Ulrich H. Kortenkamp

Extremal Properties of 0/1-Polytopes

Abstract. 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.

Geometry teaching in wireless classroom environments using Java and J2ME

Abstract Interactive geometry software is established as an important tool in geometry and math education. We present our research on possible ways to use such software in wireless classroom environments. In particular, we address user interface issues on portable devices and describe how we maintain a common code base for both desktop and mobile environments, thus increasing the stability of the application. We also report on our empirical data comparing different Java virtual machines that are available for portable devices using a prototype implementation of the Interactive Geometry Software Cinderella for J2ME.

The Future of Mathematical Software

This article gives my very personal view of the development of (mathematical) software in the past and in the future. It is based both on my experiences as a user and as an author of math software [1], and also as a non-software-using mathematician.

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

Abstract. 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.

A Dynamic Setup for Elementary Geometry

In this article we survey the theoretical background that is required to build a consistent and continuous setup of dynamic elementary geometry. Unlike in static elementary geometry in dynamic elementary geometry the elements of a construction are allowed to move around as long as the geometric constraints intended by the construction are not violated. A typical problem in such a scenario is to resolve ambiguous situations that arise from geometric operations like intersecting a circle and a line. After introducing a formal framework for dealing with dynamic geometric constructions, we will demonstrate that a suitable resolution of these ambiguities requires the consideration of complex projective spaces. We will discuss several aspects where one can benefit from such a rather general approach. Finally, we will sketch some proofs that show that several fundamental algorithmic problems arising in such a context are NP-hard or even harder.

Decision Complexity in Dynamic Geometry

Geometric straight-line programs [5,9] can be used to model geometric constructions and their implicit ambiguities. In this paper we discuss the complexity of deciding whether two instances of the same geometric straight-line program are connected by a continuous path, the Complex Reachability Problem.

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.

Interactive Geometry with Cinderella

The primary and most fundamental usage of Cinderella still is interactive geometry, often also called dynamic geometry. With a few mouse clicks you can generate constructions from elementary geometry, projective geometry or even hyperbolic geometry and fractals. The main feature of Cinderella is the so called move mode, sometimes also called drag mode. This mode gives you much more than an ordinary geometric drawing tool. You can drag the base elements of your construction and while you do this all the dependent elements follow accordingly. Using move mode you can explore many geometric constructions and observe effects and theorems on an experimental level.

Euklidische und Nicht-Euklidische Geometrie in Cinderella

ZusammenfassungWir erinnern an die Konzeption euklidischer und nicht-euklidischer Geometrie durch Klein und Cayley und weisen ihre Bedeutung für moderne Software zur interaktiven (oder dynamischen) Geometrie hin. Insbesondere zeigt es sich, dass sich komplexe Zahlen vorzüglich zu einer allgemein Behandlung hyperbolischer, elliptischer, euklidischer und anderer Geometrie eignen.AbstractWe recall the concepts of Euclidean and non-Euclidean Geometry in the sense of Klein and Cayley and illustrate their importance for modern interactive (dynamic) geometry software. In particular, the use of complex numbers turns out to be the key to the most general approach to measurements in hyperbolic, elliptic, Euclidean and other geometries.

Copy, Paste and Macros

The macro concept of Cinderella differs a little from macros available in other geometry software. We describe ways of using the macro capabilities below.