The Dragon Curve (and a bit on Fractals)

If you have ever read Michael Crichton’s “Jurassic Park” (Yes, the 1993 movie was adapted from a book as most tend to be) then you would have noticed the unorthodox chapter headings.

1st Iteration

Each chapter (or iteration as the book called it) was found to have a seemingly meaningless illustration of some lines and squares adjunct to a quotation from one the book’s characters.

The First Chapter/First Iteration contained the above image with the quote: “At the earliest drawings of the fractal curve, few clues to the underlying mathematical structure will be seen.” Ian Malcolm.

With each subsequent chapter, the quotations begin to resemble the events in the story – the idea of unpredictability; chaos theory – and the illustrations become more elaborate – and eerily reptilian.

1,2,3,4,5,7

 What we have is called a Dragon Curve.

It is a fractal with a very simple iterative process:

  1. Draw a Line
  2. Rotate a copy of the Line from Step 1 90 degrees (clockwise or anti-clockwise is arbitrary) and attach it to the end of the First Line. (You will now have two perpendicular lines – an L shape).
  3. Rotate a copy of the entire L shape from Step 2 90 degrees (continue rotating it the same direction as you did in Step 2) and attach it to the end of the First L shape. (You will now have a saucepan shape).
  4. Rotate a copy of the entire saucepan shape from Step 3 90 degrees (continue rotating it the same direction) and attach it to the end of the First saucepan shape.
  5. Rotate a copy of the entire image and attach it to the end of the original.

Each step corresponds to the panel in the image below (read from left to right, top to bottom)

1-15 horizontal

*Some things to be aware of is the overlapping that occurs from Step 5 onwards. If you’re attempting to draw the curve by hand, it would be helpful to use different colours to help yourself differentiate between the original and the rotated copy.

As you can see it is an extremely simple fractal design. You can see that the number of lines double with each iteration; this is because we are copying the previous iteration. Another thing to notice is that each section of the Dragon Curve (more noticeable in higher iterations) is reduced by a factor of {\sqrt{2}} and rotated by 45°.

rotate and reduce

Dragon Curves, like many other fractals, posses a property called Self-Similarity. For a fractal to be self-similar means that if you zoom at any specific region of the fractal, no matter you far you go, it will remain the same image. An example of this can be seen in the Sierpinski Triangle or Koch Snowflake.

Kochsim

Unlike some fractals, Dragon Curves can be produced with different iterative processes, though they are more complicated to pursue compared to the one mentioned previously.

The first method requires you to:

tumblr_mfxijqlvKl1qzxsbbo1_400

  1. Take a square
  2. Divide it in half horizontally.
  3. Translate both halves in opposite directions horizontally such that their edges touch the midpoints of the longer edge.
  4. Divide the shape in vertically in sixths
  5. Translate these divisions vertically similar to Step 3.
  6. Divide the shape in horizontally in tenths
  7. Translate these divisions horizontally similar to Step 3.
  8. and so on…

The second method requires you to:

  1. Essentially draw a lot of triangles…I have been unable to understand the integral mechanics of this particular iterative process. But mathtuber TheMathGuy’s video on the Dragon Fractal (another name for the Dragon Curve; also called the Jurassic Park Dragon, Heighway Dragon, etc…) provides a detailed demonstration on another way to draw the Dragon Curve.

While the Dragon Curve is just another good-looking fractal that can be a fairly decent (yet cliché)
desktop wallpaper, fractals have numerous applications in fields such as astronomy, graphics design, Fractal naturemeteorology, geology, economics and many more.

The iterative nature of fractals allow computer designers to map landscapes in video games and 3-D maps. Fractals can be used to understand the nature of different crystal lattice structures and correlate the iterative process to their strength. Price trends are found to possess self-similarity, just like fractals, as patterns in price fluctuations follow similar patterns in different time periods, regardless of magnitude. You can calculate the length of coastlines using fractal mathematics.

Fractals ominously occur in real life: the branches of a tree, the arrangement of flower petals, the design of a seashell, tributaries in a delta, crystalline structures of compounds and rocks (ice, diamond, halite), lightning, DNA, animal horns, and many many more.

The dragon is symbolic for its mystery and revered power, and the Dragon Curve is likewise the perfect testimonial to illustrate – and confirm – the power of fractals.

Dragon_curve_animation

Works Cited:

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