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GEODESICS IN THE SIERPINSKI CARPET AND MENGER SPONGE
Fractals ( IF 4.7 ) Pub Date : 2020-07-13 , DOI: 10.1142/s0218348x20501200
ETHAN BERKOVE 1 , DEREK SMITH 1
Affiliation  

In this paper, we study geodesics in the Sierpinski carpet and Menger sponge, as well as in a family of fractals that naturally generalize the carpet and sponge to higher dimensions. In all dimensions, between any two points we construct a geodesic taxicab path, namely a path comprised of segments parallel to the coordinate axes and possibly limiting to its endpoints by necessity. These paths are related to the skeletal graph approximations of the Sierpinski carpet that have been studied by many authors. We then provide a sharp bound on the ratio of the taxicab metric to the Euclidean metric, extending Cristea’s result for the Sierpinski carpet. As an application, we determine the diameter of the Sierpinski carpet taken over all rectifiable curves. For other members of the family, we provide a lower bound on the diameter taken over all piecewise smooth curves.

中文翻译:

谢尔宾斯基地毯和门格尔海绵中的测地线

在本文中,我们研究了谢尔宾斯基地毯和门格尔海绵中的测地线,以及一系列自然地将地毯和海绵推广到更高维度的分形。在所有维度上,我们在任意两点之间构建了一条测地出租车路径,即一条由平行于坐标轴的线段组成的路径,并且可能根据需要限制到其端点。这些路径与许多作者研究过的谢尔宾斯基地毯的骨架图近似有关。然后,我们对出租车度量与欧几里得度量的比率提供了一个明确的界限,扩展了 Cristea 对 Sierpinski 地毯的结果。作为一个应用程序,我们确定了所有可校正曲线上的谢尔宾斯基地毯的直径。对于其他家庭成员,
更新日期:2020-07-13
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