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Fractals corresponding to the metallic means sequences

Y. S. J. Liang, Darren C. Ong

Abstract

In this work, we introduce a class of fractal subsets of $[0,1]$ corresponding to the aperiodically ordered metallic means sequences. We find simple formulas for the fractal dimension for these fractals.

Fractals corresponding to the metallic means sequences

Abstract

In this work, we introduce a class of fractal subsets of corresponding to the aperiodically ordered metallic means sequences. We find simple formulas for the fractal dimension for these fractals.

Paper Structure

This paper contains 13 sections, 28 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. The similarity dimension of the $1/3$ Cantor set
  4. Hausdorff dimension of the $1/3$ Cantor set
  5. Metallic means sequences and tilings of $[0,1]$
  6. The fractal corresponding to the Fibonacci Word
  7. The Fibonacci tiling of $[0,1]$
  8. Constructing the fractal
  9. Silver ratio
  10. Silver ratio basics
  11. Fractal Dimension for Silver ratio fractals
  12. Fractals corresponding to general metallic means sequences
  13. Acknowledgements

Figures (6)

  • Figure 1: Cantor set
  • Figure 2: Tilings corresponding to the Fibonacci inflation rule.
  • Figure 3: Construction process for the $(3,0,1)$ fractal.
  • Figure 4: Construction of the $(4,1,1)$ fractal
  • Figure 5: Silver ratio tilings
  • ...and 1 more figures