mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Dragon curves

 2016-03-30 
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.
                        
(Click on one of these icons to react to this blog post)

You might also enjoy...

Comments

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> <logo>
To prove you are not a spam bot, please type "vector" in the box below (case sensitive):

Archive

Show me a random blog post
 2024 

Feb 2024

Zines, pt. 2

Jan 2024

Christmas (2023) is over
 2023 
▼ show ▼
 2022 
▼ show ▼
 2021 
▼ show ▼
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

gaussian elimination national lottery inverse matrices bodmas royal institution chess guest posts frobel world cup advent calendar exponential growth binary harriss spiral mathsjam numerical analysis golden spiral estimation numbers matrices javascript a gamut of games the aperiodical captain scarlet game show probability quadrilaterals squares propositional calculus games geometry simultaneous equations matrix of cofactors logo zines countdown signorini conditions curvature news databet fence posts pythagoras weather station tennis speed data newcastle pac-man finite group sobolev spaces london rugby latex logic boundary element methods accuracy hyperbolic surfaces reddit ucl recursion books european cup menace draughts palindromes wave scattering anscombe's quartet map projections craft sport flexagons cross stitch fractals london underground inline code machine learning runge's phenomenon oeis dataset rhombicuboctahedron folding paper game of life braiding sorting 24 hour maths royal baby manchester christmas card ternary plastic ratio preconditioning polynomials final fantasy dates nine men's morris determinants crossnumber bubble bobble realhats live stream talking maths in public matrix of minors probability noughts and crosses finite element method pascal's triangle puzzles bempp correlation statistics reuleaux polygons martin gardner geogebra hexapawn matt parker wool asteroids computational complexity electromagnetic field light gerry anderson tmip error bars folding tube maps golden ratio programming gather town platonic solids crochet graph theory interpolation phd pizza cutting triangles dragon curves misleading statistics edinburgh raspberry pi mean chalkdust magazine stirling numbers stickers hannah fry cambridge weak imposition go mathsteroids pi approximation day christmas coins python fonts football errors arithmetic logs chebyshev trigonometry sound manchester science festival big internet math-off approximation video games hats php dinosaurs convergence mathslogicbot standard deviation youtube datasaurus dozen matrix multiplication data visualisation turtles graphs people maths pi radio 4

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2024