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This chapter discusses some of the various roles fractal geometry plays, from a visual representation of convergence in the analysis classroom to constructs in the research setting. We begin with the concepts of convergence and completeness by introducing the Hausdorff metric on sets and demonstrate Cauchy sequences and completeness in terms of that metric. Fractals provide a visual way to reinforce these important concepts. We then tie the concepts of convergence and completeness to invariance and dimension. We use contraction mappings and fixed-point theory to prove invariance. We then discuss computing dimension from seed patterns. Using completeness, iterations of the seeds converge to fractals with geometric properties determined by dimension. This allows us to develop a method of four in our analysis. We use this method to develop research constructs which explore theoretical boundaries, from measures to differentiation/regularity to Thurston’s theory of conformal rigidity.

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Fractal Constructs: From the Classroom Blackboard to the Research Workbench

  • 动态工具书词条

  • 发表日期：

    2026年

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  • Stephen D. Casey,
  • Richard Laurberg,
  • Bharath Sriraman

摘要

This chapter discusses some of the various roles fractal geometry plays, from a visual representation of convergence in the analysis classroom to constructs in the research setting. We begin with the concepts of convergence and completeness by introducing the Hausdorff metric on sets and demonstrate Cauchy sequences and completeness in terms of that metric. Fractals provide a visual way to reinforce these important concepts. We then tie the concepts of convergence and completeness to invariance and dimension. We use contraction mappings and fixed-point theory to prove invariance. We then discuss computing dimension from seed patterns. Using completeness, iterations of the seeds converge to fractals with geometric properties determined by dimension. This allows us to develop a method of four in our analysis. We use this method to develop research constructs which explore theoretical boundaries, from measures to differentiation/regularity to Thurston’s theory of conformal rigidity.