Chuanming Zong - Characterization of the Three-Dimensional Multiple Tiles

This talk was part of the workshop on "Optimal Point Configurations on Manifolds" held at the ESI January 10 to 21, 2022.

In 1885, Fedorov characterized the three-dimensional lattice tiles. They are parallelotopes, hexagonal prisms, rhombic dodecahedra, elongated dodecahedra, or truncated octahedra. Through the works of Minkowski, Voronoi, Delone, Venkov and McMullen, we know that, in all dimensions, every translative tile is a lattice tile.

Recently, Mei Han, Kirati Sriamorn, Qi Yang and Chuanming Zong have made a series of discoveries in multiple tilings in two and three dimensions. In particular, in three dimensions, they proved that, if a convex body can form a two, three or fourfold translative tiling, it must be a lattice tile (a parallelohedron). In other words, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron. In this talk, we will report this progress.

Видео Chuanming Zong - Characterization of the Three-Dimensional Multiple Tiles канала Erwin Schrödinger International Institute for Mathematics and Physics (ESI)