding their way into curricula. In this article, we investigate how computers can help in providing individual teaching tools for students. Within the Visage project we focus on electronic activities that can enhance explorations with graphs and guide studients even if the teacher is not available—without taking away freedom and creativity. The software package is embedded into a standard DGS, and it offers many pre-built and teacher-customizable tools in the area of graph algorithms. Until now, there are no complete didactical concepts for teaching graph algorithms, in particular using new media. We see a huge potential in our methods, and the topic is highly requested on part of the teachers, as it introduces a modern and highly relevant part of mathematics into the curriculum.     

Every Simplicial Polytope with at Most d + 4 Vertices Is a Quotient of a Neighborly Polytope

We show that every simplicial d-polytope with d+4 vertices is a quotient of a neighborly (2d+4)-polytope with 2d+8 vertices, using the technique of affine Gale diagrams. The result is extended to matroid polytopes. Received September 27, 1995.     

Experimental mathematics and proofs in the classroom

Experimental mathematics is a serious branch of mathematics that starts gaining attention both in mathematics education and research. We given examples of using experimental techniques (not only) on the classroom. At first sight it seems that introducing experiments will weaken the formal rules and the abstractness of mathematics that are considered a valuable contribution to education as a whole. By putting proof and experiment side by side we show how this can be avoided. We also highlight consequences of experimentation for educational computer software.