 Larry Riddle, Agnes Scott College ## Different Versions of the Heighway Dragon

The Heighway dragon now appears in different horizontal and vertical orientations depending on how it is drawn.

Scaling factor $$r = \frac{1}{\sqrt 2}$$

### This website's version

 First 2 iterations Dragon IFS  $${f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/2} & { - 1/2} \\ {1/2} & {1/2} \\ \end{array}} \right]{\bf{x}}$$scale by r, rotate by 45°  $${f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - 1/2} & { - 1/2} \\ {1/2} & { - 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 1 \\ 0 \\ \end{array}} \right]$$scale by r, rotate by 135°  $${f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/2} & { - 1/2} \\ {1/2} & {1/2} \\ \end{array}} \right]{\bf{x}}$$scale by r, rotate by 45°  $${f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - 1/2} & { - 1/2} \\ {1/2} & { - 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0 \\ 1 \\ \end{array}} \right]$$scale by r, rotate by 135°

### William Harter's booklet cover

 First 2 iterations Dragon IFS  $${f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - 1/2} & { -1/2} \\ {1/2} & { - 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 1/2 \\ -1/2 \\ \end{array}} \right]$$scale by r, rotate by 135°  $${f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { 1/2} & { -1/2} \\ {1/2} & { 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 1/2 \\ -1/2 \\ \end{array}} \right]$$scale by r, rotate by 45°  $${f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - 1/2} & { -1/2} \\ {1/2} & { - 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 1/2 \\ 1/2 \\ \end{array}} \right]$$scale by r, rotate by 135°  $${f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { 1/2} & { -1/2} \\ {1/2} & { 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 1/2 \\ 1/2 \\ \end{array}} \right]$$scale by r, rotate by 45°

### Martin Gardner's Mathematical Games column

 First 2 iterations Dragon IFS  $${f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/2} & { 1/2} \\ {-1/2} & {1/2} \\ \end{array}} \right]{\bf{x}}$$scale by r, rotate by −45°  $${f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - 1/2} & { 1/2} \\ {-1/2} & { - 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 1 \\ 0 \\ \end{array}} \right]$$scale by r, rotate by −135°  $${f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {1/2} & { 1/2} \\ {-1/2} & {1/2} \\ \end{array}} \right]{\bf{x}}$$scale by r, rotate by −45°  $${f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - 1/2} & { 1/2} \\ {-1/2} & { - 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0 \\ 1 \\ \end{array}} \right]$$scale by r, rotate by −135°

### Donald Knuth-Chandler Davis article

 First 2 iterations Dragon IFS  $${f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - 1/2} & { 1/2} \\ {-1/2} & { - 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 1/2 \\ 1/2 \\ \end{array}} \right]$$scale by r, rotate by −135°  $${f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { 1/2} & { 1/2} \\ {-1/2} & { 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 1/2 \\ 1/2 \\ \end{array}} \right]$$scale by r, rotate by −45°  $${f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { - 1/2} & { 1/2} \\ {-1/2} & { - 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} -1/2 \\ 1/2 \\ \end{array}} \right]$$scale by r, rotate by −135°  $${f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} { 1/2} & { 1/2} \\ {-1/2} & { 1/2} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} -1/2 \\ 1/2 \\ \end{array}} \right]$$scale by r, rotate by −45°

### Michael Creighton's Jurassic Park novel

Each of the seven "chapters" of Jurassic Park starts with an iteration of the Heighway dragon (and a quotation from Ian Malcolm, the mathematician in the novel). The first three are in a vertical orientation and the last four are in a horizontal orientation. First Iteration Second Iteration Third Iteration Fourth Iteration Fifth Iteration Sixth Iteration Seventh Iteration

These actually come from a mixture of three different iterated function systems! The First Iteration is the 4th iterate of this website's IFS with adjusted translation vectors to make it vertical rather than horizontal. The Second Iteration and the Third Iteration are the 5th and 6th iterates, respectively, of the IFS corresponding to the William Harter's booklet cover, again with adjusted translation vectors to make them vertical. And finally, the Fourth through Seventh Iterations are, respectively, the 8th, 10th, 12th, and 14th iterates of the IFS for the version of the dragon from the Knuth-Davis article.

#### References

1. Crichton, Michael. Jurassic Park, Ballantine Books, New York 1990.
2. Davis, Chandler and Donald J. Knuth. "Number representations and dragon curves" J. Recreational Math. 3 (1970) 66-81 (Part 1), 133-149 (Part 2). Reprinted with extra addendum in Selected Papers on Fun and Games, Donald Knuth, CSLI Publications, 2011.
3. Gardner, Martin. "Mathematical Games," Scientific American, March, April, July, 1967. Also shows a copy of the cover design by William Harter for a booklet he used for a seminar on group theory at NASA's Lewis Research Center. [Reprinted in his Mathematical Magic Show, Knopf, 1977]