Abstract
In this paper, we firstly define a decreasing sequence {Pn(S)} by the the generation of the Sierpinski gasket where each Pn(S) can be obtained in finite steps. Then we prove that the Hausdorff measure Hs(S) of the Sierpinski gasket S can be approximated by {Pn(S)} with Pn(S)/1+1/2n−3)8≤Hs(S)≤Pn(S). An algorithm is presented to get Pn(S) for n≤5. As an application, we obtain the best lower bound of Hs(S) till now: Hs(S)≥0.5631.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Falconer, K. J., Fractal Geometry: Mathematical Foundations and Applications, (New York: John Wiley & Sons), 1990.
Jia Baoguo, Zhu Zhiwei and Zhou Zuoling, Hausdorff Measure of Sierpinski Gasket and Self-Product Sets of Cantor Sets (in Chinese), Zhongshan University, preprint, 2001
Marion, J., Measure de Hausdorff D'ensembles Fractals, Ann. Sci. Math. Quebec 11(1987), 111–137.
Ayer, E. and Strichartz, R. S., Exact Hausdorff Measure and Intervals of Maximum Density for Cantor Sets, Trans. Amer. Math. Soc., 351 (1999), 3725–3741.
Wang, X.H., Estimation and Cconjecture of the Hausdorff Measure of Sierpinski Gasket (in Chinese), Prog. Nat. Sci., 9(1999), 488–493.
Zhou, Z.L., The Hausdorff Measures of the Koch Ccurve and Sierpinski Gasket (in Chinese), Prog. Nat. Sci., 7(1997), 403–409.
Zhou, Z.L., Hausdorff Measure of Sierpinski Gasket, Sci. China (Series A), 40(1997), 1016–1021.
Zhou, Z.L. and Feng, L., A New Eestimate of the Hausdorff Measure of the Sierpinski Gasket, Nonlinearity, 13(2000), 479–491.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper was presented in the Fractal Satellite Conference of ICM 2002 in Nanjing.
Research supported by NSFC, grant 10041005, grant 10171045.
Rights and permissions
About this article
Cite this article
Huojun, R., Weiyi, S. An approximation method to estimate the Hausdorff measure of the Sierpinski gasket. Anal. Theory Appl. 20, 158–166 (2004). https://doi.org/10.1007/BF02901439
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02901439