We consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation-reversing three-interval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We observe that, for a particular choice of triangle tiling, certain trajectories appear to approach the Rauzy fractal, under rescaling.

中文翻译：

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三角形平铺上的平铺台球以及区间交换变换

我们考虑三角形平铺中光线的动力学，其中三角形是透明的，并且相邻三角形具有相等但相反的折射率。我们发现，三角形平铺上的轨迹的行为是通过圆上的方向反转三区间交换变换来描述的，并且给定三角形平铺上的所有轨迹的行为是通过多边形交换变换来描述的。我们观察到，对于三角形平铺的特定选择，在重新缩放时某些轨迹似乎接近劳齐分形。