Paper-Folding-Fractals
Chasing the Dragon

In order to get a computer to Chase the Dragon, we need to find a way to represent it symbolically.

## Observations of the Dragon:

• The Dragon always begins with a straight line that goes directly upward on the display.
• At any given level, each straight segment of the Dragon is the same length as every other segment.
• At the end of every segment, there is always a 90° turn (either right or left). The Dragon never turns back on itself, and it never continues straight.
• Level 1 of the dragon has 2 straight segments. Level 2 has 4 segments. Level 3 has 8 segments. Level 4 has 16 segments. Level n has 2n segments.
• Each segment in level n+1 is one-half the length of each side in level n.
• The first half of Level n+1 is an exact, scale model of level n (it even has exactly the same orientation).
• The second half of Level n+1 is also exact, scale model of level n except it is rotated 90°. Notice how this can be seen in level 3 and level 4 of figure 1 below. The green half exactaly fits inside the blue half.

Figure 1: Level 3 and 4 of the Dragon showing how the sequence of folds in the second half is a reverse copy of the sequence of folds in the first half.

## The Abstract Dragon:

The observations above are useful in creating an abstract representation of the Dragon. At a given level, all segments of the Dragon are the same length, and each segment always makes a right or left 90° turn. Therefore, the visual pattern of the Dragon can be completely represented by a string of two symbols: L for a left turn and R for a right turn. For an example, see figure 2.

Figure 2: Representing the Dragon as a series of right and left turns.

Level 1 begins with a line that goes upward on the display. It makes a left turn and ends. Level 2 begins with a line that goes upward on the display. It makes a left turn, a second left turn and finally a right turn. We can represent these figures as follows:

Level 1: L
Level 2: LLR
Level 3: LLRLLRR

Following the direction of these turns can get a bit confusing since when a segment is going up on the display, a turn to the left side of the display is a left turn, yet when a segment is going down on the display, then a turn to the left side of the display is a right turn.

Before we look at Level 4 and list the series of left and right turns, let us use the observations we have made about the Dragon to predict what the series will be:

The first half of level 4 is identical to all of level 3, therefore:

Level 4: LLRLLRR????????

The second half of level 4 is also an exact copy of all of level 3; however, where the first half began by moving upward on the display, the second half begins by moving downward on the display. Therefore, each right turn in the first half will be a left turn in the second half (and each left in the first will be a right in the second):

LLRLLRR?L??????
LLRLL
RR?LL?????
LLRL
LRR?LLR????
LLR
LLRR?LLRR???
LL
RLLRR?LLRRL??
L
LRLLRR?LLRRLR?
LLRLLRR?LLRRLRR

Finally, the center fold is a left turn. Can you convince yourself that the center fold will always be a left turn on every level of the Dragon? Can you explain why?

Thus, we have:

Level 4: LLRLLRRLLLRRLRR

Likewise,:

Level 5: LLRLLRRLLLRRLRRLLLRLLRRRLLRRLRR

This abstract representation of the Dragon has several advantages: First, in a computer, it is very easy to represent a string of L and R character. Second, the string representation made it very easy to generate each subsequent level of the Dragon.

With the Dragon represented as s string, it becomes easy to see yet another pattern for generating the next Dragon level. Notice that, we can generate level n+1 by alternately inserting an L and an R between each turn in level n. For example:

Level 3: L L R L L R R

Level 4:LLRLLRRLLLRRLRR

Personally, I find it utterly fascinating that such a simple and regular process of inserting an L and an R between each turn of the current level can, from the starting point of a single L, create a sequence that is amazingly complex and intricate!

This process is heart of the software that was used to create these pictures. If you already know something about how to write a computer program to draw lines on a screen, then you are now ready to write your own Dragon generating program.

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