Counterexamples in theory of fractal dimension for fractal structures |
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| Institution: | 1. University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, Santiago de la Ribera, Murcia 30720, Spain;2. Jan Kochanowski University in Kielce, Świetokrzyska 15, 25-406 Kielce, Poland;3. Department of Mathematics, Universidad de Almería, Almería 04120, Spain;1. School of Computer Science and Engineer, Southeast University, Nanjing 210096, China;2. School of Continuing Education, Southeast University, Nanjing 210096, China;3. School of Computer Engineering, Jiangsu University of Technology, Changzhou, Jiangsu 213001, China;4. School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China;1. School of information engineering, Guangdong Mechanical & Electrical College, Guangzhou 510550, PR China;2. Department of Information Science and Technology, Guangdong University of Foreign Studies South China Business College, Guangzhou 510545, PR China;1. School of Science, East China Jiaotong University, Nanchang , Jiangxi, Postal code 330013, PR China;2. School of Mathematics, South China Normal University, Guangzhou, Guangdong, Postal code 510631, PR China;3. School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, Guizhou, Postal code 550001, PR China |
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| Abstract: | Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and also the most accurate fractal dimension, presents the best analytical properties. Additionally, fractal structures provide an appropriate topological context where new models of fractal dimension for a fractal structure could be developed in order to generalize the classical models of fractal dimension. In this survey, we gather different definitions and counterexamples regarding these new models of fractal dimension in order to show the reader how they behave mathematically with respect to the classical models, and also to point out which features of such models can be exploited to powerful effect in applications. |
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