Minkowski dimension and content of fractals
LE3 .A278 2011
Bachelor of Science
Mathematics and Statistics
Mathematics & Statistics
This thesis focuses upon fractal geometry and how we can characterise various fractals by two properties known as the Minkowski dimension and the Minkowski content. A fractal is treated as a geometric construct of a generic set of points considered in an m-dimensional Euclidean space. We rst explain how Minkowski dimension and content are important in measur- ing the \size" of a set. Measuring the size by length if the set is 1-dimensional, area if the set is in a plane, or volume if the set is in 3-dimensional space will only work for certain simplistic sets, for example a polygon or polyhedron. Hence we need another method of measure. Chapter 1 explains how this can be done with Minkowski dimen- sion and content and how subsets of 1-dimensional Euclidean space or a straight line can be measured with Minkowski dimension and content. We then go into a section that goes through various properties that make Minkowski dimension and content useful in certain respects. The third chapter looks at sets in higher dimensions where we try to nd sets of points that have any given positive Minkowski dimension. We create a two-dimensional cut-out set where we remove various rectangles out of a larger rectangle, leaving only straight lines. Given any Minkowski dimension between 1 and 2, we can create a set using this construction. The Minkowski content is easily calculable. Then we look at a set consisting the perimeters of concentric squares and then a subset of this set consisting of only countably many points. What is special about this construction is that not only can we generate a set with any Minkowski dimension between 0 and 2, but we can do so where the set only has one limit point. We then consider an analo- gous set in m-dimensions where we construct a set consisting of the m-1-dimensional surface area of concentric m-dimensional hypercubes and then again a subset of this set consisting of only countably many points. Again, the construction of the count- ably many points only as one limit point and such a set can be constructed that has any Minkowski dimension between 0 and m. The Minkowski content is not calculated for these sets, but a di erent kind of Minkowski content was calculated.
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