The year of the magical dragon

by Burkard Polster and Marty Ross

The Age, 8 February 2010

Sunday is Chinese New Year, kicking off the Year of the Tiger. Undoubtedly the celebrations will be magnificent as always, but we're impatient for 2012. That will be the Year of the Dragon, and we have in preparation a famous and beautiful mathematical dragon.

Like the stunning dragons dancing at New Year celebrations, ours is made of paper. To make our dragon all that is required is one very long strip of paper, plus a pinch of mathemagic.

First fold the paper strip in half three times, each time folding in the same direction. Now unfold the strip, fixing all the creases at right angles. What you have is a 3-year-old baby "dragon" as pictured below, to the right of the 1-year-old and the 2-year-old babies.

In what sense are these dragons? Well, if you are youthfully imaginative or a big fan of Picasso, the 3-year-old has the suggestion of a head (to the left), and a tail and legs. For the sceptics remaining, here is a much more impressive dragon, a 12-year-old (with rounded corners):

However, do not try this with paper at home! Baby dragons are easily made, but refolding paper quickly becomes painfully difficult. The world record is 12 folds, on a kilometer strip of paper: not much chance of actually unfolding it to reveal the dragon.

None of this is a problem for mathematicians, as our magical paper has no thickness, and can be as long as we wish. And, for older dragons we can easily use greater lengths of paper. Choosing just the right lengths, we can imagine that there is just one dragon, with head and tail fixed, and growing older with each fold-unfold.

Amazingly, no matter how old, the unfolded dragon never runs into itself. However, with each fold, the dragon appears more solid. Finally, when the dragon is infinity years old (yes, mathemagical dragons can do this), our baby has fully grown into an adult, the dragon curve:

The grown up dragon has many amazing features. For example, though the dragon is constructed as a curve - just the bending of an infinitely long line - it consists of whole solid regions; mathematicians refer to such paradoxical creatures as space-filling curves. Next, the numbered regions are all exactly the same shape, forming an infinite spiral. And, infinitely many dragons fit together seamlessly, like the pieces of a jigsaw puzzle. And there is much more. A truly magical creature, waiting for the arrival of its special Year.

Puzzle to ponder

Feel free to use the comments section to suggest solutions. Later in the week we'll post our solution in the comments section.

Suppose you start with a 1 meter strip of paper, fold it 10 times, and unfold to give a 10 year old dragon. Roughly how long will the dragon be, from head to tail? What if you could fold the paper 100 times?

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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