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Distances in Sierpiński graphs and on the Sierpiński gasket. (English) Zbl 1275.28007

Author’s abstract: The well-known planar fractal called the Sierpiński gasket can be defined with the help of a related sequence of graphs \(\{G_n\}_{n\geq 0}\), where \(G_n\) is the \(n\)th Sierpiński graph, embedded in the Euclidean plane. We prove a geometric criteria that allows us to decide, whether a shortest path between two distinct vertices \(x\) and \(y\) in \(G_n\) that lie in two neighbouring elementary triangles (of the same level) goes through the common vertex of the triangles or through two distinct vertices (both distinct from the common vertex) of those triangles. We also show criteria for the analogous problem on the planar Sierpiński gasket and in the 3-dimensional Euclidean space.

MSC:

28A80 Fractals
05C12 Distance in graphs
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[1] Bandt, C., Mubarak, M.: Distribution of distances and interior distances for certain self-similar measures. Arab. J. Sci. Eng., 29, Number 2C, (2004) · Zbl 1271.28002
[2] Barlow M.T., Perkins E.A.: Brownian motion on the Sierpiński gasket. Prob. Theory Rel. Fields 79, 543-623 (1988) · Zbl 0635.60090 · doi:10.1007/BF00318785
[3] Grabner P., Tichy R.F.: Equidistribution and Brownian motion on the Sierpiński gasket. Mh. Math. 20, 147-164 (1998) · Zbl 0915.11041 · doi:10.1007/BF01332824
[4] Falconer K.J.: Fractal Geometry. Wiley, Chichester (1990) · Zbl 0689.28003
[5] Hinz A.M.: The Tower of Hanoi. L’Enseignement Mathématique, t. 35, 289-321 (1989) · Zbl 0746.05035
[6] Hinz A.M.: Shortest Paths Between Regular States of the Tower of Hanoi. Inf. Sci. 63, 173-181 (1992) · Zbl 0792.68125 · doi:10.1016/0020-0255(92)90067-I
[7] Hinz A.M., Schief A.: The average distance on the Sierpiński gasket. Prob. Theory Rel. Fields 87, 129-138 (1990) · Zbl 0688.60074 · doi:10.1007/BF01217750
[8] Jakovaca, M., Klavža, S.: Vertex-, edge-, and total-colorings of Sierpiński-like graphs. Discrete Math. 309(6), 1548-1559 (2009) · Zbl 1198.05056
[9] Mandelbrot B.: Die fraktale Geometrie der Natur. Birkhäuser Verlag, Basel (1987) · Zbl 0652.28002
[10] Romik D.: Shortest Paths in the Tower of Hanoi Graph and Finite Automata. SIAM J. Discrete Math. 20(3), 610-622 (2006) · Zbl 1127.68069 · doi:10.1137/050628660
[11] Sierpiński W.: Sur une courbe dont tout point est un point de ramification. CR Acad. Sci. Paris 160, 302-305 (1915) · JFM 45.0628.02
[12] Strichartz R.S.: Isoperimetric estimates on Sierpinski gasket type fractals. Trans. Amer. Math. Soc. 351(5), 1705-1752 (1999) · Zbl 0917.28005 · doi:10.1090/S0002-9947-99-01999-6
[13] Yamaguti, M., Hata, M., Kigami, J.: Mathematics of Fractals. Translations of Mathematical Monographs, 167, American Mathematical Society, Providence (1997) · Zbl 0888.58030
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