# Blog

## Archive

Show me a Random Blog Post**2018**

**2017**

**2016**

**2015**

**2014**

**2013**

**2012**

## Tags

folding paper folding tube maps london underground platonic solids london rhombicuboctahedron raspberry pi weather station programming python php inline code news royal baby probability game show probability christmas flexagons frobel coins reuleaux polygons countdown football world cup sport stickers tennis braiding craft wool emf camp people maths trigonometry logic propositional calculus twitter mathslogicbot oeis matt parker pac-man graph theory video games games chalkdust magazine menace machine learning javascript martin gardner reddit national lottery rugby puzzles game of life dragon curves fractals pythagoras geometry triangles european cup dates palindromes chalkdust christmas card ternary bubble bobble asteroids final fantasy curvature binary arithmetic bodmas statistics error bars estimation accuracy misleading statistics pizza cutting captain scarlet gerry anderson light sound speed manchester science festival manchester a gamut of games**2016-03-30**

## 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:

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: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.

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 | Origins of World War I | MENACE at Manchester Science Festival | Logical Contradictions |

### Comments

Comments in green were written by me. Comments in blue were not written by me.

Add a Comment