We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each “hole” is bounded by three pairwise tangent discs are called triangulated. There are 164 pairs $(r,s)$, $1>r>s$, allowing triangulated packings by discs of radii 1, r and s. In this paper, we enhance existing methods of dealing with maximal-density packings in order to find ternary triangulated packings which maximize the density among all the packings with the same disc radii. We showed for 16 pairs that the density is maximized by a triangulated ternary packing; for 16 other pairs, we proved the density to be maximized by a triangulated packing using only two sizes of discs; for 45 pairs, we found non-triangulated packings strictly denser than any triangulated one; finally, we classified the remaining cases where our methods are not applicable.

我们考虑平面三元盘填料，即使用三种不同半径的盘的填料。每个“孔”由三个成对的相切圆盘界定的填料称为三角填料。有164对$（r,s）$,$1>r>s$，允许由半径为 1、 r和s的盘组成的三角填料。在本文中，我们增强了处理最大密度填料的现有方法，以便找到使具有相同圆盘半径的所有填料中的密度最大化的三元三角填料。我们展示了 16 对三角三元填料的密度最大化；对于其他 16 对，我们证明了仅使用两种尺寸的圆盘即可通过三角填充来最大化密度；对于 45 对，我们发现非三角填料的密度严格高于任何三角填料；最后，我们对我们的方法不适用的其余情况进行了分类。