The Double Bubble Problem on the Flat Two-Torus

Optimal Packings of Two to Four Equal Circles on Any Flat Torus

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.

两个最对称的平坦环面作为八顶点纸环面

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

本文构造了实现正方形和六边形这两个最对称平坦环面的八顶点纸环面，并证明其附近所有形状均可由八顶点纸环面实现。

A Simple 3D Isometric Embedding of the Flat Square Torus

Start with Gott (2019)'s envelope polyhedron (Squares-4 around a point): a unit cube missing its top and bottom faces. Stretch by a factor of 2 in the vertical direction so its sides become (2x1 unit) rectangles. This has 8 faces (4 exterior, 4 interior), 8 vertices, and 16 edges. F-E+V = 0, implying a (toroidal) genus = 1. It is isometric to a flat square torus. Like any polyhedron it has zero intrinsic Gaussian curvature on its faces and edges. Since 4 right angled rectangles meet at each vertex, there is no angle deficit and zero Gaussian curvature there as well. All meridian and latitudina…

Optimal packings of congruent circles on a square flat torus

We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason - the problem of "super resolution of images." We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.

On the oscillation rigidity of a Lipschitz function on a high-dimensional flat torus

Given an arbitrary $1$-Lipschitz function $f$ on the torus $\mathbb{T}^n $, we find a $k$-dimensional subtorus $M \subseteq \mathbb{T}^n$, parallel to the axes, such that the restriction of $f$ to the subtorus $M$ is nearly a constant function. The $k$-dimensional subtorus $M$ is chosen randomly and uniformly. We show that when $k \leq c \log n / (\log \log n + \log 1/\varepsilon)$, the maximum and the minimum of $f$ on this random subtorus $M$ differ by at most $\varepsilon$, with high probability.

平坦环面按较小直径部分的最优划分

Optimal partitions of the flat torus into parts of smaller diameter

本文研究将二维平坦环面划分为 $m$ 个部分以最小化最大直径的问题，对 $m \leq 25$ 给出数值估计，并提出多种方法，其中 $m=3$ 时用拓扑推理证明精确值。