Dragon Curve

History

    The Dragon Curve was first discovered in June 1966. The curve was discovered by a physicist named John Heighway who worked at NASA. After discovering the Dragon Curve, he had his colleague William Harter name it. The way that it was discovered is Heighway walked into Harter’s office and told him to fold a $1 bill repeatedly in half. Heighway was thinking that by folding the $1 bill it would make a random pattern or something. After that Harter suggested folding a big piece of paper to hopefully see the pattern better. Once they folded the big paper, they still couldn’t see the pattern very well. Then they decided to fold tracing paper. They chose to do tracing paper because when they unfold it, they would be able to record the pattern that they saw. The pattern that they saw was curves and squares that kept repeating. The reason that the pattern was seen the best on tracing paper is because it is easier to crease the tracing paper in order to see the folds better. This pattern then became the Dragon Curve.

    Later, the Dragon Curve and method was studied by Davis and Knuth. Chandler Davis was an American-Canadian mathematician. Donald Knuth is an American computer scientist and mathematician. When Davis and Knuth studied the Dragon Curve, they were more looking at how it related to number systems and to paper-folding. When Davis and Knuth looked at it, they eventually wrote a theory about it in 1970.

Exploration of Mathematics

    To learn about the mathematics, we first must do a step to help visualize the mathematics. First take a strip of paper and fold it in half, then in half again, and do this several more times. Next you will unfold the paper. Now looking at the paper we have the result of "n folding" which is a sequence S_n of letters D, for a down crease that is the shape of a V, and U, for an up crease in the shape of ∧, of length 2^n-1. Then we can get S_(n+1) from S_n by “Let \[S_n=a_1 a_2 a…a_m\] where m=2^n+1 and each a_i is either D or U. Then, \[S_{(n+1)}=Da_1 Ua_2 Da_3 U…Da_m U\]” (Tabachnikov, 2014, pg.2). The reason that this works is because when you fold n+1 times that is achieved by folding the paper n times and then folding it one more time after that. Going a little deeper into this explanation, we then have S_n equal to the even numbered creases and D,U,D,U,… equal to the odd numbered creases in alternating order.

    Next while you have the paper unfolded, make the angles so that they are 90^o. After doing that we are going to round the angles so that we have a curve instead of segments that are rigid. By doing this D is now a left turn and U is a right turn. Now we are going to make S_(n+1) geometrically since we already showed it algebraically. To do this let D_n be the curve of the nth generation and let O be its end point. Next turn D_n about O \[90^o\]. Once this is done then attach the new curve that was made to the original curve. This gives the result of n+1.

    When looking more into the Dragon Curve, the research explained that it is a self-similar set that is invariant under the iterated function. Invariant is a property that remains the same when a transition is applied. There is also a well-known twin dragon which is the union of two dragon curves. The twin dragon is a topological disc. A topological disc is a surface that is homeomorphic to a disc in a plane. This is not the case for the single dragon curve. There are still questions that are not answered about the topological structure of the twin dragon. Although Davis and Knuth have discovered that the curve is non-crossing and fills the plane. They have also discovered many other properties.