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L-systems for the Boundaries of fractal dragon space filling folding curves

H. A. Verrill

Abstract

We describe an algorithm to find an L-system for the boundary of space-filling square grid based folding curves, such as the fractal dragon curves. This complements work of Dekking, Arndt, Handl, on space filling curves.

L-systems for the Boundaries of fractal dragon space filling folding curves

Abstract

We describe an algorithm to find an L-system for the boundary of space-filling square grid based folding curves, such as the fractal dragon curves. This complements work of Dekking, Arndt, Handl, on space filling curves.
Paper Structure (8 sections, 1 theorem, 6 equations, 10 figures, 6 tables, 1 algorithm)

This paper contains 8 sections, 1 theorem, 6 equations, 10 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction: plane-filling folding curves
  2. The L-system for the boundary of plane-filling curves
  3. The Algorithm
  4. Examples
  5. Proof that Algorithm \ref{['alg:1']} produces the boundary of the plane-filling curve
  6. Creation of boundary paths
  7. Reduction of paths, removing backtracking
  8. Parity / case of letters of sequences

Key Result

Theorem 1

The curves $\tau^\infty({\texttt{R}})$ and $\tau^\infty({\texttt{L}})$ form the boundary of the plane-filling curve $\sigma^\infty({\texttt{A}})$.

Figures (10)

  • Figure 1: Left: Square grid of allowed path segments. Middle: Choice of turns, $+$ or $-$ drawn on dual grid, with curved segments. Right: Heighway dragon and its boundary after 5 iterations of the L-systems.
  • Figure 2: Geometric interpretation of letters of L-system. Gray squares are even, white odd. Starting and ending points centre of squares.
  • Figure 3: Left: Path corresponding to the word LrSRrLslLrL. Middle: relationship between A and L and R, which are all paths from the mid point of an even square to the mid point of an odd square. Right: boundary segments (black), which go in diagonal directions on the same grid as ${\texttt{A}}, {\texttt{B}}$ segments (both blue), which go horizontally and vertically respectively. (Though ${\texttt{A}}, {\texttt{B}}$ are shown curved, so this is an approximation.)
  • Figure 4: Three examples of plane-filling curves and boundaries, with L-systems as in Tables \ref{['tab:fill']} and \ref{['tab:bou']}. Column 1 shows segment A (black), a right turn R (red), on the left side of A, and a left turn, L (blue), on the right side of A. Column 2 shows A, L, R, after one application of the relevant L-system. The number of iterations for columns 3 and 4 is given in brackets.
  • Figure 5: Example 3, with $\sigma({\texttt{A}})=\texttt{B}+A-{\texttt{B}}-A+{\texttt{B}}+A+{\texttt{B}}-A+{\texttt{B}}+A-{\texttt{B}}-A-{\texttt{B}}+A-{\texttt{B}}+A+{\texttt{B}}$, from Table \ref{['fig:3examples']}. (a) Application of L-system operation once to four line segments, two A and two B direction. Gray squares are even. (b) Diagram of left side boundary, in blue. Red path is prior to removal of backtracking. A, R, L start in odd squares.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Theorem 1