Abstract
We study 4 × 4-labyrinth fractals, which are self similar dendrites. For all 4 × 4-labyrinth fractals we answer the question, whether there is a curve of finite length in the fractal from one point to another point in the fractal. In the first case, between any two points in the fractal there is a unique arc a, the length of a is infinite, and the set of points, where no tangent exists to a, is dense in a. In the second case, there are also pairs of points between that there is a unique arc of finite length.
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