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Fractals as Sculptures

Şahin Koçak

Abstract

The attractors of iterated function systems are usually obtained as the Hausdorff limit of any non-empty compact subset under iteration. In this note we show that an iterated function system on a boundedly compact metric space has compact, invariant subspaces so that the attractor of the IFS can also be expressed as the intersection of a sequence of decreasing compact spaces.

Fractals as Sculptures

Abstract

The attractors of iterated function systems are usually obtained as the Hausdorff limit of any non-empty compact subset under iteration. In this note we show that an iterated function system on a boundedly compact metric space has compact, invariant subspaces so that the attractor of the IFS can also be expressed as the intersection of a sequence of decreasing compact spaces.

Paper Structure

This paper contains 1 section, 1 theorem, 2 equations, 8 figures.

Table of Contents

  1. Acknowledgment

Key Result

Proposition 1

Let $(X,d)$ be a boundedly compact metric space and $f_i:X\to X$, $i=1,2,\dots,N$ be contractions. Then, there exists a (non-empty) compact subspace $B\subset X$ with $f_i(B)\subset B$ for all $i=1,2,\dots,N.$

Figures (8)

  • Figure 1: Michelangelo chiseling away the superfluous material (Michweb)
  • Figure 2: Cantor Set obtained by successive deletions
  • Figure 3: Sierpinski Triangle obtained by successive deletions
  • Figure 4: Chiseling tunnels through a cube
  • Figure 5: Maxwell's drawing of a 3-foci ellipse (with focal points $A, B$ and $C$) with a thread (Max) (and without a computer, which was non-existent anyway!)
  • ...and 3 more figures

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Example 1