三角形T对三角形ABC的N -平铺是将ABC写成N个全等于T的三角形的并集的一种方式，仅在它们的边界处重叠。三角形T是“瓦片”。瓷砖可能与ABC相似，也可能不相似。我们希望通过完全表征三元组（ABC，T，N）来理解可能的平铺，使得ABC可以被T N -平铺。特别地，这种理解应该使我们能够指定对于哪个N存在瓦片T和由T进行N-瓦片化的三角形ABC;或者给定N，确定哪些瓦片和三角形可以用于N-瓦片化;或者给定ABC，确定哪些瓦片和N可以用于N-瓦片ABC。这是关于这个问题的四篇论文中的第二篇。在第一篇文章中，我们讨论了ABC与T相似的情况，以及T是直角三角形的情况。本文假设ABC与T不相似，T也不是直角三角形，并且如果T有120度角，则T是等腰三角形。在这些假设下，只有两族平铺。有一个等边三角形ABC由一个等腰瓷砖与底角30度的平铺，并有新发现的“三次平铺”，这是在本系列的第三篇文章中处理。在3α+ 2β= π或3β+ 2α= π且α不是π的有理数倍数的特殊情况下会出现这些问题，其中α和β是瓦片的两个最小角，瓦片的边有有理数比。当瓷砖有一个120度角，而不是等腰的情况下，采取了在随后的文件。我们使用线性代数和初等域论的技巧，在一种情况下，我们使用一些代数数论。我们使用一些计数参数和一些初等几何和三角学。
英文摘要：
An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile".
The tile may or may not be similar to ABC . We wish to understand possible tilings by completely characterizing the triples (ABC, T, N) such that ABC can be N -tiled by T. In particular, this understanding should enable us to specify for which N there exists a tile T and a triangle ABC that is N-tiled by T; or given N, determine which tiles and triangles can be used for N-tilings; or given ABC, to determine which tiles and N can be used to N-tile ABC. This is the second of four papers on this subject. In the first paper, we dealt with the case when ABC is similar to T, and the case when T is a right triangle. In this paper, we assume that ABC is not similar to T, and T is not a right triangle, and furthermore that if T has a 120 degree angle, then T is isosceles.
Under those hypotheses, there are only two families of tilings. There are tilings of an equilateral triangle ABC by an isosceles tile with base angles 30 degrees, and there are newly-discovered "triquadratic tilings", which are treated in the third paper in this series. These arise in the special case that 3α+ 2β= πor 3β+ 2α= πwith αnot a rational multiple of π, where αand βare the two smallest angles of the tile, and the sides of the tile have rational ratios.
The case when the tile has a 120 degree angle and is not isosceles is taken up in a subsequent paper.
We use techniques from linear algebra and elementary field theory, and in one case we use some algebraic number theory. We use some counting arguments and some elementary geometry and trigonometry.
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。
