arXiv subjects

Fabian Lander

Publications and source records attributed to Fabian Lander.

5 recordsLinked to original sources

arXiv2026

The two most symmetric flat tori as eight-vertex paper tori

A paper torus is a polyhedral torus embedded in $\mathbb{R}^3$ whose intrinsic metric is locally Euclidean, a torus folded from a flat sheet of paper. Many constructions are known, from dozens of vertices into the thousands, and it is natural to ask how few suffice. Schwartz proved that no paper torus has seven vertices, and built one with eight. Doyle and Schwartz then built paper tori realizing almost every shape of flat torus with eight vertices, their construction missing exactly two rays in moduli space, all the rectangular tori, and the rhombic tori of aspect ratio at least $\sqrt3$. These rays end at the two most symmetric flat tori, the square torus and the hexagonal torus. We build an eight-vertex paper torus realizing each of them. We also prove that eight-vertex paper tori realize every shape near these two, since the flat configurations around both form a smooth seventeen dimensional manifold on which the modulus is a submersion.

math.MG↗

arXiv2026

Illustrating Hyperbolic Surfaces with Mesh Embeddings

Hyperbolic geometry exhibits geometric phenomena, such as fast area growth, that are difficult to visualize faithfully in Euclidean space, and which standard models like the Poincar\'e disk can obscure. To bring hyperbolic geometry to life, we embed hyperbolic surfaces in Euclidean space by discretizing the surfaces into meshes, and minimizing a distortion energy so that the edge lengths in the embeddings match those in the hyperbolic plane. The resulting surfaces buckle and ruffle to accommodate the extra area, making visible what flat models hide. We present exemplary illustrations, such as embedded disks, equidistant strips, diverging geodesics, and also artistic organic-like renders. We discuss our use of these models, as renders and 3D prints, in research talks, public engagement, outreach, and education.

math.HO↗

arXiv2026

Symplectic Tiling Billiards on Complete Affine Tori

In 2023, Richard Schwartz introduced a new dynamical system which is a marriage of two types of familiar billiards, tiling billiards and symplectic billiards. In this paper we investigate this dynamical system played on tilings of the plane which arise from non-Euclidean geometries on the torus. We review the affine analogue of the flat conformal structures on the torus through the work of Oliver Baues and William Goldman, and define an open subset of this deformation space corresponding to markings of complete affine tilings of the plane. We make this definition precise, and provide algebraic conditions on the symmetries of the tiling to define it. We then analyze the dynamics of symplectic tiling billiards played on these types of tilings and investigate the long-term dynamics of the system to prove a stability result concerning divergent trajectories. The divergence is defined in terms of geometric invariants arising from the tiling symmetry group. We argue that in some sense this divergence is a consequence of the tiles of a non-Euclidean tiling becoming 'thin' as one moves far away in the tiling. To do so we introduce a notion of thinness that is well adapted to the non-Euclidean affine tilings.

math.DS↗

arXiv2025

Immersive Visualization of Flat Surfaces Using Ray Marching

We present an effective method for visualizing flat surfaces using ray marching. Our approach provides an intuitive way to explore translation surfaces, mirror rooms, unfolded polyhedra, and translation prisms while maintaining computational efficiency. We demonstrate the utility of the method through various examples and provide implementation insights for programmers. Finally, we discuss the use of our visualizations in outreach. We make our simulations and code available online.

cs.GR↗

arXiv2024

Symplectic billiards for pairs of polygons

We introduce symplectic billiards for pairs of possibly non-convex polygons. After establishing basic properties, we give several criteria on pairs of polygons for the symplectic billiard map to be fully periodic, i.e. $\textit{every}$ orbit is periodic. The first fully periodic examples were discovered by Albers-Tabachnikov [AT18] and Albers-Banhatti-Sadlo-Schwartz-Tabachnikov in [ABS+25]. Our criteria allow us to construct a plethora of new examples. Moreover, we provide an example of a pair of polygons where the symplectic billiard map is fully periodic while having orbits of arbitrarily large period. After giving a class of examples which provably have isolated periodic orbits (and are thus not fully periodic) we exhibit the first example without any periodic orbits at all. It is open whether having no periodic orbits at all is possible in the single polygon setting. Finally, we prove that if one replaces polygons by smooth, strictly convex curves then there are always infinitely many periodic orbits.

math.DS↗