An $N$-tiling of triangle $ABC$ by triangle $T$ (the ``tile'') is a way of writing $ABC$ as a union of $N$ copies of $T$ overlapping only at their boundaries. Let the tile $T$ have angles $(α,β,γ)$, and sides $(a,b,c)$. This paper takes up the case when $3α+ 2β= π$. Then there are (as was already known) exactly five possible shapes of $ABC$ not similar to $T$: either $ABC$ is isosceles with base angles $α$, $β$, or $α+β$, or the angles of $ABC$ are $(2α,β,α+β)$, or the angles of $ABC$ are $(2α, α, 2β)$. We prove that the tile is rational (the ratios of its sides are rational). We then exhibit several families of previously unknown tilings. We also exhibit, for each shape, a Diophantine equation in $N$ and the parameters of the tile that must have a solution whenever there is a tiling of some triangle of that shape.
By means of these Diophantine equations, some conclusions about the possible values of $N$ are drawn. We prove, for example, that there is no $N$-tiling with $N$ prime when $3α+ 2β= π$, for three of the five shapes; that question remains open for the shape $(2α,β,α+β)$ and for isosceles triangles with base angles $β$. These equations also imply that for each $N$, there is a finite set of possibilities for the tile $(a,b,c)$ and the triangle $ABC$. (Usually, but not always, there is just one possible tile.) For each of the five shapes, we give a construction method that works under certain sufficient conditions, but there remain values of $N$ that satisfy our necessary conditions and for which we do not know whether a tiling exists. All results have been formalized in Lean.
Side-to-side Tiling of the Sphere by Congruent Curvilinear Triangles
Chinese University of Hong Kong（香港中文大学）；Zhejiang Normal University（浙江师范大学）；East China Normal University（华东师范大学）；Jinhua No.4 Middle School（金华市第四中学）；Hong Kong University of Science and Technology（香港科技大学）
R. Nandakumar问是否有一个平铺的平面由成对的不全等面积和周长的三角形。最近，Kupavskii，Pach和Tardos给出了否定的答案。仍然有人可能会要求这个问题的较弱版本，或者这个问题的四边形，五边形或六边形的类似物。第一作者在以前的一篇论文中给出了几个答案。在这里，我们解决了另外三个案例。特别地，我们的主要结果表明，存在单位面积和有界周长的两两不全等三角形的顶点到顶点的平铺。
Equality case in van der Corput's inequality and collisions in multiple lattice tilings
Van der Corput给出了d维原点对称的凸体C的体积上的一个精确界vol（C）\le m 2 ^d，该凸体C的内部有整数格的2 m-1个点。对于m=1，等式情况vol（C）= m 2 ^d的特征等价于通过凸体的平移来特征化平铺的众所周知的问题。令人惊讶的是，到目前为止，对于m \ge 2，还没有相等情况的表征，尽管在1987年Gruber和Lekkerkerker的专著中可以找到对相应表征问题的提示。我们给出了一个明确的特征的平等的情况下，所有的m \ge 2。我们的结果表明，平等的情况下，m \ge 2是更严格的比$m=1$。我们还提出了我们的特点的背景下，多个晶格平铺的后果。
