Dragon curves

Take a piece of paper. Fold it in half in the same direction many times. Now unfold it. What pattern will the folds make?
I first found this question in one of Martin Gardner's books. At first, you might that the answer will be simple, but if you look at the shapes made for a few folds, you will see otherwise:
Dragon curves of orders 1 to 6.
The curves formed are called dragon curves as they allegedly look like dragons with smoke rising from their nostrils. I'm not sure I see the resemblance:
An order 10 dragon curve.
As you increase the order of the curve (the number of times the paper was folded), the dragon curve squiggles across more of the plane, while never crossing itself. In fact, if the process was continued forever, an order infinity dragon curve would cover the whole plane, never crossing itself.
This is not the only way to cover a plane with dragon curves: the curves tessellate.
When tiled, this picture demonstrates how dragon curves tessellate. For a demonstration, try obtaining infinite lives...
Dragon curves of different orders can also fit together:

Drawing dragon curves

To generate digital dragon curves, first notice that an order \(n\) curve can be made from two order \(n-1\) curves:
This can easily be seen to be true if you consider folding paper: If you fold a strip of paper in half once, then \(n-1\) times, each half of the strip will have made an order \(n-1\) dragon curve. But the whole strip has been folded \(n\) times, so is an order \(n\) dragon curve.
Because of this, higher order dragons can be thought of as lots of lower order dragons tiled together. An the infinite dragon curve is actually equivalent to tiling the plane with a infinite number of dragons.
If you would like to create your own dragon curves, you can download the Python code I used to draw them from GitHub. If you are more of a thinker, then you might like to ponder what difference it would make if the folds used to make the dragon were in different directions.

Similar posts

Dragon curves II
PhD thesis, chapter 2
Visualising MENACE's learning
Harriss and other spirals


Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 

I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "jump" in the box below (case sensitive):


Show me a random blog post

May 2021

Close encounters of the second kind

Jan 2021

Christmas (2020) is over
▼ show ▼
▼ show ▼
▼ show ▼
▼ show ▼
▼ show ▼
▼ show ▼
▼ show ▼
▼ show ▼
▼ show ▼


triangles trigonometry exponential growth statistics london machine learning hats accuracy speed matrix multiplication plastic ratio oeis wave scattering propositional calculus tmip ternary geogebra braiding national lottery folding paper noughts and crosses rugby error bars sound arithmetic news interpolation sobolev spaces big internet math-off programming flexagons advent calendar dates radio 4 asteroids logic martin gardner folding tube maps london underground final fantasy mathsjam matt parker graphs probability puzzles coins sorting data visualisation matrix of minors manchester numerical analysis frobel convergence chebyshev platonic solids gerry anderson weather station gaussian elimination video games stirling numbers go fractals squares christmas draughts computational complexity inverse matrices hexapawn rhombicuboctahedron pi captain scarlet harriss spiral games european cup electromagnetic field game of life map projections inline code wool palindromes game show probability matrices guest posts stickers matrix of cofactors latex realhats royal institution pi approximation day quadrilaterals polynomials pizza cutting approximation football mathslogicbot reuleaux polygons people maths bodmas estimation chalkdust magazine golden ratio raspberry pi sport a gamut of games geometry signorini conditions pac-man reddit books simultaneous equations phd mathsteroids logs hannah fry manchester science festival twitter golden spiral preconditioning craft misleading statistics dataset javascript pythagoras cross stitch cambridge light dragon curves talking maths in public world cup chess pascal's triangle data bubble bobble bempp christmas card numbers graph theory recursion determinants php nine men's morris the aperiodical countdown curvature finite element method weak imposition python menace royal baby boundary element methods tennis ucl binary


Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2021