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 "rebmun" backwards 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 ▼


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


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