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Jürgen Richter-Gebert

Publications and source records attributed to Jürgen Richter-Gebert.

13 recordsLinked to original sources
arXiv2026

On Chasles' Quadrilateral Theorem

Chasles' Quadrilateral Theorem is a classical statement about four tangents to a conic that simultaneously circumscribe a circle. In its various formulations, it relates the concurrence of certain lines to the existence of confocal conics or inscribed circles. We show that several classical and modern versions of this theorem are affected by subtle ambiguities arising from multiple solutions in the underlying geometric constructions. These ambiguities often enter through seemingly natural extensions of otherwise correct statements. We provide a systematic analysis of these issues and present coherent formulations of the theorem that avoid these inconsistencies. In particular, we interpret the theorem in a projective framework and relate it to the Cayley-Bacharach theorem, which explains the underlying incidence structure.

math.AG↗
arXiv2026

An almost trivial observation about the icosahedron

We consider the incidence structure formed by the twelve pentagons given by the vertex neighborhoods of the icosahedron. Interpreting this structure purely in terms of coplanarity conditions, we show that -- up to projective equivalence -- it admits exactly two realizations. Both realizations coincide with the vertex set of the regular icosahedron and interpreted as cell complex they correspond to the great dodecahedron and the small stellated dodecahedron. The key step is to reinterpret the configuration via the pentagram map. We prove that any realization gives rise to a pentagon $X$ satisfying a homothety relation $P^2(X)\sim X$, and show that this condition forces $X$ to be an affine image of either a regular pentagon or a regular pentagram. This reduces the problem to a quadratic constraint and explains the rigidity of the configuration.

math.CO↗
arXiv2026

Manifold-based Proving Methods in Projective Geometry

This article compares different proving methods for projective incidence theorems. In particular, a technique using quadrilateral tilings recently introduced by Sergey Fomin and Pavlo Pylyavskyy is shown to be at most as strong as proofs using bi-quadratic final polynomials and thus, also proofs using Ceva-Menelaus-tilings. Furthermore, we demonstrate the equivalence between quadrilateral-tiling-proofs and proofs using exclusively Menelaus configurations. We exemplify the transition between the proofs in several examples in 2D and in 3D.

math.CO↗
arXiv2025

The Smooth Power of the "Neandertal Method"

We describe an algorithmic method to transform a Euclidean wallpaper pattern into a Circle Limit-style picture à la Escher. The design goals for the method are to be mathematically sound, aesthetically pleasing and fast to compute. It turns out that a certain class of conformal maps is particularly well-suited for the problem. Moreover, in our specific application, a very simple method, sometimes jokingly called the "Neandertal Method" for its almost brutal simplicity, proves to be highly efficient, as it can easily be parallelized to be run on the GPU, unlike many other approaches.

math.MG↗
arXiv2025

Hyperbolic Structure of the Equilateral Pentagon

The combinatorial structure of the realization space of the euqilateral pentagon linkage is closely related to a tiling of the hyperbolic plane by right-angled pentagons. In this correspondence lower dimensional faces of the tiling correspond to degenerate realizations of the linkage. We extend this combinatorial correspondence to a full conformal parameterization of the space of all such linkaged controlled by one point in the hyperbolic plane. To do so we exploit the symmetry of the realization space, combine it with the Riemann mapping theorem and a normalization procedure introduced by Springborn. The resulting parameterization is "democratic" in the sense of Yoshida Masaaki: All points are treated exactly equal.

math.DG↗
arXiv2024

When Grünbaum meets Poncelet -- Infinite Classes of Movable $n_4$ Configurations

We study relations between $(n_4)$ incidence configurations and the classical Poncelet Porism. Poncelet's result studies two conics and a sequence of points and lines that inscribes one conic and circumscribes the other. Poncelet's Porism states that whether this sequence closes up after $m$ steps only depends on the conics and not on the initial point of the sequence. In other words: Poncelet polygons are movable. We transfer this motion into a flexibility statement about a large class of $(n_4)$ configurations, which are configurations where 4 (straight) lines pass through each point and four points lie on each line. A first instance of such configurations in real geometry had been given by Grünbaum and Rigby in their classical 1990 paper where they constructed the first known real geometric realisation of a well known combinatorial $(21_4)$ configuration (which had been studied by Felix Klein), now called the Grünbaum-Rigby configuration. Since then, there has been an intensive search for movable $(n_4)$ configurations, but it is very surprising that the Grünbaum-Rigby $(21_4)$ configuration admits nontrivial motions. It is well-known that the Grünbaum-Rigby configuration is the smallest example of an infinite class of $(n_4)$ configurations, the trivial celestial configurations. A major result of this paper is that we show that all trivial celestial configurations are movable via Poncelet's Porism and results about properties of Poncelet grids. Alternative approaches via geometry of billiards, in-circle nets, and pentagram maps that relate the subject to discrete integrable systems are given as well.

math.CO↗
arXiv2024

Explicit Constructions for Poncelet Polygons

We study the geometric structure of Poncelet $n$-gons from a projective point of view. In particular we present explicit constructions of Poncelet $n$-gons for certain $n$ and derive algebraic characterisations in terms of bracket polynomials. Via the connections of Poncelet polygons and $(N_4)$-configurations, the results of this article can be used to construct a large class of specific movable $(N_4)$-configurations, the trivial celestial 4-configurations, which up to this point were all thought to be rigid and to require regular polygons for their construction.

math.CO↗
arXiv2018

Bringing Together Dynamic Geometry Software and the Graphics Processing Unit

We equip dynamic geometry software (DGS) with a user-friendly method that enables massively parallel calculations on the graphics processing unit (GPU). This interplay of DGS and GPU opens up various applications in education and mathematical research. The GPU-aided discovery of mathematical properties, interactive visualizations of algebraic surfaces (raycasting), the mathematical deformation of images and footage in real-time, and computationally demanding numerical simulations of PDEs are examples from the long and versatile list of new domains that our approach makes accessible within a DGS. We ease the development of complex (mathematical) visualizations and provide a rapid-prototyping scheme for general-purpose computations (GPGPU). The possibility to program both CPU and GPU with the use of only one high-level (scripting) programming language is a crucial aspect of our concept. We embed shader programming seamlessly within a high-level (scripting) programming environment. The aforementioned requires the symbolic process of the transcompilation of a high-level programming language into shader programming language for GPU and, in this article, we address the challenge of the automatic translation of a high-level programming language to a shader language of the GPU. To maintain platform independence and the possibility to use our technology on modern devices, we focus on a realization through WebGL.

cs.MS↗
arXiv2014

Cayley-Bacharach Formulas

The Cayley-Bacharach Theorem states that all cubic curves through eight given points in the plane also pass through a unique ninth point. We write that point as an explicit rational function in the other eight.

math.AG↗
arXiv2009

Geometry of Numerical Complex Time Integration

We are studying Runge-Kutta methods along complex paths of integration from a geometric point of view. Thereby we derive special complex time grids, which applied to the problem of integrating a linear autonomous system of ordinary differential equations, can be used to achieve a classical superconvergence effect. The approach is also adapted for arbitrary ODEs. Furthermore we draw a connection from our geometric reasoning to the class of composition methods with complex coefficients. Thereby, our main goal is to introduce a new point of view on these methods.

math.NA↗
arXiv2003

First steps in tropical geometry

Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus' Theorem.

math.AG↗
arXiv1995

Realization spaces of 4-polytopes are universal

Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic set~$V$ defined over~$\Z$, there is a $4$-polytope $P(V)$ whose realization space is ``stably equivalent'' to~$V$. This implies that the realization space of a $4$-polytope can have the homotopy type of an arbitrary finite simplicial complex, and that all algebraic numbers are needed to realize all $4$- polytopes. The proof is constructive. These results sharply contrast the $3$-dimensional case, where realization spaces are contractible and all polytopes are realizable with integral coordinates (Steinitz's Theorem). No similar universality result was previously known in any fixed dimension.

math.MG↗