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Mathematical Recreations

Heighway Curve in Acheron 2.0




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Sample from Acheron of a Heighway Curve All pictures are from Acheron 2.0, a free explorer of geometrical fractals. You can download Acheron 2.0 here Acheron 2.0 Screen Overview
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The american physicist John Heighway discovered this curve while working at NASA. The Heighway curve, also called the Heighway Dragon or Harter-Heighway Dragon, was popularized in 1967 when Martin Gardner described it in American Scientific. Three years later, the first mathematical analysis was published by Chandler Davis and Donald Knuth. Although the construction of this curve is quite simple, its fascinating properties have stimulate a lot of works in different fields of mathematics.



Construction

While three methods were described for the construction of the Dragon curve, the discussion will be limited to the method used originally by Heighway.

To construct the curve, imagine that you take a trip of paper. Fold it in two equal parts by taking the right edge and adjusting it above the left edge of the trip. Repeat this operation two or three times and then unravel the paper trip so that each corner makes a right angle in the natural direction of the fold. Looking at the paper trip from the edge will reveal the Heighway curve.

Here are the result of the first five foldings:


Note that the software strips slightly the folding corners to avoid touches between folders.

Increasing the number of folding leads to more and more complex drawing and, finally, to the limiting shape, the Dragon Curve.

There are two types of folding: the folding of type 1 consists of ajusting the right edge above the left edge of the paper. The folding of type 0 is the inverse operation, as you adjust the right edge below the left one.

Mixing folding types changes dramatically the look of the curve:

Folding 11111111 ( 8 foldings of type 1)
Folding Sequence: 11111111

Folding 10101010 ( 8 foldings alternatively of type 1 and 0)
Folding Sequence: 10101010

Folding 11100111 ( 6 foldings type 1 with 2 of type 0 in the middle)
Folding Sequence: 11100111
Playing around with the folding sequence may lead to very peculiar drawings ...

Properties Back to Top

  • Curve Order


    This property derives from the construction method: a curve of order n is the combination of two curves of order n - 1.

    As an example, the following picture shows:
    • on the left, a curve obtained by using the folding sequence 111111
    • on the right, two instances of a curve obtained with a 11111 sequence, one of them being rotated to be in the correct alignment.




  • Fractal Dimension


    The fractal dimension is computed using the Hausdorff-Besicovitch equation:

      D = log (N) / log ( r)

    The self-similarity N and r values are derived from the construction of the curve using the triangle method.

      D = log(2) / log(2*sqr(2)) = 2

Variations Back to Top

All Variations described are available using Acheron 2.0

  • Folding Sequence

    The folding sequence is entered as a string of 1 and 0, up to 16 characters.

Author Biography Back to Top