Show me a random blog post


interpolation palindromes matt parker flexagons london countdown propositional calculus coins trigonometry golden ratio map projections draughts gerry anderson go fractals misleading statistics video games dragon curves hexapawn inline code sound noughts and crosses triangles reddit electromagnetic field binary hats manchester science festival news platonic solids curvature harriss spiral pythagoras accuracy chalkdust magazine graph theory puzzles plastic ratio mathsjam menace game show probability mathsteroids folding paper a gamut of games tennis craft data pizza cutting pac-man bubble bobble ternary mathslogicbot radio 4 folding tube maps javascript national lottery chess logic rugby probability royal baby geometry chebyshev error bars game of life martin gardner twitter books final fantasy approximation aperiodical big internet math-off football rhombicuboctahedron wool frobel light dates captain scarlet polynomials programming cross stitch the aperiodical manchester reuleaux polygons bodmas world cup london underground speed people maths dataset php arithmetic python oeis nine men's morris machine learning sorting golden spiral statistics asteroids christmas sport weather station stickers christmas card games realhats estimation raspberry pi latex braiding european cup


Show me a random blog post
▼ show ▼

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
Harriss and other spirals
MENACE in fiction
Building MENACEs for other games


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 "orez" backwards in the box below (case sensitive):
© Matthew Scroggs 2019